Which group prior to the Norman invasion influenced England the most?

In terms of genetics, there have been several attempts at reconstructing the influence of different groups of immigrants on England's (and Brotain's) genetic makeup. The most recent was published by Nature this past spring (summary | full study. It found that, perhaps unsurprisingly, the areas which showed evidence of migration into England from Germany were those closest to the North Sea, and also those which identified themselves as the descendants of the Angles, Saxons, and Jutes in the early middle ages. Areas that were more isolated, like Cornwall, were more genetically unique.

On the whole, Britain's current genetic makeup is (again, from the study in Nature) very homogenous, but you do still see subtle differences between regions separated by geographic boundaries. Wales has several distinct sets of small genetic difderences dividing it into regions, and these aren't shared with England. These are partly due to things like the early medieval migrations from Germany not extending into Wales, but really map very neatly to things like mountains and rivers (some of which were clearly boundaries across which few people traveled to procreate).

These studies aren't super precise, because they're based on the DNA of modern, living populations, and are looking for patterns and traits which, through mathematical models, are used to estimate how far back in time shared genes entered a population. So, in the case of Germanic immigrants, the Nature study's authors found evidence of a genetic marker that's found in modern Germany in parts of England, and their models suggested it appeared in the English population about 1500 years ago. There were other genetic markers shared eith modern Germany and found througout Britain, which seem to have come from a much earlier migration (during the Neolithic, perhaps).

Because of the nature of these studies, you can't be so precise as to spot the difference between Jutes of Saxons in modern genes.

But - and this is where I want to make my answer much more complicated - even if we were to be able to pinpoint the specific DNA of different tribes (by, say, reconstructing DNA from bones in early medieval Anglo-Saxon cemeteries, which is something scientists are beginning to do with some human remains from the early middle ages), we almost certainly wouldn't find clear-cut differences between tribes. Because, scholars mostly agree, the Angles, Saxons, and Jutes weren't distinct people groups with their own unique genetics. Instead they, and the other tribes you see mentioned in ancient and early medieval texts, were loose confederations of people who might, individually, not have much in the way of a shared history or ancestry.

There are several competing scholarly schools for understanding early medieval tribes and ethnicities, but they all agree that the evidence we have suggests these groups weren't racially homogenous people groups. Instead, they grew and shrank and formed and disbanded in response to changing political pressures and internal politics. A group like the Ostrogoths might form around a leader and, in a few generations, rule Roman Italy, but when they then told you they had a thousand-year-long history reaching back to migrations out of Scandinavia, they were being inventive.

The Saxons, Angles, and Jutes were real names of people groups that lived in Germany during the late aroman empire, but it's not clear that these groups moved en-masse into England in the fifth century as Bede would have us believe. In fact, the archaeology shows a much more gradual process of individual farmers and their families crossing the English Channel or North Sea and blending in with the local population, not migrations of whole tribes. Some migrants from Germany almost certainly immigrated while Britain was still part of the empire, perhaps serving as recruits for the army (a common occupation for barbarians in the late empire). Indeed, many are now arguing (and I believe this to be correct) that the regions of England which came to be identified with Jutish, Angle, or Saxon migrants by the time Bede wrote down his island's history (the 730s, about 300 years after the fact) invented these connections with specific, famous, named tribes in Germany well after the fact (example). That is, they picked a group of promising sounding ancestors in Germany, and said, 'we're all descended from those guys!' Some of them probably were; but all of them were products of centuries of piecemeal migration and genetic and cultural blending with England's Romano-British population, as well.

But your questions is about influence, not just ancestry, which means we need to talk about culture as well as genetics. And if we're talking about cultural influence in England, the picture above gets far more messy. Because not only did tribal groups like the Angles and Saxons lack distinguishing genetic features (and, indeed, almost certainly invented themselves after the fact); there's also growing consensus that we can't connect unique cultural practices to specific tribes.

Archaeologists have, indeed, tried to identify cultural differences between Angles, Saxons, and Jutes, and they did manage to identify some regional differences which they decided to connect with each of the three tribes. People in Kent (where Bede says the Jutes lived) tended to use particular styles of jewelry, different from that used in Wessex (which was, supposedly, all Saxons). But the methodology behind connecting these style differences with different named tribes is really sketchy, and ultimately boils down to selectively identifying patterns in the archaeology which serve to confirm pre-existing notions of where people groups 'should' have lived (according to Bede's 300-years-later, and very unreliable, description), and then slapping an ethnic label onto that pattern (Jutish brooch! Saxon pot!). It sounds very cut and dry to see a buckle described as 'Anglian' in an archaeological report, but that identification is not at all scientific, nor even especially accurate. Indeed, a recent study has shown that the patterns in objects called 'Jutish', 'Anglian,' and 'Saxon' actually match up with England's geography: the 'Anglian' objects map onto the region where the rivers connect to Scandinavia, the 'Jutish' objects are near rivers that flow toward the Channel and France, and the 'Saxon' items are concentrated near rivers that flow toward Ireland. What's that mean? These are almost certainly regional, not tribal, styles, and they tell us more about how people moved around inside England (and between England and its neighbors France, Scandinavia, and Wales/Ireland) than they do any specific tribe's cultural style or enduring 'influence.'

If we look at the culture that emerged before the Norman conquest, what we see is a complex mix of influences that varied across the Island (and it gets even more complex if you extend this to all of Britain instead just England!). At its base, England still preserved a lot of its pre-Roman farming culture. England had been intensely cultivated for thousands of years before the Romans arrived, and despite a spate of villa building that attempted to restructure parts of the British countryside in the later Roman empire, Britain's agricultural landscape seems to have remained strongly influenced by field patterns established before the Romans arrived. Though in some areas at least, the Anglo-Saxon kingdoms that emerged by the seventh century restructured the landscape. And despite the decline of Roman urban areas, England's farming economy seems to have kept calm and carried on.

In other ways, Roman influence continued to be felt. Rome had left a massive pile of ruins on the landscape, and these ruins provided both physical materials for recycling (including iron, lead, and numerous trinkets and antiques) and ideological capital for future invented identities (especially in Kent, which was closely connected with its self-styled Roman neighbor Merovongian Francia - recall that Clovis, according to Gregory of Tours, wad given the official title of a Roman consul - and which was the first to re-establish connections with the Roman church).

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