Although Jesuits are orthodox in their religious beliefs, they are well educated and tend to act as an institution in a fairly scientific and socially progressive fashion as a whole. This has brought them into conflict with politically and socially conservative forces.

They also bring a certain amount of structure and at times an almost military discipline and hierarchy to their work. They were and are capable of organizing in ways other groups are not. Add to this the fact that a disproportionate number of Jesuits have professional credentials and advanced academic degrees and you can start to see how they can be seen as troublesome by national governments even when Catholicism is the state religion. Even a Jesuit who does not hold a masters of doctorate is very likely to hold multiple undergraduate degrees or to have pursued some form of postgraduate education. It's de rigueur since the Jesuits historically have been used as troubleshooters and internal experts on a wide range of fields by the Catholic hierarchy. The corollary to this is that even a freshly ordained Jesuit priest is likely to be much more worldly and older than their counterparts coming from other religious orders and congregations.

There really are very few equivalent religious orders in any major religious group anywhere in the world. As a group they are the "Ivy League" graduates of the Catholic Church. One can see how this can be a recipe for rumours and conspiracy theories.

Beyond their education and credentials, one of the key reasons that they have been watched closely is because as an organization they have been not only qualified but also very active in society and especially active in areas that are considered the secular or state domain such as higher education, medicine, the law, social services and scientific research. Depending on the place and time this has been both very beneficial and also unwelcome.

One of the side effects was to create a power base that exists within the fold of the Catholic Church and has access to its resources, but which acts with its own interests and as a somewhat independent organization.

For these reason the Jesuits were not nearly as easily controlled and influenced, particularly in the era when the Catholic Church was still a major political power in many European States.

The Society of Jesus has also historically not been a path for promotion to high office in the Catholic Church as compared to other religious orders. (As an aside, this is one of the reasons that the election of Bergoglio as pope was seen as remarkable.)

This article discusses at length the suppression of the Jesuits and some of the political motivations: http://ieg-ego.eu/en/threads/european-media/european-media-events/christine-vogel-suppression-of-the-society-of-jesus-1758-1773

If you're not familiar with their history, this article in The Guardian is a very brief overview: http://www.theguardian.com/world/2013/mar/14/pope-francis-who-are-the-jesuits

I'm not an expert on the topic and in university I didn't study European history except at an introductory level. Hopefully somebody with deeper knowledge can elaborate and/or correct any mistakes or misinterpretation I've made.