Great question! During the third century A.D., Roman military organization undergoes several changes, the reasons for this are unclear due to a paucity of sources. I am therefore going to backdate my answer to the second century A.D., although do note most of these arrangements likely persisted throughout the early and mid third century.

During the Principate, the commander of an auxiliary unit was known as a praefectus, or prefect. Prefects were members of the Equestrian class, the “lower”(relatively speaking) tier of the Roman imperial elite. Equestrians who wished to serve in the army followed a specific career path of a succession of auxiliary commands, known as the militia equestris. Following the reign of Claudius, this consisted of a prefecture of a cohort, legionary tribunate, and prefecture of an ala. Although prestige doubtless played a role in this arrangement, increasing levels of responsibility was likely more important. It required more competence to command a cavalry ala than to be one of several legionary tribunes. Promotion along this career path was not a given- patronage and personal or familial influence mattered in securing advancement, especially in an environment of limited posts. It is estimated that only about two-thirds of cohort prefects during the second century A.D. would have progressed to a legionary tribunate, with only half of those men advancing to the prefectship of an ala. Even fewer would have commanded larger auxiliary units (which I’ll discuss in a second), or gone on to hold junior administrative posts. The further one advanced, the more imperial patronage and perceived loyalty mattered.

Auxiliary units during the Principate were classified by several different titles. The “basic” infantry unit was the cohortes quingenariae peditatae, organized like a legionary cohort. It thus had six centuries of (supposedly) eighty men. This gives a paper-strength of around 480 men, though actual strength varied. From the reign of Tiberius, mixed auxiliary units began to be created. Cohortes equitatae consisted of four turmae (sections or “companies”) of thirty to thirty-two cavalrymen, along with six centuries of infantry. Equitatae units were popular, with about 130-135 in existence during the second century, similar to the number of cohortes quingenariae peditatae. Both types of units were generally commanded by Equestrians beginning their military careers.

From the time of Nero, larger infantry units (known as cohortes milliariae peditatae) are known. These units had ten centuries, rather than six, with a total paper-strength of around 800 men. Larger mixed units, known as cohortes milliariae equitatae, contained twelve cavalry turmae in addition to the six infantry centuries (for a total of around 1,056 men). These peditatae and equitatae units were distinguished by the title of their commander- tribunus (tribune) rather than praefectus. These were commanded by men as an alternative to serving as a legionary tribune.

Alae quingenariae were purely cavalry units, with 512 men divided into sixteen turmae. More prestigious than infantry or mixed units, they were commanded by praefecti in what was likely a final military command. As there were only around ninety such units in the entirety of the army, competition was fierce. If an Equestrian officer had progressed this far, he would likely be able to pursue a procuratorship (governor of a junior province) before ending what was by all accounts a successful public career.

Only about eight or nine larger cavalry units existed during the second century. Bearing the title alae milliariae, they were comprised of twenty-four turmae, with a strength of 768 men. It is clear these were regarded as elite and highly prestigious units, with command going to a prefect holding an exceptional fourth position in his military career. Such commands were not given lightly- they were awarded to only the most highly-viewed officers. These men likely went on to hold the most senior Equestrian posts.

So, to summarize: Auxiliary officers were Equestrians, who followed a standardized career path. The number of commands was limited; most officers did not complete the full path of service. Auxiliary units were distinguished by their titles. In ascending order of precedence, these were: cohortes quingenariae peditatae, cohortes equitatae, cohortes milliariae peditatae and equitatae, alae quingenariae, and alae milliariae.

Information for this response was drawn largely from Boris Rankov’s chapter, “Military Forces” in The Cambridge Companion to Greek and Roman Warfare (Vol. II). This text outlines current scholarly consensuses and debates on a variety of topics in Greek and Roman warfare. It’s abhorrently expensive (like $400+ on Amazon or some nonsense) so don’t buy it- see if you can get it from a university or college library. For a more general (and affordable) introduction to Roman warfare, I would recommend Goldsworthy’s The Complete Roman Army. It’s the best introductory text written by an expert in the field.