Before I begin, we first need to understand what the Illuminati is.
According to Encyclopedia Britannica:
...During the 1780s Masonic lodges had begun to replace scholarly academies and agrarian societies as loci of political discussion. In the 1790s more-radical secret societies emerged, modeled after the Illuminati (“Enlightened Ones”) founded in Bavaria by Adam Weishaupt, a professor of canon law, which promoted free thought and democratic political theories... ... The political and social prerogatives of the church were also threatened by the Enlightenment, especially when it became allied with the expanding claims of an autocratic “enlightened despotism.” The brotherhood cultivated by groups such as the Freemasons and the Illuminati, a rationalist secret society, constituted a rival to the feeling of community that the church had once provided. The Masonic alternative to the Catholic mass even became the subject of an opera, The Magic Flute byMozart.
So, obviously the Illuminati weren't the only secret society, and their political theories echoed that of Freemasonry: liberty, equality, and fraternity. The Illuminati were officially formed in Bavaria in 1776, May 1 I believe, a high occult holiday.
Secondly, the Illuminati were not only a small organization; Illuminism itself is an ideology. And unlike other secret socities, the Illuminati's primary mode of influence was not organizational, but literary. The evidence suggests that even though the Illuminati organizational structure suffered from government opposition and sanctions in Bavaria and other places, Illuminism as an ideology continued to exist even to today.
Weishaupt admired Igantius of Loyola, and so the Illuminati were in many ways an imitation of the Jesuits, and their recruiting and control practices a parody of the Spiritual Exercises. Where faults are identified by the Jesuit superior through examination of conscience to be then confessed and their power over the novice thereby broken, the Illuminist parody of the examination of conscience first ferrets out dominant passions to be preserved and manipulated by the Illuminist controller, rather than extirpated through repentance and confession. Examination of conscience in the Illuminist sense of the word is used by the Illuminist confessor as an instrument of control. Once the adept had confided his vices to his superior as part of the initiation rite, his passions will be used as a way of controlling him. If he discovers the ploy and objects, his past sins will be used against him in a form of blackmail that is in many ways demonic perversion of the seal of the confessional...."
—E. Michael Jones, Libido Dominandi: Sexual Liberation and Political Control, 2000, pg. 88-89
Jones is talking about Illuminism as ideology in the above excerpt. Clearly, this is the foundation for Sigmund Freud's psychoanalysis (evidence suggests Freud was a Kabbalist, which is central to Illuminism) and, later, how so-called sex doctor Alfred Kinsey was able to secure control and funding for his corrupt sex research, by taking the sex histories of his colleagues and of Rockefeller executives and using that knowledge as a form of blackmail, which resulted in Kinsey's unfettered financial support from the Rockefeller Foundation and, ultimately, the beginnings of the modern sexual revolution as a form of political control.
Incidentally, the first sex research institute, Institut für Sexualwissenschaft, was built in the Illuminist Weimar Republic. Both Freud and Kinsey epitomize the ideology of Illuminism. Like Weishaupt, Freud and Kinsey would use the intimate knowledge of their patients—and colleagues—as a form of control over them. The ultimate end of this is political control through sexual liberation on a grand scale. Enter Freud's nephew Eddie Bernays, who engineered a way to sell products by exploiting the sexual passions of the general populations.
On January 2, 1785, the Prince Bishop of Eichstaett demanded that the Prince of Bavaria purge all Illuminati from the University of Ingolstadt. In spite of the secrecy of the Illuminati, Weishaupt was a prime suspect because of the radical Enlightenment books he had ordered for the University library. Weishaupt was removed from his chair of canon law at the University of Ingolstadt on February 11, 1784. Over the next year, the hue and cry against secret societies increased dramatically...when the Prince of Bavaria, Karl Theodore, issued his second edict, the Minerval lodge in INgolstadt, now without Weshaupt (he fled to Regensburg on Feb. 2, 1785) as its head, was dissolved...If the Bavarian authorities had left it at that, the Illuminati would most probably have been forgotten forever or at best rmained a minor footnote in a very small book. But the Bavarian government, after discovering the secret documents associated with the lodge in Munich, made a fateful decision; they decided to publish what they found and in so doing assured Weishaupt and his conspirators an influence they never could have achieved on their own.... —E. Michael Jones