Asking elsewhere, I have learnt that
Pythagoras seems to have established his system on a similar plan to that of Freemasonry.
"The schools established by Pythagoras at Crotona, and other cities, have been considered by many writers as the models after which Masonic Lodges were subsequently constructed. They undoubtedly served the Christian ascetics of the first century as a pattern for their monastic institutions, with which institutions the Freemasonry of the Middle Ages, in its operative character, was intimately connected.
Women proved to be powerful, determining influences in the life, philosophy and schools of Pythagoras. Beginning with his mother, Parthenis (an epithet of Athena, meaning ‘virgin’) who, after dedicating him to the Pythoness at Delphi (hence his name), was herself renamed Pythasis (identifying her as a priestess of the cult). Among the later women in the life of Pythagoras were:
Aristocleia who taught him;
Theano who, as well as being the head of the cult at Delphi, was already a leading mathematician and philosopher when they met. She was his finest student, they married and he named her as his successor; and Damo, their daughter, to whom they entrusted all their secrets and compositions. Included among the most famous of the Pythagorean philosophers were, according to Iamblichus, no fewer than seventeen women.
Manly Hall reports that Pythagoras was criticised for compromising his lessons by his strict adherence to the obligations he had undertaken in various mystery schools of Greece, Egypt, Persia and India, requiring at times such things as regular initiation, secrecy, years of silence, teaching in allegories or from behind a screen. However, as Hall observed, ‘Having accepted the obligations of these societies, Pythagoras had no honorable course other than to abide by their regulations.’ Having proved to be so faithful to all his obligations, Pythagoras would certainly not have admitted women to his schools if their admission was prohibited by any of the Ancient Mysteries which contributed to the wisdom he taught."