Columbia does not really have any Secret Societies (unless you count St. A's, which can be pretty $ecretive). If it does, then they're pretty damned secret because no one knows about them. Columbia does have two very low-profile Senior Societies, the Senior Society of Sachems and the Senior Society of Nacoms. Most students don't even know the societies exist.
Each society taps 15 juniors each year, the Sachems by luring the candidates into an ambush where he or she gets hit in the face with pie, the Nacoms by convincing the candidate that he or she is in trouble.
Society membership tends to reflect powerful student leaders across the campus, and some seats develop 'lineages', passing on from one holder of a leadership position to the next.
Their commitment to secrecy is questionable. Current members take it very seriously, but many alumni put it on their resumes.
There is some indication that there was a secret society nearer the beginning of Columbia's history, possibly called LNF (but what that stands for is a mystery), but little to nothing is known about it, and if it did exist, it is unlikely that it still does.