Random subgroup of a group

Question

Given a finite group $G$, Is there any known algorithm which gives a random subgroup of $G$?

Answer 1

Unless you have more information about the group like a Cayley table, or some information about the structure of the group, they the answer is obviously no because the group could literally be anything. If you do have this information available, then as drhab mentioned you could write a program that exhaustively searches for all the subgroups of $G$. You can save computation by using Lagrange's theorem to only check subsets of $G$ that divide the order of $|G|$ (so if $G$ is prime, you're in luck my friend!). I'm sure there are other ways you could also improve efficiency too, but that's the first one that comes to mind. After that's done, just label the subgroups from 1 to however many subgroups there are and choose a random number in that range.

Answer 2

Choose a random element as cyclic subgroup generator, generate the subgroup from that element, and you already have a random cyclic subgroup.