The representation theory of subfactors generalizes the representation theory of quantum groups, and thus we think of subfactors as objects which encode quantum symmetries. In one sense, subfactors of small index are the simplest examples of subfactors, and we have a complete classification of their standard invariants to index 5. I will discuss recent joint work with Morrison which classifies certain examples at index $3+\sqrt{5}$. One important ingredient is a new variation of Bigelow's jellyfish algorithm which is universal for finite depth subfactors.