Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.
There are plenty of examples of 3-super-transitive (3-ST) subfactors; Haagerup, S_4 < S_5, and others. There's exactly one known example of a 5-ST subfactor, the Haagerup-Asaeda subfactor, and one 7-ST subfactor, the extended Haagerup subfactor.
Below index 4 there are the $A_n$ and $D_n$ families, which are arbitrarily super-transitive. Ignore those; I'm just interested above index 4.
Is there anything that's even more super-transitive?
