Symmetries of subfactors motivated by Aschbacher–Guralnick conjecture

Aschbacher–Guralnick conjecture states that the number of conjugacy classes of maximal subgroups of a finite group is bounded by the number of conjugacy classes of the group. Attempting to generalize this to the more general framework of subfactors, we are led to investigate the actions of two finite symmetry groups of a subfactor on the first non-trivial relative commutant of the subfactor which are compatible with their actions on the intermediate subfactors. Our first main result is that the actions of these two groups commute, and thus we can formulate a sensible subfactor generalization of Aschbacher–Guralnick conjecture which reduces to the original Aschbacher–Guralnick conjecture in the group subfactor case. In the case of group-subgroup subfactors, our conjecture can be stated in group theory terms as a relative version of Aschbacher–Guralnick conjecture, and we prove that this conjecture is true for solvable groups.