oa.operator algebras – Is there a finite depth irreducible subfactor of prime index and not group-subgroup?

Let $N subset M$ be a finite depth unital inclusion of II$_1$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $|M:N|$ is integer then for any intermediate subfactor $N subset P subset M$, $|M:P|$ and $|P:N|$ are also integer. In particular, if $|M:N|$ is prime then there is no proper intermediate (maximal subfactor).

A finite group subfactor remembers the group but a finite group-subgroup subfactor does not always remember the (core-free) inclusion of groups (Kodiyalam-Sunder, 2000). This paper (Izumi, 2002) provides a group-theoretical characterization of isomorphic group-subgroup subfactors. It is easy to see from this characterization that a maximal group-subgroup subfactor remembers the (core-free) inclusion, because the intersection of a core-free maximal subgroup with an abelian normal subgroup is trivial (see here).

A core-free inclusion of finite groups is the same thing than a transitive permutation group, and in the maximal case, replace transitive by primitive. Obviously, at prime degree, a transitive permutation group is always primitive.

So the number of maximal group-subgroup subfactors (up to dual) of index $n$ is exactly the number of primitive permutation groups of degree $n$, see OEIS/A000019 ($1, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, dots$).

Here is a way to get the corresponding inclusion of finite groups $H subset G$ by GAP:

gap> NrPrimitiveGroups(5);
gap> for i in (1..5) do G:=PrimitiveGroup(5,i);; H:=Stabilizer(G,1);; Print((H,G)); od;
( Group( () ), C(5) )( Group( ( (2,4)(3,5) ) ), D(2*5) )( Group( ( (2,3,4,5) ) ), AGL(1, 5) )
( Group( ( (2,4)(3,5), (3,4,5) ) ), A(5) )( SymmetricGroup( ( 2 .. 5 ) ), S(5) )

The subfactors of index $2$ or $3$ are given by the groups $C_2$, $C_3$ and the inclusion $(S_2 subset S_3)$, and (dual) principal graphs $A_4$, $D_4$ and $A_6$. At index $4$, the subfactors of principal graph $E_i^{(1)}$, $i=6,7,8$ are all maximal (because $2$-supertransitive), but there is only two primitive permutation groups of degree $4$, giving the inclusion $A_3 subset A_4$ and $S_3 subset S_4$, with (dual) principal graphs $E_i^{(1)}$, $i=6,7$. So the maximal subfactor of index $4$ and (dual) principal graph $E_8^{(1)}$ is not a group-subgroup subfactor. Finally every finite depth subfactor of index $5$ is group-subgroup (Izumi-Morrison-Penneys-Peters-Snyder, 2015).

Conclusion, a finite depth maximal irreducible subfactor of integral index is not always group-subgroup, but the found counter-example (of principal graph $E_8^{(1)}$) is not of prime index. So the following question remains:

Question: Is there a finite depth irreducible subfactor of prime index and not group-subgroup?