TITLE: What is known about arbitrary subfactors of integer index?
QUESTION [11 upvotes]: Let $N\subset M$ be an inclusion of ${\rm II}_1$ factors of finite index, $[M:N]<\infty$. I would be mostly interested in the hyperfinite case, $N\simeq M\simeq R$, but let us just take them arbitrary.
There is an evolving theory about "what can be said about $N\subset M$, in the general case'', which started with Jones' index theorem, in 1983. Well-known results here include the Pimsner-Popa basis and entropy formula, the bimodule interpretation, the planar algebra formalism.

Question: assuming that we are still in the general case, but with the extra assumption that the index is an integer, $[M,N]\in\mathbb N$, what else can be said about $N\subset M$? 

To my knowledge, at least some time ago (5-10 years), the only answer here was just that the Pimsner-Popa basis is a "clean" one, I mean as in standard linear algebra. I was wondering if any advances on this question come from the recent work on subject, in small or arbitrary index I mean, perhaps as some corollaries of the theory developed there (?) I would be interested in any comment/answer here, this is actually a question that I spent some time on, long time ago.

REPLY [2 votes]: See the issue : Non-“weakly group theoretical” integral fusion categories? 
Here is proposed some integral fusion rings, such that the prospective related fusion categories, are integral but not weakly group theoretical.
Currently, we try to build some Kac algebras $\mathbb{A}$, having these fusion rings as their Grothendieck rings. 
The related subfactors $R^{\mathbb{A}} \subset R$ would be irreducible, depth 2, integer index, but not coming from cohomology and group theory.   
You can also read my answer to the issue : Are subfactor planar algebras hard to classify at index 6?