# What are the intermediate subfactors of the tensor product of two maximal subfactors?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.

Let $N_{1} \subset M_{1}$ and $N_{2} \subset M_{2}$ be two maximal subfactors :
What are the non-trivial intermediate subfactors of $N_{1} \otimes N_{2} \subset M_{1} \otimes M_{2}$ ?

Obviously, there are:
- $N_{1} \otimes N_{2} \subset N_{1} \otimes M_{2} \subset M_{1} \otimes M_{2}$
- $N_{1} \otimes N_{2} \subset M_{1} \otimes N_{2} \subset M_{1} \otimes M_{2}$
but, what about the others ?

For example, in the case of $R^{\mathbb{Z}/p\mathbb{Z}} \subset R$ and $R^{\mathbb{Z}/q\mathbb{Z}} \subset R$ with $p$ and $q$ prime numbers, their tensor product admits one (and only one) other intermediate subfactor if and only if $p=q$.

So in general we could speculate that there is one (and only one) other intermediate subfactor if and only if the initial subfactors are isomorphic, but it's false if they are 2-supertransitive and of index > 2, because in this case, even if they are isomorphic, there is no room for a single one other.
$\rightarrow$ See Watatani (1996) proposition 5.1 page 329.