Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such subfactor of prime index must be maximal, moreover, many possible indices of such subfactors can be ruled out as follows: let $\alpha_1, \dots , \alpha_n > 0$ such that $\prod_i \alpha_i \in \mathbb{N}$, then one among the $\alpha_j \not \in \mathbb{N}$ cannot be the index of such a subfactor (proof: if all of them are, you get a contradiction with Bisch's theorem by considering a direct product of subfactors). For example, there is no such subfactor of index $n^{p/q} \not \in \mathbb{N}$ (where $n, p, q \in \mathbb{N}^{*})$, or there is none of index $n(3-\sqrt{5})$, because there are ones of indices $2$ and $\phi =(3+\sqrt{5})/2$, but $2 \cdot \phi \cdot n(3-\sqrt{5}) = 4n \in \mathbb{N}$. So we can a lot retrict the set of possible such indices like that (I don't know how much).

Now there is an other way to make such restrictions coming from fusion categories, using a theorem in this paper (Etingof-Nikshych-Ostrik, 2005), which implies that the index of such subfactors must be a cyclotomic integer, i.e. an element of $\mathbb{Z}[c_d]$ for some integer $d$, with $c_d=2\cos(2\pi/d)$.

**Question**: Can Bisch's theorem be extended to cyclotomic integers of fixed degree (i.e. if the index of a finite depth ${\rm II}_1$ subfactor is in $\mathbb{Z}[c_d]$, then so is for the index of the intermediate subfactors)?

As an application, we could get even more restrictions on possible indices of such subfactors, moreover, if such a subfactor is of index $\alpha \in \mathbb{Z}[c_d]$ and if $\beta$ is the index of an intermediate subfactor then $\beta\mathbb{Z}[c_d]$ would be a divisor ideal of $\alpha\mathbb{Z}[c_d]$; in particular, if $\alpha\mathbb{Z}[c_d]$ is a prime ideal, then the index of intermediate subfactors would only be units of $\mathbb{Z}[c_d]$ or units multiple of $\alpha$.