All the subfactors $(N\subset M)$ are irreducible and finite index inclusions of II$_1$ factors.

First recall that in this paper, D. Bisch characterizes the Jones projections $e_K$ of the intermediate subfactors $(N \subset K \subset M)$ as projections $p \in N' \cap M_1$ such that $p \geq e_N$ and $\mathscr{F}(p) = \lambda q$,

with $q$ a projection, $\lambda \in \mathbb{C}$ and $ \mathscr{F} : N' \cap M_1 \to M' \cap M_2$ the Ocneanu Fourier transform.

In this paper of T. Teruya, an intermediate subfactor
$(N \subset K \subset M)$ is defined as **normal** if:

- $e_K \in Z(N' \cap M_1 )$
- $\mathscr{F}(e_K) \in Z(M' \cap M_2)$

with $Z(X)$ the center of X, and $\mathscr{F}$ as above ($\mathscr{F}(x)=[M : N]^{-3/2}E_{M'}^{N'}(xe_{M}e_{N})$).

Teruya proves that for the depth $2$ case (Kac algebra), the normal intermediate subfactors gives exactly the normal Kac subalgebras (in particular, the normal subgroups for the group subfactors).

**Remark** : If $N' \cap M_1 $ and $M' \cap M_2$ are abelian, then every intermediate subfactor is normal.

**Example**: Let $(A \subset B)$ and $(C \subset D)$ be $2$-supertransitive subfactors, $N= A \otimes B$ and $M= C \otimes D$, then $P=A \otimes D$ and $Q=B \otimes C$ are normal intermediate subfactors of $(N \subset M) $ because $N' \cap M_1 $ and $M' \cap M_2$ are $\mathbb{C}^4$, and so abelian (see Watatani prop 5.1 p329).

This result is true without the $2$-supertransitivity assumption if $\mathscr{F}_{(N\subset M)} = \mathscr{F}_{(A\subset B)} \otimes \mathscr{F}_{(C\subset D)}$.

A subfactor is **simple** if it has no non-trivial normal intermediate subfactor.

A group subfactor is simple iff the group is simple. A maximal subfactor is *a fortiori* simple.

Question: Let $(N \subset M)$ be a subfactor, and let $$ N=K_1 \subset K_2 \subset \dots \subset K_r = M $$ be a normal chain such that each subfactor $(K_i \subset K_{i+1})$ is simple, and $K_i \neq K_{i+1}$ for $0<i<r$. Then any other normal chain of $(N \subset M)$ having the same properties is equivalent to this one (i.e. the sequence of subfactors in our two chains are the same up to isomorphisms, and a permutation of the indices) ?

The rest of the post is dedicated to a reformulation of the question.

Through this comment, Benjamin Steinberg shows me that the Jordan-Hölder theorem is a generic property of modular lattices (i.e. lattices checking : $x ≤ b \Rightarrow x ∨ (a ∧ b) = (x ∨ a) ∧ b$, $\forall a$).

In the lattice theory framework, the subfactors $(A \subset B)$ and the isomorphisms, are replaced by intervals $[a,b]$ and projectivities (two intervals $[a, b]$ and $[c, d]$ are **perspective** if $b ∨ c = d$ and $b ∧ c = a$ or vice versa. **Projectivity** is the transitive closure of perspectivity). There is a well-known Jordan-Hölder theorem for modular lattices (also semimodular, see this paper of Grätzer-Nation).

So we would need, firstly to prove that the set of normal intermediate subfactors is a lattice and is modular, and secondly that projective intervals in such lattices give isomorphism of subfactors.

In this paper, Y. Watatani introduced the notion of quasi-normal intermediate subfactors (by using two commuting squares) and proved modular identites (W thm3.9 p323).

But Teruya proved that a normal intermediate subfactor is quasi-normal (T thm3.4 p377).

So it rests to prove that we have a lattice and the second point about projectivity and isomorphism.

This is like the second isomorphism theorem for groups, and it's the content of the reformulation :

Reformulation: Let $(N \subset M)$ be a subfactor, let $P$, $Q$ be normal intermediate subfactors, then:

Are $P \wedge Q$ and $P \vee Q$ normal, and $(P \wedge Q \subset Q)$ isomorphic to $(P \subset P \vee Q)$ ?

**Remark** : $P \wedge Q = P \cap Q$ and $P \vee Q = PQ=QP$.

The latticeness part seems reduced to know if $\mathscr{F}(e_P.e_Q) \in Z(M' \cap M_2)$.