Normal intermediate subfactors

Let $N\subset M$ be an irreducible inclusion of type type II${}_1$ factors with finite Jones index. We shall introduce the notion of normality for intermediate subfactors of the inclusion $N\subset M$. If the depth of $N \subset M$is2,thenanintermediatesubfactor$K$for$N \subset M$ is normal in $N\subset M$ if and only if the depths of $N\subset K$ and $K\subset M$ are both 2. In particular, if $M$ is the crossed product $N⋊G$ of a finite group $G$, then $K=N⋊H$ is normal in $N\subset M$ if and only if $H$ is a normal subgroup of $G$.