Let $M$ be a type III$_1$ factor and $N$ be the type II$_1$ subalgebra of $M$.
What is the type of $N'\cap M$? Can it be any type?
Such a relative commutant $N' \cap M$ can even be any factor. Choose any type III$_1$ factor $P$ with a type II$_1$ subfactor $N \subset P$ with trivial relative commutant $N' \cap P = \mathbb{C} 1$. When $Q$ is an arbitrary factor, we get that $P \overline{\otimes} Q$ is a type III$_1$ factor that contains the type II$_1$ subfactor $N \otimes 1$ with relative commutant $1 \otimes Q$.