Intermediate subfactors with no extra structure

If $N\subseteq P,Q\subseteq M$ are type II$_1$ factors with $N'\cap M =\mathbb{C} id$ and $[M:N]<\infty$ we show that restrictions on the standard invariants of the elementary inclusions $N\subseteq P$, $N\subseteq Q$, $P\subseteq M$ and $Q\subseteq M$ imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto $P$ and $Q$ do not commute, then $[M:N]$ is $6$ or $6+4\sqrt 2$. In the former case $N$ is the fixed point algebra for an outer action of $S_3$ on $M$ and the angle is $\pi/3$, and in the latter case the angle is $\cos^{-1}(\sqrt 2-1)$ and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.