  If $N \subset P,Q \subset M$ are type II_1 factors with $N' \cap M = C id$
and $[M:N]$ finite we show that restrictions on the standard invariants of the
elementary inclusions $N \subset P$, $N \subset Q$, $P \subset M$ and $Q
\subset M$ 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.
