Let $G$ be a finite group with the property that all of its proper subgroups are abelian. Let $N$ be a normal subgroup of $G$. Prove that either $N$ is contained in the center of $G$ or else $G$ has a normal abelian subgroup of prime index.
I think $G$ is solvable. http://crazyproject.wordpress.com/2010/06/08/every-finite-group-whose-every-proper-subgroup-is-abelian-is-solvable/. I hope that idea maybe usefull.
Help me some hints.
Thanks a lot.
P/s: This is a question comes from a qualifying exam in Algebra ( Wisconsin August $1979$ )