This is some kind of continuation of an earlier MO post: Existence of maximal subgroups of even order which are not normal
It appeared in a comment in the above post. I believe the following is true:
Given a finite non-solvable group $G$, there exist two conjugate maximal subgroups $M$ and $N$ of $G$ such that there exist two elements $a \in M\setminus N$, and $b \in N\setminus M$ such that the subgroup generated by $a$ and $b$ is a proper subgroup of $G$.
If $G$ is simple, this is easy to prove. I am stuck with the non-simple, non-solvable case.
