Let $ G $ be a minimal non-solvable group. Why $ G/\Phi(G) $ is a minimal simple group, where $ \Phi(G) $ is the Frattini subgroup of $ G $ ?
If $G/\Phi(G)$ is not simple, then let $N/\Phi(G)$ be a minimal normal subgroup. Then $N < G$, so $N$ is solvable. Let $M$ be a maximal subgroup of $G$ not containing $N$ (such an $M$ must exist since $N$ is not contained in $\Phi(G)$). Then $MN=G$, and since $N$ is solvable but $G$ is not, $M$ cannot be solvable, contradicting $G$ being minimal non-solvable.
So $G/\Phi(G)$ is simple and is clearly minimal simple.