Extensions of solvable groups are solvable

Statement

Let $G$ be a group and let $N$ be a normal subgroup of $G$. If $N$ and $G/N$ are solvable, then $G$ is solvable.

Remarks

If $(G/N)^{(k)}$ is trivial, then $G^{(k)} \subseteq N$, so $G^{(k+m)} \subseteq N^{(m)}$ for each $m$. Choose $m$ with $N^{(m)} = \{e\}$; then $G^{(k+m)} = \{e\}$. With subgroups and quotients, this gives: $G$ is solvable if and only if $N$ and $G/N$ are solvable. In particular, direct products and semidirect products of solvable groups are solvable.

Explanations

L1 human

If the bottom part and the top part of a group can both be built from abelian layers, then so can the whole group. You stack the layers of the top part on the layers of the bottom part. For example, $S_4$ is built from $V$, $A_4/V$ and $S_4/A_4$.