Every finite group is generated by a solvable subgroup and one element

Statement
Suppose $$G$$ is a finite group. Then, there exists a solvable subgroup $$H$$ of $$G$$ and an element $$x \in G$$ such that $$G = \langle H,x \rangle$$.

In other words, every finite group has a solvable fact about::1-completed subgroup.

Related facts

 * Finite minimal simple implies 2-generated
 * Finite simple implies 2-generated
 * Finite almost simple implies 3-generated
 * Solvability is 2-local for finite groups