Maximal normal iff normal of prime index in solvable

Statement
Suppose $$G$$ is a (finite or infinite) fact about::solvable group. The following are equivalent for a normal subgroup $$H$$ of $$G$$:


 * 1) $$H$$ is a maximal normal subgroup of $$G$$.
 * 2) $$H$$ is a normal subgroup of prime index in $$G$$.
 * 3) $$G/H$$ is a group of prime order.