Maximal normal iff normal of prime index in solvable

Statement

Suppose $G$ is a (finite or infinite) 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.