  In this paper we describe some properties of groups $G$ that contain a
solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3).
We prove that if $G$ is a non-solvable group that contains a solvable subgroup
of index $p^{\alpha}$ (for some prime $p$), then the quotient $G/\mbox{rad}(G)$
of $G$ over the solvable radical is asymptotically small in comparison to
$p^{\alpha}!$ (Theorem 4).
