Theorem 2 in my preprint "Group Orders That Imply Existence of Nontrivial Normal $p$-Subgroups" shows that if $|G| = p^s m$ and $p \nmid \Gamma(m)$ (defined therein), then either $O_p (G) \ne 1$, or $G$ is not solvable.
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Edited to include information about how this answer addresses the question
Alex M.
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