Order Equal to 392

If the order of a group G is $392 = 2^{3} \times 7^{2}$ the index of a Syl₇ subgroup is equal to 8. Therefore, by Theorem 0.2, the group Gis isomorphic to a subgroup of S₈. But this would imply that $\mid G \mid$ divides the order of S₈ = 4690. This is not true and so it follows that G is not simple. A simliar result holds for a group of order 225, because $\mid G \mid$ does not divide the order of S₉ and the index of a Syl₂₅ in G is 9.