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The only examples I have seen have had an alternating group in the composition series.

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    The smallest if the group $L_3(2) \cong L_2(7)$ of order 168. It is isomorphic to the multiplicative group of $3 \times 3$ non-singular matrices with entries in the finite field of order 2.2012-04-10

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Any nonabelian finite simple group that is not isomorphic to an alternating group.

For example, any of the simple sporadic groups such as the Mathieu groups, the Monster, of the Baby Monster. Or almost all finite simple groups of Lie type (see Wikipedia), as there are only a couple of exceptions where they correspond to alternating groups (just like $A_3$ is alternating and simple, but not unsolvable).