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The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under semidirect products?

This arose out of a conversation I was having with a friend a while back. At the time, I thought I saw a simple argument that this class of groups was in fact the class of all finite groups, but that argument turned out to be gibberish. (EDIT: As Arturo points out below, this class cannot possibly consist of all finite groups.)

Thanks in advance for the help!

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    @User1729: The Sylow p-subgroups of symmetric groups are in the collection, and every group is a subgroup of a symmetric group. The class is closed under direct products and then semi-direct products, so it is closed under wreath products, and the Sylow p-subgroup of a symmetric group is a direct product of iterated wreath products of cyclic groups of order p.2012-04-03

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