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What do we know about the soluble subgroups of finite classical simple groups (and maybe their automorphism groups)? For example, are their maximal soluble subgroups explicitly known?

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    There are nontrivial bounds for soluble subgroups of $GL(n,q)$. See for example A. Mann's paper "Soluble subgroups of symmetric and linear groups" and its references. In this paper, the author said a joint paper with Z. Arad and Y. Segev about the maximal order of soluble subgroups of simple groups is in preparation, but I don't know whether such a papaer is published.2012-11-20

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