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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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This is, I think, too general a question. For example, any subgroup of the simple group $\,A_5\,$ is soluble...! Can you focus a little more on some cases? –  DonAntonio Nov 19 '12 at 16:00
    
Yes, there are two many soluble subgroups of a finite simple group, so I only ask for some informations rather than a entire list. However, I still expect the determination of maximal soluble subgroups of finite classical simple groups. –  Binzhou Xia Nov 19 '12 at 16:32
    
That depends on what you mean by "maximal soluble subgroup". There are two possible meanings! The maximal subgroups of the finite classical groups that are soluble are essentially known, although I do not believe that anyone has made a list! Are you familiar with the book by Kleidman and Liebeck, "The subgroup structure of the finite classical groups"? All the information need is in there! –  Derek Holt Nov 19 '12 at 17:54
    
By "maximal soluble subgroup" I mean the maximal one among the soluble subgroups. If consider the maximal subgroups which are soluble (I prefer to call them "soluble maximal subgroups"), then things are clearer since maximal subgroups in the almost simple class $\mathcal{S}$ are not soluble. –  Binzhou Xia Nov 20 '12 at 2:31
    
In that case I think the question is far too difficult, and nothing much is known in general, so you would have to provide more information to be able to make progress! –  Derek Holt Nov 20 '12 at 9:27
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