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How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by Liebeck and Kleidman but I think this mostly deals with the maximal subgroups (please correct me if I'm wrong) and I'd like to know about the other subgroups (if possible).

Thank you.

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Did you see this: ? – Mark Sapir Jul 12 '13 at 0:06
up vote 2 down vote accepted

Despite the the fact that it deals only with maximal subgroups, I am afraid that I am unable to restrain myself from mentioning (i.e. advertising) the forthcoming book

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups. John N. Bray, Derek F. Holt,Colva M. Roney-Dougal, London Mathematical Society Lecture Note Series, CUP,

which is due to be published in September of this year. It includes tables of all of the maximal subgroups of all finite classical groups of dimensions up to 12, which of course includes the orthogonal groups. In fact the book by Keidman and Liebeck only deals with the subgroups of geopmetric type, and the maximality proofs are only valid in dimensions greater than 12.

The maximal subgroups are typically either parabolic, or they are direct or wreath products, or they are themselves close to being nonabelian simple. So many of their subgroups could be described recursively.

Since you cannot reasonably hope for a complete description of all subgroups, I think you need to refine your question, and ask what kind of specific information you want to know about the subgroups. (For example, you might want to know which subgroups have some specific group as a composition factor.)

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Thanks for the reply. – MRD1729 Jul 13 '13 at 15:26

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