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Is there a nice way to classify/decompose finite groups? The classification of finite simple groups lets us determine all possible composition factors in the composition series but there are nonisomorphic groups with the same composition factors in their respective composition series e.g. $ C_{6} $ and $ D_6 $. If not, is there an analogous theory for a group to be 'completely reducible'?

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There has been a lot of research on this topic! I would recommend you look at the article

A millennium project: constructing small groups, Hans Ulrich Besche, Bettina Eick, E.A. O'Brien, Internat. J. Algebra Comput. 12 (5), 623-644, 2002.

You can download it from O'Brien's webpage:

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That pdf looks like a ransom note. – anon Oct 24 '11 at 9:00
@anon, what do you mean? – lhf Oct 24 '11 at 10:58

There is another way to approach this: in 1940 Philip Hall introduced isoclinism, an equivalence relation on groups coarser than isomorphism (being isomorphic implies being isoclinic, but not vice versa). The concept of isoclinism was introduced to classify p-groups, although the concept is applicable to all groups. Isoclinism can be extended to isologism, which is similar to isoclinism, but then w.r.t. a variety of groups. There exists a vast literature on isoclinism and isologism.

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