Introduction This paper is a contribution to the study of the subgroup structure of simple algebraic groups of exceptional type. The maximal closed connected subgroups of these groups were determined in [Se2], subject to some mild restrictions on the characteristic p of the underlying field. Here we take the study further, and investigate arbitrary closed connected reductive subgroups X of an exceptional algebraic group G, again with mild characteristic restrictions (in particular, p 0 or p 7 covers all the restrictions). We obtain results which determine the embeddings of arbitrary closed connected semisimple subgroups in G. We show that if X is such a subgroup, then X is em- bedded in an explicit way in a "subsystem subgroup" of G - that is, a semisimple subgroup which is normalized by a maximal torus of G. Subsystem subgroups are constructed naturally from subsystems of the root system of G this therefore de- termines the embedding of X in G. As a consequence, when p 0 there are only finitely many conjugacy classes of such subgroups X, whereas there are infinitely many when p 0. The connection with subsystem subgroups is useful in various ways. For example, it is very helpful in finding centralizers of subgroups and in restricting representations. We present tables which give all the conjugacy classes of simple subgroups X of G of rank at least 2, their connected centralizers, and their actions on £(G), the Lie algebra of G. For subgroups X of type Ai, we associate with each such subgroup a labelled Dynkin diagram, and prove that the conjugacy class of X is determined by its labelled diagram. Our proofs are based on Theorem 1, which states that if the reductive subgroup X lies in a parabolic subgroup P = QL of G, with unipotent radical Q and Levi subgroup i , then some conjugate of X lies in L. We also use this result to prove that CG(X) is always reductive. For simple subgroups X we establish that with essentially one exception, X is determined up to (Aut G)-conjugacy by its composition factors on L(G) and that if X is of rank at least 2, and p is a good prime for G, then CUQ\{X) L(CG(X)). Some of our proofs require detailed information concerning the restrictions of certain G-modules to various subgroups of G, such as maximal rank subgroups. Many results of this type can be found in Section 2. We now state our results in detail. Throughout, let G be a simple algebraic group of exceptional type over an algebraically closed field K of characteristic p. In order to specify our assumptions on p, we define, for certain simple subgroups X of G, an integer N(X, G), as given in the following table. Received by the editors April 13, 1994 Second author supported by an NSF grant and an SERC Visiting Fellowship
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