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As is well-known, the compact groups Spin(7) and SO(7) both have a singleconjugacy class of compact subgroups of exceptional type G_2. We first showthat if H is a subgroup of Spin(7), and if each element of H is conjugate tosome element of G_2, then H itself is conjugate to a subgroup of G_2. Theanalogous statement for SO(7) turns out be false, and our main result is aclassification of all the exceptions. They are the following groups, embeddedin each case in SO(7) in a very specific way: GL_2(Z/3Z), SL_2(Z/3Z), Z/4Z xZ/2Z, as well as the nonabelian subgroups of GO_2(C) with compact closure,similitude factors group {-1,1}, and which are not isomorphic to the dihedralgroup of order 8. More generally, we consider the analogous problems in whichthe Euclidean space is replaced by a quadratic space of dimension 7 over anarbitrary field.  This type of questions naturally arises in some formulation of a conversestatement of Langlands' global functoriality conjecture, to which the resultsabove have thus some applications. Moreover, we give necessary and sufficientlocal conditions on a cuspidal algebraic regular automorphic representation ofGL_7 over a totally real number field so that its associated \ell-adic Galoisrepresentations can be conjugate into G_2(\bar{Q_\ell}).

Subgroups of Spin(7) or SO(7) with each element conjugate to some element of G_2, and applications to automorphic forms