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Post Undeleted by David Jordan
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Maybe I'm going to make a dumb mistake here, since I'm writing this down rather quickly. Tell me what you think.

I think I claim this can happen all the time.

What about taking G an answer to be a finite simple group, letting C=Rep(G), and letting D=C^* Vyour question is given in http://arxiv.org/pdf/0810.0032, where C^*V means take theorem 1.1.

Subcategories of the dual category relative to V=category double D(G) are given by pairs of Vector spacesnormal subgroups K,N in G which centralize each other, regarded as a module category over C via together with the fiber functor. Now, D will be pointed, as it will be equal to representations data of the dual hopf algebra a bicharacter K\times N \to C[G], which is commutativeC^\times.A pointed fusion category is necessarily equal to Rep(H,w), where H is a finite abelian group

So in particular if G has no normal subgroups and w is a cocycle. HoweverH does, this is actually then you're going to be trivialfind that D(G) has no nontrivial subcategories, because you already realized D as Rep(C[G]^while D(H) . So that means D is just Rep[H] for H abelianwill (one can take, and K=the normal subgroup in particular not simple. By definition, these share an invertible bimodule categoryH, namely VecN={id}, so and the bicharacter K\to C^* to be trivial I think this answers your questionguess.

Post Deleted by David Jordan
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