Question

This is a question on group theory. We know that f:G->G such that f(h)=ghg^(-1) for some g in G, is an inner automorphism. Clearly, it is onto. Also, a subgroup maps to a subgroup. So, for every subgroup H, there is a preimage H'=g^(-1)Hg such that f[H']=H. And as f is one one onto, |H|=|H'|. So is every subgroup of order k of the form gHg^(-1) where |H|=k???