Let L/K be a finite Galois extension of fields with group Γ. Associated to each Hopf-Galois structure on L/K is a group G of the same order as the Galois group Γ. The type of the Hopf-Galois structure is by definition the isomorphism type of G. We investigate the extent to which general properties of either of the groups Γ and G constrain those of the other. Specifically, we show that if G is nilpotent then Γ is soluble, and that if Γ is abelian then G is soluble. In contrast to these results, we give some examples where the groups Γ and G have different composition factors. In particular, we show that a soluble extension may admit a Hopf-Galois structure of insoluble type.