Hopf-Galois structures on non-normal extensions of degree related to Sophie Germain primes

We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions L / K of squarefree degree n . If E is the normal closure then G = Gal ( ) can be viewed as a permutation group show that has derived length at most 4, but many groups and 2 cannot occur. investigate in detail case where p q ≥ 3 + 1 are both prime. (Thus Sophie Germain prime safeprime). list which arise, we enumerate for each There six such corresponding admit possible types.