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

Hopf-Galois Structures on Non-Normal Extensions of Degree Related to Sophie Germain Primes