Constructing Hopf-Galois structures and skew bracoids of small degree

Using the fact that Hopf-Galois structures on separable extensions and skew bracoids are both intrinsically connected to transitive subgroups of the holomorph of a finite group, we present algorithms to classify and enumerate these objects for small degree, and apply them to obtain significant extensions to existing results. We also explore the classifications of these structures of degree $2pq$, where $p$ and $q$ are distinct odd primes. We conclude with some enumeration-inspired observations and a conjecture.