Characteristic Subgroup Lattices and Hopf-Galois Structures

The Hopf-Galois structures on normal extensions $K\mathrm{/}k$ with $G=Gal(K\mathrm{/}k)$ are in one-to-one correspondence with the set of regular subgroups $N\leq B=Perm(G)$thatarenormalizedbytheleftregularrepresentation$\lambda(G)\leq B$\mathrm{.}Eachsuch$N$correspondstoaHopfalgebra$H_N=(K[N])^G$thatactson$K/k$\mathrm{.}Suchregularsubgroups$N$neednotbeisomorphicto$G$butmusthavethesameorder\mathrm{.}Onecansubdividethetotalityofallsuch$N$intocollections$R(G,[M])$whichisthesetofthoseregular$N$normalizedby$\lambda(G)$andisomorphictoagivenabstractgroup$M$where$|M|=|G|$. There arises an injective correspondence between the characteristic subgroups of a given $N$ an d the set of subgroups of $G$ stemming from the Galois correspondence between sub-Hopf algebras of $H_N$ and intermediate fields $k\subseteq F\subseteq K$. We utilize this correspondence to show that for certain pairings $(G,[M])$, the collection $R(G,[M])$ must be empty.