  The Hopf-Galois structures on normal extensions $K/k$ with $G=Gal(K/k)$ are
in one-to-one correspondence with the set of regular subgroups $N\leq
B=Perm(G)$ that are normalized by the left regular representation
$\lambda(G)\leq B$. Each such $N$ corresponds to a Hopf algebra $H_N=(K[N])^G$
that acts on $K/k$. Such regular subgroups $N$ need not be isomorphic to $G$
but must have the same order. One can subdivide the totality of all such $N$
into collections $R(G,[M])$ which is the set of those regular $N$ normalized by
$\lambda(G)$ and isomorphic to a given abstract group $M$ 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.
