  Let $G$ be a finite group, $\pi$ be a set of primes, and define $H_{\pi}(G)$
to be the subgroup generated by all elements of $G$ which do not have prime
order for every prime in $\pi$. In this paper, we investigate some basic
properties of $H_{\pi}(G)$ and its relationship to the Hughes subgroup. We show
that for most groups, only one of three possibilities occur: $H_{\pi}(G) = 1$,
$H_{\pi}(G)=G$, or $H_{\pi}(G) = H_{p}(G)$ for some prime $p \in \pi$. There is
only one other possibility: $G$ is a Frobenius group whose Frobenius complement
has prime order $p$, and whose Frobenius kernel, $F$, is a nonabelian $q$-group
such that $H_{\pi}(G)$ arises as a proper and nontrivial Hughes subgroup of
$F$. We investigate a few restrictions on the possible choices of the primes
$p$ and $q$.
