We generalize a result of Moon on the fibering of certain 3-manifolds over
the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold.
Suppose that $G=\pi_1(M)$ contains a finitely generated group $U$ of infinite
index in $G$ which contains a non-trivial subnormal subgroup $N\neq \mathbb{Z}$
of $G$, and suppose that $N$ has a composition series of length $n$ in which at
least $n-1$ terms are finitely generated. Suppose that $N$ intersects
nontrivially the fundamental groups of the splitting tori given by the
Geometrization Theorem and that the intersections of $N$ with the fundamental
groups of the geometric pieces are non-trivial and not isomorphic to
$\mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $\mathbb{S}$
with fiber a compact surface $F$ such that $\pi_1(F)$ and $U$ are
commensurable.