  In this paper we describe a portion of the subsemigroup lattice of the
\emph{full transformation semigroup} $\Omega^\Omega$, which consists of all
mappings on the countable infinite set $\Omega$. Gavrilov showed that there are
five maximal subsemigroups of $\Omega^\Omega$ containing the symmetric group
$\sym(\Omega)$. The portion of the subsemigroup lattice of $\Omega^\Omega$
which we describe is that between the intersection of these five maximal
subsemigroups and $\Omega^\Omega$. We prove that there are only 38
subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$
subsemigroups between $\sym(\Omega)$ and $\Omega^\Omega$.
