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In this paper we describe a portion of the subsemigroup lattice of the\emph{full transformation semigroup} $\Omega^\Omega$, which consists of allmappings on the countable infinite set $\Omega$. Gavrilov showed that there arefive 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 maximalsubsemigroups and $\Omega^\Omega$. We prove that there are only 38subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$subsemigroups between $\sym(\Omega)$ and $\Omega^\Omega$.

A finite interval in the subsemigroup lattice of the full transformation monoid