  In this paper we classify the maximal subsemigroups of the \emph{full
transformation semigroup} $\Omega^\Omega$, which consists of all mappings on
the infinite set $\Omega$, containing certain subgroups of the symmetric group
$\sym(\Omega)$ on $\Omega$. In 1965 Gavrilov showed that there are five maximal
subsemigroups of $\Omega^\Omega$ containing $\sym(\Omega)$ when $\Omega$ is
countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary
cardinality.
  We classify the maximal subsemigroups of $\Omega^\Omega$ on a set $\Omega$ of
arbitrary infinite cardinality containing one of the following subgroups of
$\sym(\Omega)$: the pointwise stabiliser of a non-empty finite subset of
$\Omega$, the stabiliser of an ultrafilter on $\Omega$, or the stabiliser of a
partition of $\Omega$ into finitely many subsets of equal cardinality. If $G$
is any of these subgroups, then we deduce a characterisation of the mappings
$f,g\in \Omega^\Omega$ such that the semigroup generated by $G\cup \{f,g\}$
equals $\Omega^\Omega$.
