  We study the structure of nilpotent subsemigroups in the semigroup
$M(n,\mathbb{F})$ of all $n\times n$ matrices over a field, $\mathbb{F}$, with
respect to the operation of the usual matrix multiplication. We describe the
maximal subsemigroups among the nilpotent subsemigroups of a fixed nilpotency
degree and classify them up to isomorphism. We also describe isolated and
completely isolated subsemigroups and conjugated elements in $M(n,\mathbb{F})$.
