In this paper we are concerned with the following question: for a semigroup $S$, what is the largest size of a subsemigroup $T \leq S$ where $T$ has a given property? The semigroups $S$ that we consider are the full transformation semigroups; all mappings from a finite set to itself under composition of mappings. The subsemigroups $T$ that we consider are of one of the following types: left zero, right zero, completely simple, or inverse. Furthermore, we find the largest size of such subsemigroups $U$ where the least rank of an element in $U$ is specified. Numerous examples are given.