 Solution

Conjugacy classes in the symmetric group are specified by cycle type, equivalently a partition of $4$. If $m_j$ counts cycles of length $j$, the number of permutations of that cycle type is $4!/{\prod }_j(j^{m_j}{m}_j!)$: cyclic rotation of each cycle and interchange of equal-length cycles do not change the permutation. This givesThe sizes sum to $24$. Since the number of complex irreducible representations of a finite group equals its number of conjugacy classes, $S_4$ has $\boxed{5}$ inequivalent complex irreducible representations.