The least primary factor of the multiplicative group

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group Mₙ = ( Z/n Z)^×. We give an asymptotic formula, with order of magnitude x/(log x)1/2, for the counting function of those integers n for which S(n) ≠ 2. We also give an asymptotic formula, for any prime power q, for the counting function of those integers n for which $S(n) = q$. This group-theoretic problem can be reduced to problems of counting integers with restrictions on their prime factors, allowing it to be addressed by classical techniques of analytic number theory.