Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition series for $M$. Clearly ${\cal C}(M)$ includes all the primes associated to $M$. But it can contain other primes as well. For example, let $R$ be the local ring of a singular point on an irreducible curve and let $M$ be the maximal ideal of $R$. Then $R/M$ appears as a quotient in every composition series for $M$, but $M$ (as an ideal) is not associated to $M$ (as a module).

So: Is there a nice characterization of the set ${\cal C}(M)$?