Let $S$ be the ring $\mathbb{C}[X_0,...,X_n]$. Let $X$ be a smooth projective scheme of the form $\mathrm{Proj}(S/I_X)$ for some ideal $I_X$. Let $C$ be a scheme associated to a Cartier divisor on $X$. Suppose $C$ is of the form $\mathrm{Proj}(S/I_C)$ for some ideal $I_C$. Is there any relation between the Castelnuovo-Mumford regularity of the coherent sheaf $\mathcal{O}_X(-C)$ and that of the module $I_C/I_X$?

EDIT Assume $X$ is irreducible but $C$ is not. $C$ might not even be reduced.