I think the centralizer in $\mathbb{C}S_{N}$ is isomorphic to $M_{(N-1)!}(\mathbb{C}H)$, that is the full matrix ring of size $(N-1)! \times (N-1)!$, where the individual matrix entries are elements of the group algebra $\mathbb{C}H$.

An alternative description in terms of matrix algebras: The centralizer of $H$ iin $\mathbb{C}S_{N}$ is a direct sum of $N$ (mutually annihilating) copies of $M_{(N-1)!}(\mathbb{C})$. Both results follow from consideration of the regular module for $\mathbb{C}S_{N}$ as a module for $\mathbb{C}H$ (visibly, the direct sum of $(N-1)!$ copies of the regular module for $\mathbb{C}H$).