For a group $G$ and a field $K$ let $S(G,K)$ be the sum of the dimensions of the irreducible K representations of $G$. Note that $S(G,\mathbb{C})< |G|$. It's not difficult to prove that if $n \ge 6$ then $S(S_n,\mathbb{C}) < (n-2)!(n-2)-n$. I'm interested in "good" bounds (not necessarily the best but at least significantly better than $|G|$).

I want to know if there is a relation between $S(G,\mathbb{Q})$ and $S(G,\mathbb{C})$. I need this result for the subgroups of $S_n$ isomorphic to $S_j \times S_{n-j}$. Specifically I need a bound for $S(S_k \times S_{n-k}, \mathbb{Q})$ the best as possible.