Number of Distinct Unordered Factorization

Let $C(n)$ denote the number of factorizations of $n$ into distinct factors greater than 1.

For example, $C(24)=5$ since $24=2\cdot 12=2\cdot 3\cdot 4=3\cdot 8=4\cdot 6$.

Let $S(n):=\sum _{i=2}^{n}C(n)$. You are given $S(10)=12$ and $S({10}^4)=99733$.

Find $S({10}^{12})$.