Eccoci.

Define $G(N) = \sum_S \operatorname{lcm}(S)$ where $S$ ranges through all subsets of $\{1, \dots, N\}$ and $\operatorname{lcm}$ denotes the lowest common multiple. Note that the $\operatorname{lcm}$ of the empty set is $1$.

You are given $G(5) = 528$ and $G(20) = 8463108648960$.

Find $G(800)$. Give your answer modulo $10^9 + 7$.