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

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

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