HMMT 2017 A5

Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $a b c=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute

$$\sum_{\substack{a b c=2310 \\ a, b, c \in \mathbb{N}}}(a+b+c)$$

where $\mathbb{N}$ denotes the positive integers.