What is the correct way to properly align the subscript with mathcal?

I am writing mathcal{H_S} in code, and this is what renders:

It seems to me that the spacing between the H and S is excessive, and indeed mathcal{H_{!S}} seems to produce a more natural spacing. I suspect this is not the correct way to do it though. What do you recommend?

Note, I have the following in my preamble.

usepackage{amsfonts}
usepackage{amsmath}
usepackage{amssymb}