Why does the definition of a fixed field specify a subgroup of the Galois group, not just a subset?

Why does the definition for the fixed field of H specify that H must be a subgroup of the Galois group of the field extension? As far as I can see, if H is simply a subset, the ‘fixed field’ of H is still a field, as we have closure under all 4 operations.

So, am I missing something, or is the requirement for H to be a subgroup not necessary for the fixed field of H to actually be a field?