This question is partly inspired by the recent question on <a href="https://mathoverflow.net/questions/211507/measurability-and-axiom-of-choice">measurability and the axiom of choice</a>.

Suppose I come up with a way to define a subgroup of a group, in a way that involves "no arbitrary choices" in some sense.  Then chances are, the subgroup will be normal.  For example, the commutator subgroup of a group is normal, as is the commutator subgroup of the commutator subgroup.

I seem to recall that there are general theorems that say something like, any subgroup that is first-order definable without parameters is normal.  Is this true?  If so, are there even stronger theorems of this type?

The motivation is that such a meta-theorem could allow me, when I'm introducing a new kind of subgroup, to sidestep an explicit verification that the subgroup is normal.