In the following, let $G=\operatorname{GL}\_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}\_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more generally ask the question for a complex, (reductive, affine,) algebraic group - but since this generalization is not of importance to me, I leave it as a bonus objective.

**Question:** I am asking whether there is a scheme that parametrizes the closed, algebraic subgroups of $G$ in the same way that the Hilbert scheme classifies all closed subvarieties.

Now, I am quite certain that $G$ has a Hilbert scheme $\operatorname{\mathfrak Hilb}\_G$. Then, I might refine the question by asking whether the closed subgroups of $G$ form a (possibly closed) subscheme of $\operatorname{\mathfrak Hilb}\_G$. 

Finally, of course I would like to know as much as possible about how that scheme, if it exists, looks like (as in, geometric properties of any kind are welcome).

**My thoughts:** Denoting by $\mu:G\times G\to G$ the group operation and by $\iota:G\to G$ inversion, it seems plausible to me that the $H\in\operatorname{\mathfrak Hilb}\_G$ which satisfy $\mu(H\times H)=H$ and $\iota(H)=H$ should form a closed subscheme, which would then be exactly what I am looking for. However, I also feel that I might only have a naive and vague idea of what is going on.