Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.)
In this answer, it is shown that the index is infinite. It also follows from the conjunction of this and this answer. But which cardinal is it? I'm almost certain that it's $\mathfrak c,$ but I have no idea how to prove it.
More generally, which cardinals are the indexes $[\mathrm{GL}_n(\Bbb C):G],$ where $G$ runs over the set of subgroups of $\mathrm{GL}_n(\Bbb C)?$ According to the latter two linked answers, there shouldn't be any finite cardinals there. I suspect all of these cardinals are actually $\mathfrak c$, but I wouldn't know how to prove it either (and this suspicion is much weaker than the previous one).
The group $GL_n(\mathbb{C})$ is divisible, because every invertible matrix $A$ can be written $A = \exp(B) = (\exp(\tfrac{1}{n}B))^n$. Hence every quotient of $GL_n(\mathbb{C})$ is divisible. In particular every finite quotient is trivial.
There does exist a countable quotient of $GL_n(\mathbb{C})$, because $\mathbb{Q}$ is a quotient of $\mathbb{R}$ (use axiom of choice to take $\mathbb{Q}$-basis of $\mathbb{R}$), and $\mathbb{R} \simeq \mathbb{R}_{>0}$ is a quotient of $\mathbb{C}^\times$, and the latter is a quotient of $GL_n(\mathbb{C})$.
EDIT : take $\Gamma$ a $\mathbb{Q}$-subvector space of $\mathbb{R}$ of codimension $1$ (hence $\mathbb{R}/\Gamma \simeq \mathbb{Q}$). Let $H$ be the subgroup $GL_n(\mathbb{C})$ consisting of matrices $A$ such that $\log(|\det A|) \in \Gamma$. Then the map $$GL_n(\mathbb{C})/H \longrightarrow \mathbb{R}/\Gamma, \quad (A \mod H) \mapsto (\log(|\det A|) \mod \Gamma)$$ is an isomorphism.