Let $K\subset L$ be a Galois extension of fields with Galois group $G$. Then the intermediate fields of $K\subset L$ correspond bijectively to the closed subgroups of $G$.
More precisely, the maps
defined by
and
are bijective and inverse to each other. This correspondence reverses the inclusion relation: $K$ corresponds to $G$ and $L$ to $\{{\mathrm{id}}_{L}\}$.
If $E$ corresponds to $H$, then we have
$K\subset E$ is finite precisely if $H$ is open (in the profinite topology on $G$)
$[E:K]\simeq \mathrm{index}[G:K]$ if $H$ is open;
$E\subset L$ is Galois with $\mathrm{Gal}(L/E)\simeq H$ (as topological groups);
for every $\sigma \in G$ we have that $\sigma [E]$ corresponds to $\sigma H{\sigma}^{-1}$;
$L\subset E$ is Galois precisely if $H$ is a normal subgroup of $G$;
$\mathrm{Gal}(E/K)\simeq G/H$ (as topological groups) if $K\subset E$ is Galois.
This appears for instance as Lenstra, theorem 2.3.
This suggests that more fundamental than the subgroups of a Galois group $G$ are its quotients by subgroups, which can be identified with transitive $G$-sets. This naturally raises the question of what corresponds to non-transitive $G$-sets.