Let $K / F$ be a finite Galois extension. There is a bijection $$ \begin{aligned} \{ \text{sub-extensions } F \subseteq E \subseteq K \} &\longleftrightarrow \{ \text{subgroups } H \leq G \} \\ E &\longmapsto \mathrm{Gal}(K / E) \\ K^{H} &\longleftarrow\!\shortmid H \end{aligned} $$ with the two maps inverse to each other.