 Solution

The Fundamental theorem of Galois theory gives an inclusion-reversing bijection for a finite Galois extension $L/K$ with group $G$:It satisfies $[L:L^H]=|H|$ and $[L^H:K]=[G:H]$. An intermediate extension $M/K$ is Galois precisely when its corresponding subgroup $H$ is normal, in which case restriction induces $\mathrm{Gal}(M/K)\cong G/H$.