I am stuck with the following problem. I am sure it cannot be that hard since it is intuitively true, but I can't find a way to prove it.
Let $A,B,C \leq G$ where $G$ is a finite group. Suppose moreover that $B \leq A$. Then $\vert A:B \vert \geq \vert C \cap A : C \cap B \vert$.