A basis for the profinite topology

Question

Let $G$ be a group, and let $$C =\{H\leq G :\ [G:H]<\infty \}$$ $$B_R =\{Hx\leq G :\ H\in C ,x\in G\}$$ $$B_N =\{Hx\leq G :\ H\in C ,H\ is\ normal\ in\ G,x\in G\}$$ I want to show that $B_R$ and $B_N$ are bases of the same topology on $G$. I proved that both of them are bases for $G$ and one can easily check that $B_R$ contains $B_N$. Hence it suffices to show that $\forall H \in C \ \exists A\subseteq B_N $ s.t. $\cup A =H$ , but I have no idea more. Thanks in advance.