Let $f$ be an irreducible polynomial of degree $n$ over $\mathbb{Q}$ and let $F$ be its splitting field. Then we can normally extend $\mathbb{Q}$ to $F$ and $\operatorname{Gal}(F/\mathbb{Q}) = G$ will corrspond to a subgroup of $S_n$, while $\operatorname{Gal}(F/F) = \{e\}$.
Now if $H$ is any non-trivial normal subgroup of $G$ in $S_n$, what will be an intermediate normal extension $E$ of $\mathbb{Q}$ corresponding to $H$?
I see that there could be several normal extensions $E$ of $\mathbb{Q}$ corresponding to $H$ but none of them will have any root of $f$. Is there any relationship between $f$ and the polynomials that will have their splitting field as $E$ since $f$ is certainly not one of them?