Fix a positive integer $n=p_1^{a_1}\cdots p_k^{a_k}$. Consider the cyclotomic field $K=\Bbb Q(\zeta_n)$.
I know that only $p_1,\ldots,p_k$ are ramified in $K/\Bbb Q$, and that the ramification index of $p_i$ is $\varphi(p_i^{a_i})$.
How do the primes $p_1,\ldots,p_k$ ramify in the subfield lattice of $K/\Bbb Q$?
For example, if $n=20$ then the ramification is \begin{matrix} &&\Bbb Q(\zeta_{20})\\ &\llap{2}\huge\diagup&\llap{\varnothing}\huge|&\rlap{5}\huge\diagdown\\ \Bbb Q(\zeta_5)&&\Bbb Q(\zeta_{20}+\zeta_{20}^{-1})&&\Bbb Q(i,\sqrt5)\\ &\rlap{5}\huge\diagdown&\llap{2,5}\huge|&\llap{2}\huge\diagup&\llap{\varnothing}\huge|&\rlap{5}\huge\diagdown\\ &&\Bbb Q(\sqrt5)&&\Bbb Q(\sqrt{-5})&&\Bbb Q(i)\\ &&&\rlap{5}\huge\diagdown&\llap{2,5}\huge|&\llap{2}\huge\diagup\\ &&&&\Bbb Q \end{matrix} I suspect that there might be a way to read this ramification data off from the corresponding subgroup lattice \begin{matrix} &&\{1\}\\ &\huge\diagup&\huge|&\huge\diagdown\\ \{1,11\}&&\{1,19\}&&\{1,9\}\\ &\huge\diagdown & \huge| & \huge\diagup & \huge| & \huge\diagdown\\ &&\{1,9,11,19\}&&\{1,3,7,9\}&&\{1,9,13,17\}\\ &&&\huge\diagdown&\huge|&\huge\diagup\\ &&&&\{1,3,7,9,11,13,17,19\} \end{matrix} In this case, the inertia subgroups are $I_2=\{1,11\}$ and $I_5=\{1,9,13,17\}$.
In general the inertia subgroups are $(\Bbb Z/p_i^{a_i}\Bbb Z)^\times\leq(\Bbb Z/n\Bbb Z)^\times$. However, I don't think that knowing the inertia subgroups is enough to know the entire ramification picture.