I am desperately trying to use the Lattice Isomorphism Theorem to find all subgroups $\left\{H | \:SL_n\left(F_7\right)\le H\le GL_n\left(F_7\right)\right\}$, (where $F_7$ is the prime field of 7 elements). But I just don't understand how.
I noticed $SL_n\left(F_7\right)$ is normal in $GL_n\left(F_7\right)$, and so, exists an isomorphism $\phi(H)=H/SL_n(F_7)$... But I have no idea what to do with it.
I would appreciate your help in solving it(spent lots of time on it...) thanks