I am working through a problem where I have shown that a group homomorphism $\phi :G \longrightarrow H$ is uniquely determined by the image of its generators, and also by $\text{ker}\;\phi$ and the induced homomorphism $\bar\phi :G/\text{ker}\phi \longrightarrow H$.
I now need to list the normal subgroups of $S_4$ (without proof) and find the number of surjective homomorphisms from $S_4$ to $C_2$, $S_3$, and $A_4$. [Given the hint that $S_3$ has $6$ isomorphisms]
I know this could done by analysing the possibilities of the kernel as a normal subgroup using the first isomorphic theorem, but is it possible to prove the results without the theorem, and instead with the previously shown results? Would this method be a roundabout way of proving the first isomorphism theorem?