Moderate# Subgroups of S_4 under Homomorphism

ABSALG-YJEJ4H

Let $S_4$ be the group of permutations of the set $T_4=\{1,2,3,4\}$.

Let $T_3$ be the set of partitions of $T_4$ of the form:

$$\{1,2\}\cup\{3,4\},\quad \{1,3\}\cup\{2,4\},\quad \{1,4\}\cup\{2,3\}$$

Let $S_3$ be the group of permutations of the $3$ elements of $T_3$.

Define the homomorphism $f:S_4\rightarrow S_3$ by letting $f(\sigma)$, $\sigma\in S_4$, be the action:

...on $T_3$.

Which of the following are true? Select **ALL** that apply.