$A = \langle (1,2,3),(1,2)\rangle$
$B = \langle (1,2,4),(1,2)\rangle$
$C = \langle (1,3,4),(1,3)\rangle$
$D = \langle (2,3,4),(2,3)\rangle$
I want to proof that these subgroups of $S_4$ ( which are $\cong S_3$)are conjugate to one another but the computations are tedious! :'( These subgroups aren't Sylow 2-supgroups so I can't say they are conjugate to one another....however they're 2-subgroups. Is there some theorem/lemma that says p-subgroups are conjugate? Or is there a smarter way to do this? or am I compelled to use brute force.
Thanks :D
UPDATE:
I computed $A$= $\{(1),(123),(132),(13),(12),(23)\}$
and $B$=$\{(1),(142),(124),(14),(12),(24)\}$
Following @Alex suggestion, $(14)A(14)=B$ Sample computation with $\sigma =(14) $ $ (14)(123)(14) = (\sigma(1),\sigma(2),\sigma(3))=(423)=(234)$ $(14)A(14)= \{(1),(234).........\}$ I can see already that this isn't equal to $B$.
So I tried $(14)B(14)$ so
$(14)B(14)=\{(1),(124).....\}$ which again is not $A$
What am I doing wrong?