What should be the mental process for quickly evaluating representative Sylow subgroups of $A_5$?

Question

This Groupprops Wiki page has a nice chart classifying the subgroups of $A_5$ upto automorphism.

It shows the various representative subgroups. However, say if I were told to manually find the representative subgroups corresponding to 2-Sylow, 3-Sylow and 5-Sylow, how should I go about it?

Trying to list all possible subgroups of $A_5$ and then eliminating the unnecessary ones is a rather tedious process and more so in an examination hall. Memorizing all the representative subgroups also seems hard. Most textbooks I saw don't really explain any better method though. Could someone please clarify? Like, what would your mental process be like if you were told to write down the representative subgroups corresponding to 3-Sylow of $A_5$, i.e., $\{(), (1, 2, 3), (1, 3, 2)\}$?

Answer 1

Since the order of the group $A_5$ equals $60=2^2\cdot 3\cdot 5$, its Sylow subgroups should consist of $2^2$, $3$, and $5$ elements. There are unique up to an isomorphism groups of prime orders $3$ and $5$, which are cyclic. The respective Sylow subgroups of $A_5$ are generated by elements of order $3$ (for instance, $(123)$) and order $5$ (for instance, $(12345)$).

Let $G$ be a group of order $2^2$. If $G$ contains an element of order $4$ then $G$ is cyclic. In this case $G$ is not a subgroup of $A_5$ because elements of $S_5$ of order $4$ are cycles, which have length $4$ and so (as it is easy to check) are odd permutations.

If $G$ has no elements of order $4$ then each its non-identity element has order $2$ and this uniquely determines the multiplication table for $G$:

| e a b c
----------
e| e a b c
a| a e c b
b| b c e a
c| c b a e

Thus $G=V_4$ and can be realized in $A_5$, for instance, as $\{(), (12)(34), (13)(24), (14)(23)\}$.