We have $\displaystyle |G| = 3^2 13^2 = 1521 $.

Since the order of G has 2 distinct prime factors, 3 and 13, Sylow Theorem I (henceforth to be referred to as Sylow I) tells us that G has at least 1 Sylow-3 subgroup and at least one 1 Sylow-13 subgroup. Let's find out how many of each.

We will denote the number of Sylow-3 subgroups and Sylow-13 subgroups by $\displaystyle n_3 \text{ and } n_{13} $, respectively.

Sylow III tells us that the number of subgroups of G of order $\displaystyle 3^2 $ is congruent to 1 modulo 3 and is a factor of $\displaystyle \frac{|G|}{3^2} = 169 $. Likewise, the number of subgroups of order $\displaystyle 13^2 $ is congruent to 1 modulo 13 and is a factor of $\displaystyle \frac{|G|}{13^2} = 9 $. Moreover, since a subgroup of a prime power order is a Sylow-p subgroup (and thus is isomorphic to every other Sylow-p subgroup), we have

$\displaystyle n_3 \equiv 1 \pmod3 \text{ and } n_3 | 169

\implies n_3 = 1, 13, \text{ or } 169 $

$\displaystyle n_{13} \equiv 1 \pmod3 \text{ and } n_{13} | 9

\implies n_{13} = 1 $

There are 13 isomorphism types of finite groups of order 1521. They are:

C1521

C169 : C9

C39 x C39

C507 x C3

C117 x C13

C3 x (C169 : C3)

C3 x ((C13 x C13) : C3)

C3 x ((C13 x C13) : C3)

C13 x (C13 : C9)

(C13 x C13) : C9

(C13 x C13) : C9

(C13 : C3) x (C13 : C3)

C39 x (C13 : C3)