This is what I've done so far. For the first part of the problem, we decompose 1350. into a product of primes as $\displaystyle 3^3.2.5^2$. Then we have

$\displaystyle \mathbb{Z}_{3^3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5^2}$

$\displaystyle \mathbb{Z}_{3^2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5^2}$

$\displaystyle \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5^2}$

$\displaystyle \mathbb{Z}_{3^3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5} \oplus \mathbb{Z}_{5}$

$\displaystyle \mathbb{Z}_{3^2} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5} \oplus \mathbb{Z}_{5}$

$\displaystyle \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{3} \oplus \mathbb{Z}_{2} \oplus \mathbb{Z}_{5} \oplus \mathbb{Z}_{5}$

Is this right?

Yes it is
And what do I need to do for the second part? How do I identify the ones which are isomorphic to $\displaystyle \mathbb{Z}_{6} \oplus \mathbb{Z}_{45} \oplus \mathbb{Z}_{5}$?

In the last direct product in my list, gcd(3,3,3,2,5,5)=1 i.e. they are relatively prime. Does this mean it is isomorphic to the given group?