Groups of order 10

There are, up to isomorphism, two groups of order 10, indicated in the table below:

That these are the only two possibilities can be shown via the classification of groups of order a product of two distinct primes. Since $$10 = 5 \cdot 2$$ and $$2 \mid (5 - 1)$$, the number $$10$$ falls in the two isomorphism classes case in that classification.