Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order $pq$. I think here are two non isomorphic of order $122$.
Please check my answer, it is right or not.
Best Answer
The general classifiaction of groups of order $pq$ ($p < q$ prime) is the following:
There is always the cyclic group $\mathbb Z/pq\mathbb Z$.
For $q\not\equiv 1$ mod $p$, this is the only isomorphism type of groups of order $pq$.
For $q \equiv 1$ mod $p$, there is a single additional isomorphism type, which has the form $\mathbb Z/q\mathbb Z\rtimes \mathbb Z/p\mathbb Z$. If $p = 2$ (and $q$ is odd), a suitable representative of this second isomorphism type is the Dihedral group $D_q$.
So in your case of order $122$, there is $$\mathbb Z/122\mathbb Z \qquad \text{and} \qquad D_{61}.$$