Is there a number $n$, such that there are $22$ groups of order $n$?

Question

Denot : $N(n)$ : the number of groupfs of order $n$ ?

Is there a number $n$ with $N(n)=22$ ?

Checking the first about $2000$ numbers, I noticed that there is no $n\in [1,2000]$ with $N(n)=22$. Does such an $n$ exist ? And if no, why ?

Generalization : Given a number $k$, can we determine whether there is a $n$ with $N(n)=k$ in a reasonable time ?

Answer 1

According to Maple, there are a number of values of $n < 50000$ for which $N(n) = 22$:

> with( GroupTheory ):
> select( n -> NumGroups( n ) = 22, [seq]( 1 .. 50000 ) );

        [6321, 9075, 9765, 18135, 18669, 19215, 27075, 31017, 31605, 35685, 40053, 45045, 46431, 47565, 49539]

There is a conjecture that $N$ is surjective, but to my knowledge, there is little progress on this problem.