Description of a class of finite groups

The Cayley classification problem, which is to give a complete classification of all groups whose orders are equal to a given natural number $n$, is solved in two ways. First, it is order fixing and studying non-Abelian groups proceeding from the size of the center or from a normality of a Sylow subgroup or other characteristics of the group.
The second direction is to consider the whole class of groups of order $n$ with a certain canonical decomposition of its order. For example, we know that if $n$ is a prime number, there exists a unique group of this order. A classical example of the description of groups of order $n=pq$, where $p$ and $q$ are different prime numbers, is implemented using Sylow theorems. The problem in the general case has no rational solutions; at present, in connection with this, it has undergone some changes. One of new formulations is as follows: to describe groups of order $ap$, where $a$ is a factor (in the general case, not prime) such that $(a,p)=1$.
The author describes a group of order with the condition of normality of its Sylow $p$-subgroup. Note that the order 23 is the first one that presents the full range of groups. In addition to a cyclic group, which exists for any order, this order is inherent to two Abelian noncyclic groups and two non-Abelian groups.