Number of groups of given order

Definition

Let $n$ be a natural number. The number of groups of order $n$ is defined as the number of isomorphism classes of groups whose order is $n$.

This is a finite number and is bounded by $n^{n^2}$ for obvious reasons. The function is not strictly increasing in $n$ and depends heavily on the nature of the prime factorization of $n$.

Initial values

The ID of the sequence of these numbers in the Online Encyclopedia of Integer Sequences is A000001

$n$	Number of groups of order $n$	Reason/explanation
1	1
2	1	prime number
3	1	prime number
4	2	square of a prime; see classification of groups of prime-square order
5	1	prime number
6	2	form $pq$ where $p,q$ primes, $q\mid p-1$
7	1	prime number
8	5	prime cube: classification of groups of prime-cube order, also see groups of order 8
9	2	prime square; see classification of groups of prime-square order

Orders 10 to 36. [SHOW MORE]

We omit the prime numbers since there is only one group of each such order.

$n$	Number of groups of order $n$	Reason/explanation
10	2	form $pq$ where $p,q$ primes, $q\mid p-1$
12	5	see groups of order 12
14	2	form $pq$ where $p,q$ primes, $q\mid p-1$
15	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q-1$, $q$ doesn't divide $p-1$
16	14	see groups of order 16
18	5	see groups of order 18
20	5	see groups of order 20
21	2	form $pq$ where $p,q$ primes, $q\mid p-1$
22	2	form $pq$ where $p,q$ primes, $q\mid p-1$
24	15	see groups of order 24
25	2	prime square; see classification of groups of prime-square order
26	2	form $pq$ where $p,q$ primes, $q\mid p-1$
27	5	see classification of groups of prime-cube order
28	4	see groups of order 28
30	4	see groups of order 30
32	51	see groups of order 32
33	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q-1$, $q$ doesn't divide $p-1$
34	2	form $pq$ where $p,q$ primes, $q\mid p-1$
35	1	form $pq$ ($p,q$ primes) where $p$ doesn't divide $q-1$, $q$ doesn't divide $p-1$
36	14	see groups of order 36

Orders greater than 36. We omit prime numbers, squares of primes, and numbers of the form

$pq$

where

$p,q$

both primes, since these are covered by standard cases.[SHOW MORE]

$n$	Number of groups of order $n$	Reason/explanation
40	14	see groups of order 40
42	6	see groups of order 42
44	4	see groups of order 44
45	2	both Sylow subgroups are normal, so it is a direct product.
48	52	see groups of order 48
50	5	see groups of order 50
52	5	see groups of order 52

Facts

Basic facts

Value of $n$	What we can say about the number of groups of order $n$	Explanation
1	1	only the trivial group
$p$ a prime number	1	only the group of prime order. See equivalence of definitions of group of prime order
$p^2$, $p$ prime	2	only the cyclic group of prime-square order and the elementary abelian group of prime-square order
$p^3$, $p$ prime	5	see classification of groups of prime-cube order
$2^4=16$	14	see classification of groups of order 16, also groups of order 16 for summary information.
$p^4$, $p$ odd prime	15	see classification of groups of prime-fourth order for odd prime
$2^5=32$	51
$3^5=243$	67
$p^5$, prime $p\ge 5$	$2p+61+2\mathrm{g}\mathrm{c}\mathrm{d}(p-1,3)+\mathrm{g}\mathrm{c}\mathrm{d}(p-1,4)$
$p^6$, prime $p\ge 5$	$3p^2+39p+344+24\mathrm{g}\mathrm{c}\mathrm{d}(p-1,3)+11\mathrm{g}\mathrm{c}\mathrm{d}(p-1,4)+2\mathrm{g}\mathrm{c}\mathrm{d}(p-1,5)$
product $p_1{p}_2\dots p_n$, $p_i$ distinct primes with no $p_i$ dividing $p_j-1$	1	the cyclic group of that order. See classification of cyclicity-forcing numbers
product $pq$, $p,q$ primes with $p$ dividing $q-1$	2
product $4p$, $p$ prime, $p>3$, $p\equiv -1(\mathrm{mod}4)$	4
product $4p$, $p$ prime, $p\equiv 1(\mathrm{mod}4)$	5

Asymptotic facts and conjectures

• Higman-Sims asymptotic formula on number of groups of prime power order: This states that the number of groups of order $p^n$ is about $p^{(2n^3/27)+O(n^{8/3}}$.
• Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity
• Higman's PORC conjecture states that the number of groups of order $p^n$ is a PORC function in $p$ for fixed $n$.

Properties

Supermultiplicativity

If $n=ab$ with $a$ and $b$ relatively prime, the number of groups of order $n$ is bounded from below by the product of the number of groups of orders $a$ and $b$ respectively. This is because we can take direct products for every pair of a group of order $a$ and a group of order $b$.