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

Numbers up till 100

$n$	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. We omit the prime numbers since there is only one group of each such order.[SHOW MORE]

$n$	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

Small powers of small primes

(For general formulas, see the next section).

Powers of 2:[SHOW MORE]

Exponent $n$	Value $2^{n}$	Number of groups of order $2^{n}$	Reason/Explanation/List
1	2	1	Only cyclic group:Z2; see [[equivalence of definitions of group of prime order
2	4	2	Only cyclic group:Z4 and Klein four-group; see also groups of order 4 and classification of groups of prime-square order
3	8	5	See groups of order 8; see classification of groups of prime-cube order
4	16	14	See groups of order 16
5	32	51	See groups of order 32
6	64	267	See groups of order 64
7	128	2328	See groups of order 128
8	256	56092	See groups of order 256
9	512	10494213	See groups of order 512
10	1024	49487365422	See groups of order 1024

Powers of 3: [SHOW MORE]

Exponent $n$	Value $3^{n}$	Number of groups of order $3^{n}$	Reason/Explanation/List
1	3	1	only cyclic group:Z3; see equivalence of definitions of group of prime order
2	9	2	cyclic group:Z9 and elementary abelian group:E9; see groups of order 9; see also classification of groups of prime-square order
3	27	5	see groups of order 27; see also classification of groups of prime-cube order
4	81	15	see groups of order 81; see also classification of groups of prime-fourth order for odd prime
5	243	67	see groups of order 243
6	729	504	see groups of order 729
7	2187	9310	see groups of order 2187

Powers of 5: [SHOW MORE]

Exponent $n$	Value $5^{n}$	Number of groups of order $5^{n}$	Reason/Explanation/List
1	5	1	only cyclic group:Z5; see equivalence of definitions of group of prime order
2	25	2	cyclic group:Z25 and elementary abelian group:E25; see also groups of order 25 and classification of groups of prime-square order
3	125	5	see groups of order 125 and classification of groups of prime-cube order
4	625	15	see groups of order 625 and classification of groups of prime-fourth order for odd prime
5	3125	77	see groups of order 3125, see also the PORC formula for order $p^{5}$ in the table in the next section.
6	15625	684	see groups of order 15625, see also the PORC formula for order $p^{6}$ in the table in the next section.

Powers of 7: <toggledisplay>

Exponent $n$	Value $5^{n}$	Number of groups of order $5^{n}$	Reason/Explanation/List
1	7	1	only cyclic group:Z7; see equivalence of definitions of group of prime order
2	49	2	cyclic group:Z49 and elementary abelian group:E49; see also groups of order 49 and classification of groups of prime-square order
3	343	5	see groups of order 343 and classification of groups of prime-cube order
4	2401	15	see groups of order 2401 and classification of groups of prime-fourth order for odd prime
5	16807	83	see groups of order 16807 and also the PORC formula in the table in the next section.
6	117649	860	see groups of order 117649 and also the PORC formula in the table in the next section.

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 $p\geq 5$	$2p+61+2\operatorname {gcd} (p-1,3)+\operatorname {gcd} (p-1,4)$
$p^{6}$, prime $p\geq 5$	$3p^{2}+39p+344+24\operatorname {gcd} (p-1,3)+11\operatorname {gcd} (p-1,4)+2\operatorname {gcd} (p-1,5)$
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, $p>3$, $p\equiv -1{\pmod {4}}$	4
product $4p$, $p$ prime, $p\equiv 1{\pmod {4}}$	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 $p^{(2n^{3}/27)+O(n^{8/3}}$.
• Conjecture that most finite groups are nilpotent
• 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 $b$.