Group #21477531 — subgroups

Subgroups — C₃₁×#20772 #21477531

Order 7936.

1062 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 462 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1839	C₂	1	8	yes	maximal, complemented
3	C₃₁×#1728	C₂	1	6	yes	maximal
4	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
5	C₂²×C₄²×C₃₁	C₄	1	4	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
7	C₄³×C₃₁	C₄	1	4	yes	—
8	C₄³×C₃₁	C₂²	1	4	yes	—
9	C₃₁×#336	C₂²	1	24	yes	—
10	C₂³×C₄×C₃₁	C₂×C₄	1	6	yes	—
11	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
12	C₂×M₁₆×C₃₁	—	2	24	no	—
13	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
14	C₂×C₄²×C₃₁	—	2	12	no	—
15	C₂×C₄²×C₃₁	—	2	16	no	—
16	C₂×C₄²×C₃₁	C₂×C₄	1	12	yes	—
17	C₂×C₄²×C₃₁	C₂×C₄	1	6	yes	—
18	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
20	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	—	2	12	no	—
22	C₂²×C₄×C₃₁	—	2	24	no	—
23	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
24	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	#41	1	12	yes	—
26	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
27	M₁₆×C₃₁	—	4	16	no	—
28	C₂×C₈×C₃₁	—	2	24	no	—
29	C₄²×C₃₁	—	2	12	no	—
30	C₄²×C₃₁	—	2	24	no	—
31	C₄²×C₃₁	—	2	12	no	—
32	C₄²×C₃₁	M₁₆	1	16	yes	complemented
33	#20772	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₃₁	—	2	4	no	—
35	C₂³×C₃₁	#117	1	6	yes	—
36	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	—	2	48	no	—
38	C₂×C₄×C₃₁	—	2	12	no	—
39	C₂×C₄×C₃₁	—	2	12	no	—
40	C₂×C₄×C₃₁	#117	1	6	yes	—
41	C₂×C₄×C₃₁	#126	1	16	yes	—
42	C₂×C₄×C₃₁	C₂×M₁₆	1	24	yes	—
43	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
44	C₈×C₃₁	—	4	8	no	—
45	C₂×C₄³	C₂×C₃₁	1	1	yes	characteristic
46	#1839	C₂×C₃₁	1	8	yes	complemented
47	#1728	C₂×C₃₁	1	6	yes	—
48	C₂²×C₃₁	—	2	12	no	—
49	C₂²×C₃₁	#383	1	4	yes	—
50	C₂²×C₃₁	C₂×#117	1	3	yes	—
51	C₂²×C₃₁	C₂²×M₁₆	1	4	yes	—
52	C₄×C₃₁	—	2	24	no	—
53	C₄×C₃₁	#383	1	4	yes	—
54	C₄×C₃₁	C₂×#126	1	4	yes	—
55	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
56	C₂²×C₄²	C₄×C₃₁	1	4	yes	—
57	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
58	C₄³	C₄×C₃₁	1	4	yes	—
59	C₄³	C₂²×C₃₁	1	4	yes	—
60	#336	C₂²×C₃₁	1	24	yes	—
61	C₂×C₃₁	—	2	4	no	—
62	C₂×C₃₁	C₂×#383	1	1	yes	characteristic
63	C₂×C₃₁	#2920	1	6	yes	—
64	C₂³×C₄	C₂×C₄×C₃₁	1	6	yes	—
65	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
66	C₂×M₁₆	—	2	24	no	—
67	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
68	C₂×C₄²	—	2	12	no	—
69	C₂×C₄²	—	2	16	no	—
70	C₂×C₄²	C₂×C₄×C₃₁	1	6	yes	—
71	C₂×C₄²	C₂×C₄×C₃₁	1	12	yes	—
72	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
73	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
74	C₃₁	#20772	1	1	yes	characteristic, direct product factor, Sylow
75	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
79	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
80	C₂²×C₄	#41×C₃₁	1	12	yes	—
81	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
82	M₁₆	—	4	16	no	—
83	C₂×C₈	—	2	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	24	no	—
86	C₄²	—	2	12	no	—
87	C₄²	M₁₆×C₃₁	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	C₃₁×#117	1	6	yes	—
90	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	C₃₁×#117	1	6	yes	—
95	C₂×C₄	C₃₁×#126	1	16	yes	—
96	C₂×C₄	C₂×M₁₆×C₃₁	1	24	yes	—
97	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
98	C₈	—	4	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	C₃₁×#383	1	4	yes	—
101	C₂²	C₂×C₃₁×#117	1	3	yes	—
102	C₂²	C₂²×M₁₆×C₃₁	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	C₃₁×#383	1	4	yes	—
105	C₄	C₂×C₃₁×#126	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×C₃₁×#383	1	1	yes	characteristic
108	C₂	C₃₁×#2920	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄³×C₃₁

C₂

yes

maximal, characteristic

C₃₁×#1839

C₂

yes

maximal, complemented

C₃₁×#1728

C₂

yes

maximal

C₂²×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₄²×C₃₁

C₄

yes

—

C₂²×C₄²×C₃₁

C₂²

yes

—

C₄³×C₃₁

C₄

yes

—

C₄³×C₃₁

C₂²

yes

—

C₃₁×#336

C₂²

yes

—

C₂³×C₄×C₃₁

C₂×C₄

yes

—

C₂³×C₄×C₃₁

C₂³

yes

characteristic

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

C₂³

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

—

C₂⁴×C₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

characteristic

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

M₁₆

yes

complemented

#20772

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

#117

yes

—

C₂³×C₃₁

C₂³×C₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#117

yes

—

C₂×C₄×C₃₁

#126

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

—

C₂×C₄×C₃₁

C₂×#41

yes

—

C₈×C₃₁

—

C₂×C₄³

C₂×C₃₁

yes

characteristic

#1839

C₂×C₃₁

yes

complemented

#1728

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

#383

yes

—

C₂²×C₃₁

C₂×#117

yes

—

C₂²×C₃₁

C₂²×M₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

#383

yes

—

C₄×C₃₁

C₂×#126

yes

—

C₂²×M₁₆

C₂²×C₃₁

yes

—

C₂²×C₄²

C₄×C₃₁

yes

—

C₂²×C₄²

C₂²×C₃₁

yes

—

C₄³

C₄×C₃₁

yes

—

C₄³

C₂²×C₃₁

yes

—

#336

C₂²×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#383

yes

characteristic

C₂×C₃₁

#2920

yes

—

C₂³×C₄

C₂×C₄×C₃₁

yes

—

C₂³×C₄

C₂³×C₃₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂³×C₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

—

C₃₁

#20772

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂²×C₄×C₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

C₂²×C₄

#41×C₃₁

yes

—

C₂²×C₄

C₂⁴×C₃₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

M₁₆×C₃₁

yes

complemented

C₂³

—

C₂³

C₃₁×#117

yes

—

C₂³

C₂³×C₄×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃₁×#117

yes

—

C₂×C₄

C₃₁×#126

yes

—

C₂×C₄

C₂×M₁₆×C₃₁

yes

—

C₂×C₄

C₂×#41×C₃₁

yes

—

C₈

—

C₂²

—

100

C₂²

C₃₁×#383

yes

—

101

C₂²

C₂×C₃₁×#117

yes

—

102

C₂²

C₂²×M₁₆×C₃₁

yes

—

103

C₄

—

104

C₄

C₃₁×#383

yes

—

105

C₄

C₂×C₃₁×#126

yes

—

106

C₂

—

107

C₂

C₂×C₃₁×#383

yes

characteristic

108

C₂

C₃₁×#2920

yes

—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₃₁

C₂⁴

commutator

C₂³

C₂³×C₄×C₃₁

Frattini, cosocle

C₂²×C₄

C₂⁴×C₃₁

socle

C₂³×C₃₁

C₂³×C₄

prime radical

C₂⁴×C₃₁

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₃₁ ◁ G

N	H ≅ G/N	type
2. C₃₁×#1839	C₂	semidirect
32. C₄²×C₃₁	M₁₆	semidirect
33. #20772	C₃₁	direct (N × H)
46. #1839	C₂×C₃₁	semidirect
74. C₃₁	#20772	direct (N × H)
87. C₄²	M₁₆×C₃₁	semidirect

H ≅ G/N

type

2. C₃₁×#1839

C₂

semidirect

32. C₄²×C₃₁

M₁₆

semidirect

33. #20772

C₃₁

direct (N × H)

46. #1839

C₂×C₃₁

semidirect

74. C₃₁

#20772

direct (N × H)

87. C₄²

M₁₆×C₃₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄³×C₃₁	2
C₃₁×#1839	2
C₃₁×#1728	2
#20772	31

subgroup

index

C₂×C₄³×C₃₁

C₃₁×#1839

C₃₁×#1728

#20772

Subgroups — C₃₁×#20772 #21477531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#20772 #21477531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20772 #21477531