Group #21472706 — subgroups

Subgroups — C₃₁×#15947 #21472706

Order 7936.

6146 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 6146 conjugacy classes of subgroups — too many to draw. Only its 494 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	C₂²×C₃₁×#120	C₂	1	6	yes	maximal, complemented
2	C₃₁×#1438	C₂	1	9	yes	maximal, complemented
3	C₂⁶×C₃₁	C₂²	1	2	yes	complemented
4	C₂³×D₄×C₃₁	C₂²	1	9	yes	—
5	C₂×C₃₁×#120	—	2	72	no	—
6	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	18	yes	complemented
7	C₂×C₃₁×#95	C₂²	1	6	yes	complemented
8	C₂⁵×C₃₁	—	2	36	no	—
9	C₂⁵×C₃₁	—	4	12	no	—
10	C₂⁵×C₃₁	C₂³	1	6	yes	—
11	C₂²×D₄×C₃₁	—	2	72	no	—
12	C₂²×D₄×C₃₁	—	4	18	no	—
13	C₂²×D₄×C₃₁	D₄	1	36	yes	complemented
14	C₂³×C₄×C₃₁	C₂³	1	9	yes	—
15	C₃₁×#120	—	2	192	no	—
16	C₂×C₂²⋊C₄×C₃₁	—	2	36	no	—
17	C₂×C₂²⋊C₄×C₃₁	—	2	72	no	—
18	C₃₁×#95	—	4	6	no	—
19	C₂⁴×C₃₁	—	2	36	no	—
20	C₂⁴×C₃₁	—	2	36	no	—
21	C₂⁴×C₃₁	—	2	48	no	—
22	C₂⁴×C₃₁	—	4	144	no	—
23	C₂⁴×C₃₁	—	4	72	no	—
24	C₂⁴×C₃₁	—	4	16	no	—
25	C₂⁴×C₃₁	—	4	24	no	—
26	C₂⁴×C₃₁	C₂×D₄	1	36	yes	—
27	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
28	C₂×D₄×C₃₁	—	2	144	no	—
29	C₂×D₄×C₃₁	—	4	36	no	—
30	C₂×D₄×C₃₁	—	4	144	no	—
31	C₂²×C₄×C₃₁	—	2	36	no	—
32	C₂²×C₄×C₃₁	—	2	18	no	—
33	C₂²×C₄×C₃₁	C₂×D₄	1	18	yes	—
34	C₂²⋊C₄×C₃₁	—	2	144	no	—
35	#15947	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂³×C₃₁	—	2	36	no	—
37	C₂³×C₃₁	—	2	36	no	—
38	C₂³×C₃₁	—	2	144	no	—
39	C₂³×C₃₁	—	4	288	no	—
40	C₂³×C₃₁	—	4	192	no	—
41	C₂³×C₃₁	—	4	12	no	—
42	C₂³×C₃₁	—	4	36	no	—
43	C₂³×C₃₁	—	4	36	no	—
44	C₂³×C₃₁	—	4	12	no	—
45	C₂³×C₃₁	#120	1	24	yes	complemented
46	C₂³×C₃₁	C₂²×D₄	1	9	yes	—
47	C₂³×C₃₁	H(5,𝔽₂)	1	6	yes	—
48	D₄×C₃₁	—	4	144	no	—
49	C₂×C₄×C₃₁	—	2	36	no	—
50	C₂×C₄×C₃₁	—	2	18	no	—
51	C₂×C₄×C₃₁	—	4	36	no	—
52	C₂²×#120	C₂×C₃₁	1	6	yes	complemented
53	#1438	C₂×C₃₁	1	9	yes	complemented
54	C₂²×C₃₁	—	2	72	no	—
55	C₂²×C₃₁	—	4	128	no	—
56	C₂²×C₃₁	—	4	72	no	—
57	C₂²×C₃₁	—	4	72	no	—
58	C₂²×C₃₁	C₂×#120	1	6	yes	—
59	C₂²×C₃₁	#534	1	9	yes	—
60	C₂²×C₃₁	D₄²	1	18	yes	—
61	C₂²×C₃₁	H(3,𝔽₄)	1	2	yes	—
62	C₄×C₃₁	—	4	18	no	—
63	C₂⁶	C₂²×C₃₁	1	2	yes	complemented
64	C₂³×D₄	C₂²×C₃₁	1	9	yes	—
65	C₂×#120	—	2	72	no	—
66	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	18	yes	complemented
67	C₂×#95	C₂²×C₃₁	1	6	yes	complemented
68	C₂×C₃₁	—	4	24	no	—
69	C₂×C₃₁	#2403	1	9	yes	—
70	C₂×C₃₁	#2679	1	6	yes	—
71	C₂⁵	—	2	36	no	—
72	C₂⁵	—	4	12	no	—
73	C₂⁵	C₂³×C₃₁	1	6	yes	—
74	C₂²×D₄	—	2	72	no	—
75	C₂²×D₄	—	4	18	no	—
76	C₂²×D₄	D₄×C₃₁	1	36	yes	complemented
77	C₂³×C₄	C₂³×C₃₁	1	9	yes	—
78	#120	—	2	192	no	—
79	C₂×C₂²⋊C₄	—	2	36	no	—
80	C₂×C₂²⋊C₄	—	2	72	no	—
81	#95	—	4	6	no	—
82	C₃₁	#15947	1	1	yes	characteristic, direct product factor, Sylow
83	C₂⁴	—	2	36	no	—
84	C₂⁴	—	2	36	no	—
85	C₂⁴	—	2	48	no	—
86	C₂⁴	—	4	144	no	—
87	C₂⁴	—	4	72	no	—
88	C₂⁴	—	4	16	no	—
89	C₂⁴	—	4	24	no	—
90	C₂⁴	C₂×D₄×C₃₁	1	36	yes	—
91	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
92	C₂×D₄	—	2	144	no	—
93	C₂×D₄	—	4	36	no	—
94	C₂×D₄	—	4	144	no	—
95	C₂²×C₄	—	2	36	no	—
96	C₂²×C₄	—	2	18	no	—
97	C₂²×C₄	C₂×D₄×C₃₁	1	18	yes	—
98	C₂²⋊C₄	—	2	144	no	—
99	C₂³	—	2	36	no	—
100	C₂³	—	2	36	no	—
101	C₂³	—	2	144	no	—
102	C₂³	—	4	288	no	—
103	C₂³	—	4	192	no	—
104	C₂³	—	4	12	no	—
105	C₂³	—	4	36	no	—
106	C₂³	—	4	36	no	—
107	C₂³	—	4	12	no	—
108	C₂³	C₃₁×#120	1	24	yes	complemented
109	C₂³	C₂²×D₄×C₃₁	1	9	yes	—
110	C₂³	C₃₁×H(5,𝔽₂)	1	6	yes	—
111	D₄	—	4	144	no	—
112	C₂×C₄	—	2	36	no	—
113	C₂×C₄	—	2	18	no	—
114	C₂×C₄	—	4	36	no	—
115	C₂²	—	2	72	no	—
116	C₂²	—	4	128	no	—
117	C₂²	—	4	72	no	—
118	C₂²	—	4	72	no	—
119	C₂²	C₂×C₃₁×#120	1	6	yes	—
120	C₂²	C₃₁×#534	1	9	yes	—
121	C₂²	D₄²×C₃₁	1	18	yes	—
122	C₂²	C₃₁×H(3,𝔽₄)	1	2	yes	—
123	C₄	—	4	18	no	—
124	C₂	—	4	24	no	—
125	C₂	C₃₁×#2403	1	9	yes	—
126	C₂	C₃₁×#2679	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₃₁×#120

C₂

yes

maximal, complemented

C₃₁×#1438

C₂

yes

maximal, complemented

C₂⁶×C₃₁

C₂²

yes

complemented

C₂³×D₄×C₃₁

C₂²

yes

—

C₂×C₃₁×#120

—

C₂²×C₂²⋊C₄×C₃₁

C₂²

yes

complemented

C₂×C₃₁×#95

C₂²

yes

complemented

C₂⁵×C₃₁

—

C₂⁵×C₃₁

—

C₂⁵×C₃₁

C₂³

yes

—

C₂²×D₄×C₃₁

—

C₂²×D₄×C₃₁

—

C₂²×D₄×C₃₁

D₄

yes

complemented

C₂³×C₄×C₃₁

C₂³

yes

—

C₃₁×#120

—

192

—

C₂×C₂²⋊C₄×C₃₁

—

C₂×C₂²⋊C₄×C₃₁

—

C₃₁×#95

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

144

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

—

C₂⁴×C₃₁

C₂×D₄

yes

—

C₂⁴×C₃₁

C₂⁴

yes

characteristic

C₂×D₄×C₃₁

—

144

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

144

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂²⋊C₄×C₃₁

—

144

—

#15947

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

144

—

C₂³×C₃₁

—

288

—

C₂³×C₃₁

—

192

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

#120

yes

complemented

C₂³×C₃₁

C₂²×D₄

yes

—

C₂³×C₃₁

H(5,𝔽₂)

yes

—

D₄×C₃₁

—

144

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂²×#120

C₂×C₃₁

yes

complemented

#1438

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

128

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×#120

yes

—

C₂²×C₃₁

#534

yes

—

C₂²×C₃₁

D₄²

yes

—

C₂²×C₃₁

H(3,𝔽₄)

yes

—

C₄×C₃₁

—

C₂⁶

C₂²×C₃₁

yes

complemented

C₂³×D₄

C₂²×C₃₁

yes

—

C₂×#120

—

C₂²×C₂²⋊C₄

C₂²×C₃₁

yes

complemented

C₂×#95

C₂²×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

#2403

yes

—

C₂×C₃₁

#2679

yes

—

C₂⁵

—

C₂⁵

—

C₂⁵

C₂³×C₃₁

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

D₄×C₃₁

yes

complemented

C₂³×C₄

C₂³×C₃₁

yes

—

#120

—

192

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₃₁

#15947

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

144

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×D₄×C₃₁

yes

—

C₂⁴

C₂⁴×C₃₁

yes

characteristic

C₂×D₄

—

144

—

C₂×D₄

—

C₂×D₄

—

144

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂²⋊C₄

—

144

—

C₂³

—

100

C₂³

—

101

C₂³

—

144

—

102

C₂³

—

288

—

103

C₂³

—

192

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₃₁×#120

yes

complemented

109

C₂³

C₂²×D₄×C₃₁

yes

—

110

C₂³

C₃₁×H(5,𝔽₂)

yes

—

111

D₄

—

144

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂²

—

116

C₂²

—

128

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×C₃₁×#120

yes

—

120

C₂²

C₃₁×#534

yes

—

121

C₂²

D₄²×C₃₁

yes

—

122

C₂²

C₃₁×H(3,𝔽₄)

yes

—

123

C₄

—

124

C₂

—

125

C₂

C₃₁×#2403

yes

—

126

C₂

C₃₁×#2679

yes

—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₃₁

C₂⁴

commutator, Frattini, cosocle

C₂⁴

C₂⁴×C₃₁

LCS

G ▷ C₂⁴ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₂²×C₃₁×#120	C₂	semidirect
2. C₃₁×#1438	C₂	semidirect
3. C₂⁶×C₃₁	C₂²	semidirect
6. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
7. C₂×C₃₁×#95	C₂²	semidirect
13. C₂²×D₄×C₃₁	D₄	semidirect
35. #15947	C₃₁	direct (N × H)
45. C₂³×C₃₁	#120	semidirect
52. C₂²×#120	C₂×C₃₁	semidirect
53. #1438	C₂×C₃₁	semidirect
63. C₂⁶	C₂²×C₃₁	semidirect
66. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
67. C₂×#95	C₂²×C₃₁	semidirect
76. C₂²×D₄	D₄×C₃₁	semidirect
82. C₃₁	#15947	direct (N × H)
108. C₂³	C₃₁×#120	semidirect

H ≅ G/N

type

1. C₂²×C₃₁×#120

C₂

semidirect

2. C₃₁×#1438

C₂

semidirect

3. C₂⁶×C₃₁

C₂²

semidirect

6. C₂²×C₂²⋊C₄×C₃₁

C₂²

semidirect

7. C₂×C₃₁×#95

C₂²

semidirect

13. C₂²×D₄×C₃₁

D₄

semidirect

35. #15947

C₃₁

direct (N × H)

45. C₂³×C₃₁

#120

semidirect

52. C₂²×#120

C₂×C₃₁

semidirect

53. #1438

C₂×C₃₁

semidirect

63. C₂⁶

C₂²×C₃₁

semidirect

66. C₂²×C₂²⋊C₄

C₂²×C₃₁

semidirect

67. C₂×#95

C₂²×C₃₁

semidirect

76. C₂²×D₄

D₄×C₃₁

semidirect

82. C₃₁

#15947

direct (N × H)

108. C₂³

C₃₁×#120

semidirect

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

subgroup	index
C₂²×C₃₁×#120	2
C₃₁×#1438	2
#15947	31

subgroup

index

C₂²×C₃₁×#120

C₃₁×#1438

#15947

Subgroups — C₃₁×#15947 #21472706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#15947 #21472706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#15947 #21472706