Group #21463915 — subgroups

Subgroups — C₃₁×#7156 #21463915

Order 7936.

234 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 142 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×C₃₁×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#334	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₃₁	C₄	1	2	yes	—
4	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#346	—	2	2	no	—
7	C₃₁×#334	—	2	2	no	—
8	C₃₁×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₃₁	—	4	2	no	—
10	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₃₁	—	2	2	no	—
14	C₂×C₁₆×C₃₁	—	2	2	no	—
15	C₂×C₁₆×C₃₁	C₈	1	4	yes	complemented
16	C₃₁×C₄⋊C₈	—	2	4	no	—
17	C₃₁×C₄⋊C₈	—	4	2	no	—
18	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₃₁	D₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
22	C₂×C₈×C₃₁	—	2	2	no	—
23	C₂×C₈×C₃₁	—	4	4	no	—
24	C₂×C₈×C₃₁	—	8	4	no	—
25	C₂×C₈×C₃₁	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₃₁	M₁₆	1	2	yes	—
30	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
31	C₄²×C₃₁	D₈	1	1	yes	characteristic
32	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
33	C₁₆×C₃₁	—	2	4	no	—
34	#7156	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₃₁	#106	1	1	yes	characteristic
36	C₂×C₄×C₃₁	—	2	4	no	—
37	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
38	C₂×C₄×C₃₁	#102	1	2	yes	—
39	C₂×C₄×C₃₁	#103	1	2	yes	—
40	C₂×C₄×C₃₁	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
42	C₂×C₄×C₃₁	#108	1	2	yes	—
43	C₈×C₃₁	—	2	2	no	—
44	C₈×C₃₁	—	8	4	no	—
45	C₈×C₃₁	C₄×C₈	1	1	yes	characteristic
46	C₈×C₃₁	#97	1	1	yes	characteristic
47	C₈×C₃₁	C₂²⋊C₈	1	2	yes	—
48	C₂×#346	C₂×C₃₁	1	1	yes	characteristic
49	C₂×#334	C₂×C₃₁	1	2	yes	—
50	C₂²×C₃₁	#334	1	2	yes	—
51	C₂²×C₃₁	#340	1	1	yes	characteristic
52	C₂²×C₃₁	#365	1	4	yes	—
53	C₄×C₃₁	—	2	2	no	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	#335	1	2	yes	—
56	C₄×C₃₁	#336	1	1	yes	characteristic
57	C₄×C₃₁	#341	1	1	yes	characteristic
58	C₂²×C₁₆	C₄×C₃₁	1	2	yes	—
59	C₂×C₄⋊C₈	—	2	2	no	—
60	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
61	#346	—	2	2	no	—
62	#334	—	2	2	no	—
63	#334	C₄×C₃₁	1	4	yes	—
64	C₂×C₃₁	#1277	1	1	yes	characteristic
65	C₂×C₃₁	#1368	1	4	yes	—
66	C₂×C₃₁	#1383	1	2	yes	—
67	C₂²×C₈	—	4	2	no	—
68	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
69	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
70	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₈×C₃₁	1	4	yes	complemented
74	C₄⋊C₈	—	2	4	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
77	C₄×C₈	D₄×C₃₁	1	2	yes	—
78	C₃₁	#7156	1	1	yes	characteristic, direct product factor, Sylow
79	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	—	8	4	no	—
84	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
86	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
87	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
88	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
90	C₄²	D₈×C₃₁	1	1	yes	characteristic
91	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
92	C₁₆	—	2	4	no	—
93	C₂³	C₃₁×#106	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
96	C₂×C₄	C₃₁×#102	1	2	yes	—
97	C₂×C₄	C₃₁×#103	1	2	yes	—
98	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
99	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
100	C₂×C₄	C₃₁×#108	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	8	4	no	—
103	C₈	C₄×C₈×C₃₁	1	1	yes	characteristic
104	C₈	C₃₁×#97	1	1	yes	characteristic
105	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
106	C₂²	C₃₁×#334	1	2	yes	—
107	C₂²	C₃₁×#340	1	1	yes	characteristic
108	C₂²	C₃₁×#365	1	4	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	C₃₁×#335	1	2	yes	—
112	C₄	C₃₁×#336	1	1	yes	characteristic
113	C₄	C₃₁×#341	1	1	yes	characteristic
114	C₂	C₃₁×#1277	1	1	yes	characteristic
115	C₂	C₃₁×#1368	1	4	yes	—
116	C₂	C₃₁×#1383	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#346

C₂

yes

maximal, characteristic

C₂×C₃₁×#334

C₂

yes

maximal

C₂²×C₁₆×C₃₁

C₄

yes

—

C₂×C₃₁×C₄⋊C₈

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#346

—

C₃₁×#334

—

C₃₁×#334

C₄

yes

—

C₂²×C₈×C₃₁

—

C₂²×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₃₁

Q₈

yes

characteristic

C₂×C₄²×C₃₁

D₄

yes

characteristic

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

C₈

yes

complemented

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄²

yes

characteristic

C₂×C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

M₁₆

yes

—

C₄²×C₃₁

C₂²⋊C₄

yes

—

C₄²×C₃₁

D₈

yes

characteristic

C₄²×C₃₁

Q₁₆

yes

characteristic

C₁₆×C₃₁

—

#7156

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#106

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#95

yes

characteristic

C₂×C₄×C₃₁

#102

yes

—

C₂×C₄×C₃₁

#103

yes

—

C₂×C₄×C₃₁

C₄⋊C₈

yes

—

C₂×C₄×C₃₁

#107

yes

characteristic

C₂×C₄×C₃₁

#108

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄×C₈

yes

characteristic

C₈×C₃₁

#97

yes

characteristic

C₈×C₃₁

C₂²⋊C₈

yes

—

C₂×#346

C₂×C₃₁

yes

characteristic

C₂×#334

C₂×C₃₁

yes

—

C₂²×C₃₁

#334

yes

—

C₂²×C₃₁

#340

yes

characteristic

C₂²×C₃₁

#365

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#335

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂²×C₁₆

C₄×C₃₁

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

—

C₄×C₃₁

yes

—

C₂×C₃₁

#1277

yes

characteristic

C₂×C₃₁

#1368

yes

—

C₂×C₃₁

#1383

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₃₁

yes

characteristic

C₂²×C₈

Q₈×C₃₁

yes

characteristic

C₂×C₄²

D₄×C₃₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₃₁

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

—

C₃₁

#7156

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₃₁

yes

characteristic

C₂²×C₄

SD₁₆×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₃₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₃₁

yes

characteristic

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

C₂×C₈×C₃₁

yes

—

C₂×C₈

M₁₆×C₃₁

yes

—

C₄²

C₂²⋊C₄×C₃₁

yes

—

C₄²

D₈×C₃₁

yes

characteristic

C₄²

Q₁₆×C₃₁

yes

characteristic

C₁₆

—

C₂³

C₃₁×#106

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃₁×#95

yes

characteristic

C₂×C₄

C₃₁×#102

yes

—

C₂×C₄

C₃₁×#103

yes

—

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

C₂×C₄

C₃₁×#107

yes

characteristic

100

C₂×C₄

C₃₁×#108

yes

—

101

C₈

—

102

C₈

—

103

C₈

C₄×C₈×C₃₁

yes

characteristic

104

C₈

C₃₁×#97

yes

characteristic

105

C₈

C₃₁×C₂²⋊C₈

yes

—

106

C₂²

C₃₁×#334

yes

—

107

C₂²

C₃₁×#340

yes

characteristic

108

C₂²

C₃₁×#365

yes

—

109

C₄

—

110

C₄

—

111

C₄

C₃₁×#335

yes

—

112

C₄

C₃₁×#336

yes

characteristic

113

C₄

C₃₁×#341

yes

characteristic

114

C₂

C₃₁×#1277

yes

characteristic

115

C₂

C₃₁×#1368

yes

—

116

C₂

C₃₁×#1383

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#106
commutator	C₈	C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₃₁

#106

commutator

C₈

C₄×C₈×C₃₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃₁

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
15. C₂×C₁₆×C₃₁	C₈	semidirect
34. #7156	C₃₁	direct (N × H)
73. C₂×C₁₆	C₈×C₃₁	semidirect
78. C₃₁	#7156	direct (N × H)

H ≅ G/N

type

15. C₂×C₁₆×C₃₁

C₈

semidirect

34. #7156

C₃₁

direct (N × H)

73. C₂×C₁₆

C₈×C₃₁

semidirect

78. C₃₁

#7156

direct (N × H)

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

subgroup	index
C₂×C₃₁×#346	2
C₂×C₃₁×#334	2
#7156	31

subgroup

index

C₂×C₃₁×#346

C₂×C₃₁×#334

#7156

Subgroups — C₃₁×#7156 #21463915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#7156 #21463915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7156 #21463915