Group #21463916 — subgroups

Subgroups — C₃₁×#7157 #21463916

Order 7936.

214 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₃₁×#1564	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₃₁	C₄	1	2	yes	—
4	C₃₁×#431	—	2	2	no	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#346	—	2	2	no	—
7	C₃₁×#335	—	2	2	no	—
8	C₃₁×#334	C₄	1	4	yes	—
9	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₃₁	—	2	2	no	—
13	C₂×C₁₆×C₃₁	—	2	2	no	—
14	C₂×C₁₆×C₃₁	C₈	1	4	yes	complemented
15	C₃₁×C₄⋊C₈	—	2	4	no	—
16	C₃₁×C₂²⋊C₈	—	4	2	no	—
17	C₄×C₈×C₃₁	—	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	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	4	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	SD₁₆	1	2	yes	—
32	C₁₆×C₃₁	—	2	4	no	—
33	#7157	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₃₁	#107	1	1	yes	characteristic
35	C₂×C₄×C₃₁	—	4	1	no	—
36	C₂×C₄×C₃₁	—	4	1	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	#106	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	#1564	C₂×C₃₁	1	2	yes	—
50	C₂²×C₃₁	—	2	1	no	—
51	C₂²×C₃₁	—	2	1	no	—
52	C₂²×C₃₁	#335	1	2	yes	—
53	C₂²×C₃₁	#340	1	1	yes	characteristic
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	#334	1	2	yes	—
57	C₄×C₃₁	#336	1	1	yes	characteristic
58	C₄×C₃₁	#341	1	1	yes	characteristic
59	C₂²×C₁₆	C₄×C₃₁	1	2	yes	—
60	#431	—	2	2	no	—
61	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
62	#346	—	2	2	no	—
63	#335	—	2	2	no	—
64	#334	C₄×C₃₁	1	4	yes	—
65	C₂×C₃₁	—	2	2	no	—
66	C₂×C₃₁	#1277	1	1	yes	characteristic
67	C₂×C₃₁	#1384	1	1	yes	characteristic
68	C₂×C₃₁	#1385	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₈×C₃₁	1	4	yes	complemented
75	C₄⋊C₈	—	2	4	no	—
76	C₂²⋊C₈	—	4	2	no	—
77	C₄×C₈	—	4	2	no	—
78	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
79	C₄×C₈	D₄×C₃₁	1	2	yes	—
80	C₃₁	#7157	1	1	yes	characteristic, direct product factor, Sylow
81	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
82	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
83	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₃₁	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
89	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
90	C₂×C₈	M₁₆×C₃₁	1	2	yes	—
91	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
92	C₄²	SD₁₆×C₃₁	1	2	yes	—
93	C₁₆	—	2	4	no	—
94	C₂³	C₃₁×#107	1	1	yes	characteristic
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
98	C₂×C₄	C₃₁×#102	1	2	yes	—
99	C₂×C₄	C₃₁×#103	1	2	yes	—
100	C₂×C₄	C₃₁×C₄⋊C₈	1	2	yes	—
101	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
102	C₂×C₄	C₃₁×#108	1	2	yes	—
103	C₈	—	2	2	no	—
104	C₈	—	8	4	no	—
105	C₈	C₄×C₈×C₃₁	1	1	yes	characteristic
106	C₈	C₃₁×#97	1	1	yes	characteristic
107	C₈	C₃₁×C₂²⋊C₈	1	2	yes	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	C₃₁×#335	1	2	yes	—
111	C₂²	C₃₁×#340	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	2	no	—
114	C₄	C₃₁×#334	1	2	yes	—
115	C₄	C₃₁×#336	1	1	yes	characteristic
116	C₄	C₃₁×#341	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	C₃₁×#1277	1	1	yes	characteristic
119	C₂	C₃₁×#1384	1	1	yes	characteristic
120	C₂	C₃₁×#1385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#346

C₂

yes

maximal, characteristic

C₃₁×#1564

C₂

yes

maximal

C₂²×C₁₆×C₃₁

C₄

yes

—

C₃₁×#431

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#346

—

C₃₁×#335

—

C₃₁×#334

C₄

yes

—

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×C₂²⋊C₈

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₃₁

D₈

yes

characteristic

C₂²×C₄×C₃₁

Q₁₆

yes

characteristic

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

SD₁₆

yes

—

C₁₆×C₃₁

—

#7157

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#107

yes

characteristic

C₂×C₄×C₃₁

—

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

#106

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

#1564

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#335

yes

—

C₂²×C₃₁

#340

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#334

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂²×C₁₆

C₄×C₃₁

yes

—

#431

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

—

C₄×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

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

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

—

C₃₁

#7157

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄⋊C₄×C₃₁

yes

characteristic

C₂²×C₄

D₈×C₃₁

yes

characteristic

C₂²×C₄

Q₁₆×C₃₁

yes

characteristic

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

SD₁₆×C₃₁

yes

—

C₁₆

—

C₂³

C₃₁×#107

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃₁×#95

yes

characteristic

C₂×C₄

C₃₁×#102

yes

—

C₂×C₄

C₃₁×#103

yes

—

100

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

101

C₂×C₄

C₃₁×#106

yes

characteristic

102

C₂×C₄

C₃₁×#108

yes

—

103

C₈

—

104

C₈

—

105

C₈

C₄×C₈×C₃₁

yes

characteristic

106

C₈

C₃₁×#97

yes

characteristic

107

C₈

C₃₁×C₂²⋊C₈

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₃₁×#335

yes

—

111

C₂²

C₃₁×#340

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

C₃₁×#334

yes

—

115

C₄

C₃₁×#336

yes

characteristic

116

C₄

C₃₁×#341

yes

characteristic

117

C₂

—

118

C₂

C₃₁×#1277

yes

characteristic

119

C₂

C₃₁×#1384

yes

characteristic

120

C₂

C₃₁×#1385

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#106
commutator	C₈	C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
socle	C₂²×C₃₁	#340
prime radical	C₂³×C₃₁	#107
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

#106

commutator

C₈

C₄×C₈×C₃₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃₁

C₂²×C₃₁

C₂³×C₃₁

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
14. C₂×C₁₆×C₃₁	C₈	semidirect
33. #7157	C₃₁	direct (N × H)
74. C₂×C₁₆	C₈×C₃₁	semidirect
80. C₃₁	#7157	direct (N × H)

H ≅ G/N

type

14. C₂×C₁₆×C₃₁

C₈

semidirect

33. #7157

C₃₁

direct (N × H)

74. C₂×C₁₆

C₈×C₃₁

semidirect

80. C₃₁

#7157

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
C₃₁×#1564	2
#7157	31

subgroup

index

C₂×C₃₁×#346

C₃₁×#1564

#7157

Subgroups — C₃₁×#7157 #21463916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#7157 #21463916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7157 #21463916