Group #21470279 — subgroups

Subgroups — C₃₁×#13520 #21470279

Order 7936.

238 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₃₁×#2161	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2151	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1953	C₂	1	2	yes	maximal
4	C₃₁×#1385	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1381	C₂	1	2	yes	maximal
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#430	—	2	2	no	—
9	C₂×C₃₁×#108	C₂²	1	2	yes	—
10	C₂×C₃₁×#108	C₂²	1	2	yes	—
11	C₃₁×#405	C₂²	1	1	yes	characteristic
12	C₃₁×#368	—	2	2	no	—
13	C₃₁×C₄⋊C₁₆	—	2	2	no	—
14	C₃₁×#344	—	2	2	no	—
15	C₂×M₁₆×C₃₁	—	2	2	no	—
16	C₂×M₁₆×C₃₁	—	2	2	no	—
17	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×#117	D₄	1	1	yes	characteristic
20	C₃₁×M₃₂	—	2	2	no	—
21	C₂×C₁₆×C₃₁	—	2	1	no	—
22	C₂×C₁₆×C₃₁	—	2	2	no	—
23	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
24	C₃₁×#108	—	2	2	no	—
25	C₃₁×#108	—	2	2	no	—
26	C₃₁×#108	—	2	2	no	—
27	C₃₁×#108	—	4	4	no	—
28	C₃₁×#108	Q₈	1	4	yes	—
29	C₃₁×#97	—	2	1	no	—
30	C₄×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₃₁	—	2	4	no	—
33	M₁₆×C₃₁	—	2	2	no	—
34	M₁₆×C₃₁	—	4	2	no	—
35	M₁₆×C₃₁	—	4	2	no	—
36	M₁₆×C₃₁	—	4	2	no	—
37	C₂×C₈×C₃₁	—	2	1	no	—
38	C₂×C₈×C₃₁	—	4	2	no	—
39	C₂×C₈×C₃₁	—	4	2	no	—
40	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₃₁	#41	1	2	yes	—
43	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂²⋊C₄×C₃₁	Q₁₆	1	2	yes	—
46	C₄²×C₃₁	—	2	1	no	—
47	C₁₆×C₃₁	—	2	2	no	—
48	C₁₆×C₃₁	—	4	1	no	—
49	#13520	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
53	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
54	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
55	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
56	C₈×C₃₁	—	4	1	no	—
57	C₈×C₃₁	—	4	4	no	—
58	C₈×C₃₁	—	8	2	no	—
59	C₈×C₃₁	#123	1	1	yes	characteristic
60	C₈×C₃₁	#125	1	2	yes	—
61	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
62	#2161	C₂×C₃₁	1	1	yes	characteristic
63	#2151	C₂×C₃₁	1	1	yes	characteristic
64	#1953	C₂×C₃₁	1	2	yes	—
65	#1385	C₂×C₃₁	1	1	yes	characteristic
66	#1381	C₂×C₃₁	1	2	yes	—
67	C₂²×C₃₁	—	4	1	no	—
68	C₂²×C₃₁	#467	1	1	yes	characteristic
69	C₂²×C₃₁	#477	1	2	yes	—
70	C₄×C₃₁	—	4	2	no	—
71	C₄×C₃₁	#398	1	1	yes	characteristic
72	C₄×C₃₁	#469	1	1	yes	characteristic
73	C₄×C₃₁	#478	1	2	yes	—
74	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
75	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
76	#430	—	2	2	no	—
77	C₂×#108	C₂²×C₃₁	1	2	yes	—
78	C₂×#108	C₂²×C₃₁	1	2	yes	—
79	#405	C₂²×C₃₁	1	1	yes	characteristic
80	#368	—	2	2	no	—
81	C₄⋊C₁₆	—	2	2	no	—
82	#344	—	2	2	no	—
83	C₂×C₃₁	—	2	1	no	—
84	C₂×C₃₁	—	2	2	no	—
85	C₂×C₃₁	#2087	1	1	yes	characteristic
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
89	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
90	#117	D₄×C₃₁	1	1	yes	characteristic
91	M₃₂	—	2	2	no	—
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
95	#108	—	2	2	no	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	4	4	no	—
99	#108	Q₈×C₃₁	1	4	yes	—
100	#97	—	2	1	no	—
101	C₄×C₈	—	2	1	no	—
102	C₃₁	#13520	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
104	M₁₆	—	2	4	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
113	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
114	C₂×C₈	#41×C₃₁	1	2	yes	—
115	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Q₁₆×C₃₁	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₁₆	—	2	2	no	—
120	C₁₆	—	4	1	no	—
121	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
124	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
126	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
127	C₈	—	4	1	no	—
128	C₈	—	4	4	no	—
129	C₈	—	8	2	no	—
130	C₈	C₃₁×#123	1	1	yes	characteristic
131	C₈	C₃₁×#125	1	2	yes	—
132	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
133	C₂²	—	4	1	no	—
134	C₂²	C₃₁×#467	1	1	yes	characteristic
135	C₂²	C₃₁×#477	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	C₃₁×#398	1	1	yes	characteristic
138	C₄	C₃₁×#469	1	1	yes	characteristic
139	C₄	C₃₁×#478	1	2	yes	—
140	C₂	—	2	1	no	—
141	C₂	—	2	2	no	—
142	C₂	C₃₁×#2087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2161

C₂

yes

maximal, characteristic

C₃₁×#2151

C₂

yes

maximal, characteristic

C₃₁×#1953

C₂

yes

maximal

C₃₁×#1385

C₂

yes

maximal, characteristic

C₃₁×#1381

C₂

yes

maximal

C₂×C₃₁×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₃₁

C₂²

yes

characteristic

C₃₁×#430

—

C₂×C₃₁×#108

C₂²

yes

—

C₂×C₃₁×#108

C₂²

yes

—

C₃₁×#405

C₂²

yes

characteristic

C₃₁×#368

—

C₃₁×C₄⋊C₁₆

—

C₃₁×#344

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

D₄

yes

characteristic

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₃₁×#117

D₄

yes

characteristic

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

D₄

yes

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#108

—

C₃₁×#108

Q₈

yes

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

C₂×Q₈

yes

—

C₂×C₈×C₃₁

#41

yes

—

C₂×C₈×C₃₁

#41

yes

characteristic

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

Q₁₆

yes

—

C₄²×C₃₁

—

C₁₆×C₃₁

—

C₁₆×C₃₁

—

#13520

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#121

yes

characteristic

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₃₁

#135

yes

characteristic

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

#123

yes

characteristic

C₈×C₃₁

#125

yes

—

C₈×C₃₁

C₄⋊Q₈

yes

characteristic

#2161

C₂×C₃₁

yes

characteristic

#2151

C₂×C₃₁

yes

characteristic

#1953

C₂×C₃₁

yes

—

#1385

C₂×C₃₁

yes

characteristic

#1381

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

#467

yes

characteristic

C₂²×C₃₁

#477

yes

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂×M₃₂

C₂²×C₃₁

yes

characteristic

C₂²×C₁₆

C₂²×C₃₁

yes

characteristic

#430

—

C₂×#108

C₂²×C₃₁

yes

—

C₂×#108

C₂²×C₃₁

yes

—

#405

C₂²×C₃₁

yes

characteristic

#368

—

C₄⋊C₁₆

—

#344

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#2087

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

D₄×C₃₁

yes

characteristic

C₂²×C₈

C₂³×C₃₁

yes

characteristic

#117

D₄×C₃₁

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₃₁

yes

—

—

—

—

—

Q₈×C₃₁

yes

—

100

#97

—

101

C₄×C₈

—

102

C₃₁

#13520

yes

characteristic, direct product factor, Sylow

103

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

113

C₂×C₈

C₂×Q₈×C₃₁

yes

—

114

C₂×C₈

#41×C₃₁

yes

—

115

C₂×C₈

#41×C₃₁

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

Q₁₆×C₃₁

yes

—

118

C₄²

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

C₂×Q₁₆×C₃₁

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

C₃₁×#121

yes

characteristic

124

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

125

C₂×C₄

C₃₁×#135

yes

characteristic

126

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

C₃₁×#123

yes

characteristic

131

C₈

C₃₁×#125

yes

—

132

C₈

C₃₁×C₄⋊Q₈

yes

characteristic

133

C₂²

—

134

C₂²

C₃₁×#467

yes

characteristic

135

C₂²

C₃₁×#477

yes

—

136

C₄

—

137

C₄

C₃₁×#398

yes

characteristic

138

C₄

C₃₁×#469

yes

characteristic

139

C₄

C₃₁×#478

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₃₁×#2087

yes

characteristic

	subgroup	G/H
center	C₄×C₃₁	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂×C₃₁	#2087
prime radical	C₂³×C₃₁	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

subgroup

G/H

center

C₄×C₃₁

#469

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle

C₂×C₃₁

#2087

prime radical

C₂³×C₃₁

C₂×Q₁₆

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

N	H ≅ G/N	type
49. #13520	C₃₁	direct (N × H)
102. C₃₁	#13520	direct (N × H)

H ≅ G/N

type

49. #13520

C₃₁

direct (N × H)

102. C₃₁

#13520

direct (N × H)

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

subgroup	index
C₃₁×#2161	2
C₃₁×#2151	2
C₃₁×#1953	2
C₃₁×#1385	2
C₃₁×#1381	2
#13520	31

subgroup

index

C₃₁×#2161

C₃₁×#2151

C₃₁×#1953

C₃₁×#1385

C₃₁×#1381

#13520

Subgroups — C₃₁×#13520 #21470279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#13520 #21470279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13520 #21470279