Group #16842009 — subgroups

Subgroups — C₃₁×#2076 #16842009

Order 3968.

256 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 96 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×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#380	C₂	1	2	yes	maximal
4	C₃₁×#340	C₂	1	2	yes	maximal, complemented
5	C₃₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
12	C₃₁×#107	—	2	2	no	—
13	C₃₁×C₂²⋊C₈	—	2	2	no	—
14	C₃₁×#95	—	2	2	no	—
15	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	—	2	1	no	—
17	C₂²×C₄×C₃₁	—	2	1	no	—
18	C₂²×C₄×C₃₁	—	2	2	no	—
19	C₂²×C₄×C₃₁	—	2	2	no	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	—	2	2	no	—
22	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	1	no	—
26	C₂×C₈×C₃₁	—	2	2	no	—
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂²⋊C₄×C₃₁	—	4	4	no	—
33	C₂³×C₃₁	—	2	1	no	—
34	C₂³×C₃₁	—	2	2	no	—
35	C₂³×C₃₁	—	2	1	no	—
36	C₂³×C₃₁	—	2	2	no	—
37	C₂³×C₃₁	Q₁₆	1	2	yes	—
38	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	—	2	1	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	4	1	no	—
42	C₂×C₄×C₃₁	—	4	1	no	—
43	C₂×C₄×C₃₁	—	4	2	no	—
44	C₂×C₄×C₃₁	—	4	2	no	—
45	C₂×C₄×C₃₁	—	4	2	no	—
46	C₂×C₄×C₃₁	—	4	2	no	—
47	C₂×C₄×C₃₁	—	4	2	no	—
48	C₂×C₄×C₃₁	—	4	2	no	—
49	C₂×C₄×C₃₁	—	4	2	no	—
50	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₁	#41	1	2	yes	—
52	C₂×C₄×C₃₁	#41	1	2	yes	—
53	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
54	C₈×C₃₁	—	2	2	no	—
55	C₈×C₃₁	—	4	1	no	—
56	#2076	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₃₁	—	2	2	no	—
58	C₂²×C₃₁	—	2	4	no	—
59	C₂²×C₃₁	—	4	1	no	—
60	C₂²×C₃₁	—	4	1	no	—
61	C₂²×C₃₁	—	4	2	no	—
62	C₂²×C₃₁	#121	1	1	yes	characteristic
63	C₂²×C₃₁	#123	1	2	yes	—
64	C₂²×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂²×C₃₁	#135	1	1	yes	characteristic
66	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
67	C₄×C₃₁	—	4	1	no	—
68	C₄×C₃₁	—	8	2	no	—
69	C₄×C₃₁	—	8	2	no	—
70	C₄×C₃₁	#123	1	1	yes	characteristic
71	C₄×C₃₁	#124	1	1	yes	characteristic
72	C₄×C₃₁	#126	1	2	yes	—
73	C₂×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
74	C₂×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
75	#380	C₂×C₃₁	1	2	yes	—
76	#340	C₂×C₃₁	1	1	yes	characteristic, complemented
77	#340	C₂×C₃₁	1	2	yes	complemented
78	C₂×C₃₁	—	4	2	no	—
79	C₂×C₃₁	#397	1	1	yes	characteristic
80	C₂×C₃₁	#467	1	1	yes	characteristic
81	C₂×C₃₁	#469	1	1	yes	characteristic
82	C₂×C₃₁	#482	1	2	yes	—
83	C₂×C₃₁	#484	1	2	yes	—
84	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
85	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
89	C₂×C₂²⋊C₄	—	2	2	no	—
90	#107	—	2	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	2	2	no	—
93	C₃₁	#2076	1	1	yes	characteristic, direct product factor, Sylow
94	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
102	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	D₄×C₃₁	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	4	4	no	—
112	C₂³	—	2	1	no	—
113	C₂³	—	2	2	no	—
114	C₂³	—	2	1	no	—
115	C₂³	—	2	2	no	—
116	C₂³	Q₁₆×C₃₁	1	2	yes	—
117	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
130	C₂×C₄	#41×C₃₁	1	2	yes	—
131	C₂×C₄	#41×C₃₁	1	2	yes	—
132	C₂×C₄	#41×C₃₁	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	4	1	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	2	no	—
140	C₂²	C₃₁×#121	1	1	yes	characteristic
141	C₂²	C₃₁×#123	1	2	yes	—
142	C₂²	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
143	C₂²	C₃₁×#135	1	1	yes	characteristic
144	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
145	C₄	—	4	1	no	—
146	C₄	—	8	2	no	—
147	C₄	—	8	2	no	—
148	C₄	C₃₁×#123	1	1	yes	characteristic
149	C₄	C₃₁×#124	1	1	yes	characteristic
150	C₄	C₃₁×#126	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	C₃₁×#397	1	1	yes	characteristic
153	C₂	C₃₁×#467	1	1	yes	characteristic
154	C₂	C₃₁×#469	1	1	yes	characteristic
155	C₂	C₃₁×#482	1	2	yes	—
156	C₂	C₃₁×#484	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#107

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₃₁×#380

C₂

yes

maximal

C₃₁×#340

C₂

yes

maximal, complemented

C₃₁×#340

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₃₁

—

C₃₁×#107

—

C₃₁×C₂²⋊C₈

—

C₃₁×#95

—

C₂⁴×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

Q₁₆

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

#2076

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#121

yes

characteristic

C₂²×C₃₁

#123

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂×#107

C₂×C₃₁

yes

characteristic, complemented

C₂×C₂²⋊C₈

C₂×C₃₁

yes

characteristic

#380

C₂×C₃₁

yes

—

#340

C₂×C₃₁

yes

characteristic, complemented

#340

C₂×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#484

yes

—

C₂³×C₄

C₂²×C₃₁

yes

characteristic

C₂²×C₈

C₂²×C₃₁

yes

characteristic

C₂²×C₈

C₂²×C₃₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂×C₂²⋊C₄

—

#107

—

C₂²⋊C₈

—

#95

—

C₃₁

#2076

yes

characteristic, direct product factor, Sylow

C₂⁴

D₄×C₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

D₄×C₃₁

yes

characteristic

102

C₂²×C₄

C₂³×C₃₁

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

D₄×C₃₁

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

Q₁₆×C₃₁

yes

—

117

C₂³

C₂×D₄×C₃₁

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×D₄×C₃₁

yes

characteristic

130

C₂×C₄

#41×C₃₁

yes

—

131

C₂×C₄

#41×C₃₁

yes

—

132

C₂×C₄

#41×C₃₁

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₃₁×#121

yes

characteristic

141

C₂²

C₃₁×#123

yes

—

142

C₂²

C₂×Q₁₆×C₃₁

yes

characteristic

143

C₂²

C₃₁×#135

yes

characteristic

144

C₂²

C₃₁×AGL(1,ℤ/8)

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₃₁×#123

yes

characteristic

149

C₄

C₃₁×#124

yes

characteristic

150

C₄

C₃₁×#126

yes

—

151

C₂

—

152

C₂

C₃₁×#397

yes

characteristic

153

C₂

C₃₁×#467

yes

characteristic

154

C₂

C₃₁×#469

yes

characteristic

155

C₂

C₃₁×#482

yes

—

156

C₂

C₃₁×#484

yes

—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₁

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₃₁

prime radical

C₂⁴×C₃₁

D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₂×C₃₁×#107	C₂	semidirect
4. C₃₁×#340	C₂	semidirect
5. C₃₁×#340	C₂	semidirect
56. #2076	C₃₁	direct (N × H)
73. C₂×#107	C₂×C₃₁	semidirect
76. #340	C₂×C₃₁	semidirect
77. #340	C₂×C₃₁	semidirect
93. C₃₁	#2076	direct (N × H)

H ≅ G/N

type

1. C₂×C₃₁×#107

C₂

semidirect

4. C₃₁×#340

C₂

semidirect

5. C₃₁×#340

C₂

semidirect

56. #2076

C₃₁

direct (N × H)

73. C₂×#107

C₂×C₃₁

semidirect

76. #340

C₂×C₃₁

semidirect

77. #340

C₂×C₃₁

semidirect

93. C₃₁

#2076

direct (N × H)

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

subgroup	index
C₂×C₃₁×#107	2
C₂×C₃₁×C₂²⋊C₈	2
C₃₁×#380	2
C₃₁×#340	2
#2076	31

subgroup

index

C₂×C₃₁×#107

C₂×C₃₁×C₂²⋊C₈

C₃₁×#380

C₃₁×#340

#2076

Subgroups — C₃₁×#2076 #16842009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#2076 #16842009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2076 #16842009