Group #31913672 — subgroups

Subgroups — #31913672

Order 18176.

434 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22589951	C₂	1	2	yes	maximal
2	#22586424	C₂	1	4	yes	maximal, complemented
3	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	C₂²	1	4	yes	—
5	C₂²⋊(C₇₁⋊C₁₆)	—	2	4	no	—
6	C₂²×M₁₆×C₇₁	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×#117×C₇₁	C₄	1	1	yes	characteristic
9	#405×C₇₁	—	2	4	no	—
10	C₂×#97×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
12	#14556068	—	2	8	no	—
13	C₂×C₇₁⋊C₁₆	—	2	4	no	—
14	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₇₁	—	2	2	no	—
16	C₂×M₁₆×C₇₁	—	2	4	no	—
17	C₂²×C₈×C₇₁	—	2	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₇₁	D₄	1	4	yes	—
21	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
22	#117×C₇₁	—	2	2	no	—
23	#117×C₇₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₇₁	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₇₁	C₈	1	2	yes	—
26	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
27	#97×C₇₁	—	2	4	no	—
28	C₄×C₈×C₇₁	—	2	4	no	—
29	C₇₁⋊C₁₆	—	4	4	no	—
30	C₂⁴×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	—	2	2	no	—
32	C₂²×C₄×C₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
37	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	M₁₆×C₇₁	—	2	8	no	—
39	C₂×C₈×C₇₁	—	2	2	no	—
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	2	2	no	—
42	C₂×C₈×C₇₁	—	2	8	no	—
43	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
45	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₇₁	#41	1	4	yes	—
47	C₄⋊C₄×C₇₁	—	2	4	no	—
48	C₂²⋊C₄×C₇₁	—	2	4	no	—
49	C₄²×C₇₁	—	2	2	no	—
50	C₄²×C₇₁	—	2	2	no	—
51	C₂³×C₇₁	—	2	2	no	—
52	C₂³×C₇₁	—	4	2	no	—
53	C₂³×C₇₁	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	2	2	no	—
58	C₂×C₄×C₇₁	—	2	4	no	—
59	C₂×C₄×C₇₁	—	2	4	no	—
60	C₂×C₄×C₇₁	—	4	4	no	—
61	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#117	1	2	yes	—
64	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₇₁	—	2	2	no	—
67	C₈×C₇₁	—	2	4	no	—
68	C₈×C₇₁	#123	1	4	yes	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	—	2	4	no	—
72	C₂²×C₇₁	—	4	4	no	—
73	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₇₁	#431	1	2	yes	—
75	C₄×C₇₁	—	2	2	no	—
76	C₄×C₇₁	—	4	4	no	—
77	C₄×C₇₁	#381	1	1	yes	characteristic
78	C₄×C₇₁	#407	1	1	yes	characteristic
79	C₄×C₇₁	#432	1	2	yes	—
80	#12847	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	2	no	—
82	C₂×C₇₁	—	4	2	no	—
83	C₂×C₇₁	#1817	1	1	yes	characteristic
84	C₂×C₇₁	#2115	1	2	yes	—
85	C₂×#405	D₇₁	1	1	yes	characteristic
86	#2112	—	71	2	no	—
87	#1377	—	71	4	no	—
88	C₇₁	#12847	1	1	yes	characteristic, complemented, Sylow
89	C₂²×M₁₆	Dic₇₁	1	1	yes	characteristic
90	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
91	C₂×#117	Dic₇₁	1	1	yes	characteristic
92	C₂×M₃₂	—	71	4	no	—
93	#405	—	2	4	no	—
94	C₂×#97	C₂×D₇₁	1	2	yes	—
95	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
96	C₂²⋊C₁₆	—	142	4	no	—
97	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
103	C₂²×C₈	C₂²⋊D₇₁	1	4	yes	—
104	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₇₁⋊C₈	1	2	yes	—
108	C₂×C₂²⋊C₄	C₇₁⋊C₈	1	2	yes	—
109	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
110	M₃₂	—	142	8	no	—
111	C₂×C₁₆	—	142	4	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₇₁⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
118	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	C₂×C₇₁⋊C₈	1	4	yes	—
120	C₂²×C₄	#11765849	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
128	C₂×C₈	#11765849	1	4	yes	—
129	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
130	C₂×C₈	#11765865	1	4	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	284	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
139	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
148	C₂×C₄	#14556087	1	2	yes	—
149	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
150	C₂×C₄	C₂×#11765849	1	2	yes	—
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#14556111	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
159	C₂²	#17986052	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#17985862	1	1	yes	characteristic
163	C₄	#17985963	1	1	yes	characteristic
164	C₄	#17986056	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	#22588408	1	1	yes	characteristic
168	C₂	#22589961	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22589951

C₂

yes

maximal

#22586424

C₂

yes

maximal, complemented

C₂×#405×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₂²⋊(C₇₁⋊C₁₆)

—

C₂²×M₁₆×C₇₁

C₄

yes

characteristic

C₂³×C₈×C₇₁

C₂²

yes

characteristic

C₂×#117×C₇₁

C₄

yes

characteristic

#405×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₄

yes

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂³×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

D₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

#117×C₇₁

—

#117×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₈

yes

—

C₂×C₂²⋊C₄×C₇₁

C₈

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₇₁⋊C₁₆

—

C₂⁴×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

#41

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊C₈

yes

—

C₂³×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₂×C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#123

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

#431

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#12847

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#1817

yes

characteristic

C₂×C₇₁

#2115

yes

—

C₂×#405

D₇₁

yes

characteristic

#2112

—

#1377

—

C₇₁

#12847

yes

characteristic, complemented, Sylow

C₂²×M₁₆

Dic₇₁

yes

characteristic

C₂³×C₈

C₂×D₇₁

yes

characteristic

C₂×#117

Dic₇₁

yes

characteristic

C₂×M₃₂

—

#405

—

C₂×#97

C₂×D₇₁

yes

—

C₂×C₄×C₈

Dic₇₁

yes

—

C₂²⋊C₁₆

—

142

—

C₂³×C₄

C₂×Dic₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

102

C₂²×C₈

C₂×Dic₇₁

yes

—

103

C₂²×C₈

C₂²⋊D₇₁

yes

—

104

C₂²×C₈

C₂²×D₇₁

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₇₁⋊C₈

yes

—

108

C₂×C₂²⋊C₄

C₇₁⋊C₈

yes

—

109

C₂×C₄²

C₂×Dic₇₁

yes

—

110

M₃₂

—

142

—

111

C₂×C₁₆

—

142

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₇₁⋊C₈

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂²⋊Dic₇₁

yes

—

118

C₂²×C₄

C₂×C₇₁⋊C₈

yes

characteristic

119

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

120

C₂²×C₄

#11765849

yes

—

121

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

C₂²⋊Dic₇₁

yes

—

128

C₂×C₈

#11765849

yes

—

129

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

130

C₂×C₈

#11765865

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

284

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊(C₇₁⋊C₈)

yes

—

139

C₂³

C₂²×C₇₁⋊C₈

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

148

C₂×C₄

#14556087

yes

—

149

C₂×C₄

C₂×#11765849

yes

characteristic

150

C₂×C₄

C₂×#11765849

yes

—

151

C₈

—

152

C₈

—

153

C₈

#14556111

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊(C₇₁⋊C₈)

yes

characteristic

159

C₂²

#17986052

yes

—

160

C₄

—

161

C₄

—

162

C₄

#17985862

yes

characteristic

163

C₄

#17985963

yes

characteristic

164

C₄

#17986056

yes

—

165

C₂

—

166

C₂

—

167

C₂

#22588408

yes

characteristic

168

C₂

#22589961

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₁
commutator	C₂³×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#405×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇₁	C₂×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#12847
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₇₁

commutator

C₂³×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#405×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₇₁

C₂×C₈

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #22586424	C₂	semidirect
88. C₇₁	#12847	semidirect

H ≅ G/N

type

2. #22586424

C₂

semidirect

88. C₇₁

#12847

semidirect

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

subgroup	index
#22589951	2
#22586424	2
C₂×#405×C₇₁	2
#12847	71

subgroup

index

#22589951

#22586424

C₂×#405×C₇₁

#12847

Subgroups — #31913672

Conjugacy classes of subgroups

Complements

Maximal subgroups