Group #31906667 — subgroups

Subgroups — #31906667

Order 18176.

306 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22590143	C₂	1	2	yes	maximal, complemented
2	#22586282	C₂	1	2	yes	maximal, complemented
3	#22586266	C₂	1	2	yes	maximal
4	C₇₁×#2922	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
6	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
7	#17985775	—	2	2	no	—
8	C₄×#41×C₇₁	C₄	1	1	yes	characteristic
9	#433×C₇₁	—	2	2	no	—
10	#431×C₇₁	—	2	2	no	—
11	#424×C₇₁	C₄	1	1	yes	characteristic
12	#424×C₇₁	C₂²	1	1	yes	characteristic
13	#408×C₇₁	C₄	1	1	yes	characteristic
14	#408×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₇₁	C₄	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₇₁⋊C₁₆	—	2	4	no	—
18	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
23	C₄×Q₈×C₇₁	C₈	1	2	yes	—
24	C₄×D₄×C₇₁	—	2	2	no	—
25	C₄×D₄×C₇₁	C₈	1	2	yes	—
26	#117×C₇₁	—	2	1	no	—
27	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	—	2	1	no	—
29	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₇₁	—	2	2	no	—
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₂²⋊C₈×C₇₁	—	2	2	no	—
33	C₂²⋊C₈×C₇₁	—	2	2	no	—
34	#97×C₇₁	—	2	1	no	—
35	C₄×C₈×C₇₁	—	2	1	no	—
36	C₄×C₈×C₇₁	D₄	1	2	yes	—
37	C₄×C₈×C₇₁	D₄	1	2	yes	—
38	C₇₁⋊C₁₆	—	4	4	no	—
39	#41×C₇₁	—	4	1	no	—
40	#41×C₇₁	—	4	1	no	—
41	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₂×D₄×C₇₁	—	2	1	no	—
43	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	—	2	2	no	—
45	C₂²×C₄×C₇₁	—	2	1	no	—
46	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₇₁	—	4	1	no	—
49	M₁₆×C₇₁	—	4	1	no	—
50	C₂×C₈×C₇₁	—	2	1	no	—
51	C₂×C₈×C₇₁	—	2	1	no	—
52	C₂×C₈×C₇₁	—	4	1	no	—
53	C₂×C₈×C₇₁	—	4	1	no	—
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
56	C₄⋊C₄×C₇₁	—	2	2	no	—
57	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
58	C₂²⋊C₄×C₇₁	—	2	1	no	—
59	C₂²⋊C₄×C₇₁	—	2	2	no	—
60	C₄²×C₇₁	—	2	2	no	—
61	C₄²×C₇₁	—	2	1	no	—
62	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×C₈	1	2	yes	—
65	C₂³×C₇₁	—	2	1	no	—
66	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	Q₈×C₇₁	—	4	1	no	—
68	D₄×C₇₁	—	4	1	no	—
69	D₄×C₇₁	—	4	2	no	—
70	C₂×C₄×C₇₁	—	2	1	no	—
71	C₂×C₄×C₇₁	—	2	1	no	—
72	C₂×C₄×C₇₁	—	2	2	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	—	4	2	no	—
75	C₂×C₄×C₇₁	—	4	4	no	—
76	C₂×C₄×C₇₁	—	4	1	no	—
77	C₂×C₄×C₇₁	—	4	1	no	—
78	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₈×C₇₁	—	4	1	no	—
85	C₈×C₇₁	—	4	1	no	—
86	C₂²×C₇₁	—	2	2	no	—
87	C₂²×C₇₁	—	2	1	no	—
88	C₂²×C₇₁	—	4	2	no	—
89	C₂²×C₇₁	—	4	1	no	—
90	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	—	4	2	no	—
95	C₄×C₇₁	—	4	1	no	—
96	C₄×C₇₁	#410	1	1	yes	characteristic
97	C₄×C₇₁	#413	1	1	yes	characteristic
98	#11463	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	—	4	1	no	—
100	C₂×C₇₁	—	8	1	no	—
101	C₂×C₇₁	#1458	1	1	yes	characteristic
102	C₂×C₇₁	#2117	1	2	yes	—
103	#2922	D₇₁	1	1	yes	characteristic
104	#2163	—	71	2	no	—
105	#1323	—	71	2	no	—
106	#1315	—	71	2	no	—
107	C₇₁	#11463	1	1	yes	characteristic, complemented, Sylow
108	C₄×#41	Dic₇₁	1	1	yes	characteristic
109	#433	—	2	2	no	—
110	#431	—	2	2	no	—
111	#424	Dic₇₁	1	1	yes	characteristic
112	#424	C₂×D₇₁	1	1	yes	characteristic
113	#408	Dic₇₁	1	1	yes	characteristic
114	#408	C₂×D₇₁	1	1	yes	characteristic
115	C₄×M₁₆	Dic₇₁	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
117	C₄⋊C₁₆	—	142	2	no	—
118	C₂²⋊C₁₆	—	71	4	no	—
119	#346	—	142	2	no	—
120	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
121	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
122	C₂×M₁₆	C₂×Dic₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
125	C₄×Q₈	C₇₁⋊C₈	1	2	yes	—
126	C₄×D₄	—	2	2	no	—
127	C₄×D₄	C₇₁⋊C₈	1	2	yes	—
128	#117	—	2	1	no	—
129	#117	C₂×Dic₇₁	1	1	yes	characteristic
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
132	C₂×C₁₆	—	142	4	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₂²⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	2	2	no	—
137	#97	—	2	1	no	—
138	C₄×C₈	—	2	1	no	—
139	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
140	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
141	#41	—	4	1	no	—
142	#41	—	4	1	no	—
143	C₂×Q₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	#11765849	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
150	M₁₆	—	4	1	no	—
151	M₁₆	—	4	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	1	no	—
164	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
165	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
166	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
167	C₁₆	—	284	4	no	—
168	C₂³	—	2	1	no	—
169	C₂³	C₂×#11765849	1	1	yes	characteristic
170	Q₈	—	4	1	no	—
171	D₄	—	4	1	no	—
172	D₄	—	4	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
182	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	4	1	no	—
188	C₈	—	4	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	1	no	—
193	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	1	no	—
199	C₄	#17985973	1	1	yes	characteristic
200	C₄	#17985983	1	1	yes	characteristic
201	C₂	—	4	1	no	—
202	C₂	—	8	1	no	—
203	C₂	#22586654	1	1	yes	characteristic
204	C₂	#22589968	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal, complemented

#22586282

C₂

yes

maximal, complemented

#22586266

C₂

yes

maximal

C₇₁×#2922

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

—

C₂²⋊(C₇₁⋊C₁₆)

C₂²

yes

—

#17985775

—

C₄×#41×C₇₁

C₄

yes

characteristic

#433×C₇₁

—

#431×C₇₁

—

#424×C₇₁

C₄

yes

characteristic

#424×C₇₁

C₂²

yes

characteristic

#408×C₇₁

C₄

yes

characteristic

#408×C₇₁

C₂²

yes

characteristic

C₄×M₁₆×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×#41×C₇₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄×Q₈×C₇₁

C₈

yes

—

C₄×D₄×C₇₁

—

C₄×D₄×C₇₁

C₈

yes

—

#117×C₇₁

—

#117×C₇₁

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₈

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₈

yes

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₂²⋊C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

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

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

100

C₂×C₇₁

—

101

C₂×C₇₁

#1458

yes

characteristic

102

C₂×C₇₁

#2117

yes

—

103

#2922

D₇₁

yes

characteristic

104

#2163

—

105

#1323

—

106

#1315

—

107

C₇₁

#11463

yes

characteristic, complemented, Sylow

108

C₄×#41

Dic₇₁

yes

characteristic

109

#433

—

110

#431

—

111

#424

Dic₇₁

yes

characteristic

112

#424

C₂×D₇₁

yes

characteristic

113

#408

Dic₇₁

yes

characteristic

114

#408

C₂×D₇₁

yes

characteristic

115

C₄×M₁₆

Dic₇₁

yes

characteristic

116

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

117

C₄⋊C₁₆

—

142

—

118

C₂²⋊C₁₆

—

119

#346

—

142

—

120

C₂×#41

C₂×Dic₇₁

yes

characteristic

121

C₂×M₁₆

C₂×Dic₇₁

yes

characteristic

122

C₂×M₁₆

C₂×Dic₇₁

yes

characteristic

123

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

124

C₂²×C₈

C₂²×D₇₁

yes

characteristic

125

C₄×Q₈

C₇₁⋊C₈

yes

—

126

C₄×D₄

—

127

C₄×D₄

C₇₁⋊C₈

yes

—

128

#117

—

129

#117

C₂×Dic₇₁

yes

characteristic

130

C₂×C₄²

—

131

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

132

C₂×C₁₆

—

142

—

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₂²⋊C₈

—

136

C₂²⋊C₈

—

137

#97

—

138

C₄×C₈

—

139

C₄×C₈

C₂²⋊D₇₁

yes

—

140

C₄×C₈

C₂²⋊D₇₁

yes

—

141

#41

—

142

#41

—

143

C₂×Q₈

C₂×C₇₁⋊C₈

yes

characteristic

144

C₂×D₄

—

145

C₂×D₄

C₂×C₇₁⋊C₈

yes

characteristic

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

#11765849

yes

—

149

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

150

M₁₆

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₂²⋊Dic₇₁

yes

—

157

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

158

C₄⋊C₄

—

159

C₄⋊C₄

C₂×C₇₁⋊C₈

yes

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₄²

—

163

C₄²

—

164

C₄²

C₂²⋊Dic₇₁

yes

characteristic

165

C₄²

C₂²⋊Dic₇₁

yes

characteristic

166

C₄²

C₂×C₇₁⋊C₈

yes

—

167

C₁₆

—

284

—

168

C₂³

—

169

C₂³

C₂×#11765849

yes

characteristic

170

Q₈

—

171

D₄

—

172

D₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

182

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

183

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

184

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₂×C₂²⋊(C₇₁⋊C₈)

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

#17985973

yes

characteristic

200

C₄

#17985983

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

#22586654

yes

characteristic

204

C₂

#22589968

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2922	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2922	D₇₁
hypocenter	C₇₁	#11463
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2922

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₇₁

C₂×C₄

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2922

N	H ≅ G/N	type
1. #22590143	C₂	semidirect
2. #22586282	C₂	semidirect
107. C₇₁	#11463	semidirect

H ≅ G/N

type

1. #22590143

C₂

semidirect

2. #22586282

C₂

semidirect

107. C₇₁

#11463

semidirect

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

subgroup	index
#22590143	2
#22586282	2
#22586266	2
C₇₁×#2922	2
#11463	71

subgroup

index

C₇₁×#2922

Subgroups — #31906667

Conjugacy classes of subgroups

Complements

Maximal subgroups