Group #31902201 — subgroups

Subgroups — #31902201

Order 18176.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22588468	C₂	1	1	yes	maximal, characteristic
2	#22588345	C₂	1	2	yes	maximal
3	#22586224	C₂	1	2	yes	maximal
4	#22586175	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14556052	—	2	2	no	—
8	#17985869	C₂²	1	2	yes	—
9	#17985762	—	2	2	no	—
10	#17985761	—	2	2	no	—
11	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#336×C₇₁	—	2	2	no	—
15	C₂²×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Dic₇₁	—	2	2	no	—
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	#14556052	—	4	4	no	—
22	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	—	2	1	no	—
24	C₂²×C₈×C₇₁	D₄	1	2	yes	—
25	C₄×Q₈×C₇₁	—	2	2	no	—
26	C₄×Q₈×C₇₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	—	2	1	no	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	—	2	2	no	—
33	C₂²×Dic₇₁	—	4	2	no	—
34	C₂×C₇₁⋊C₈	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	C₄⋊Dic₇₁	—	4	2	no	—
39	C₄⋊Dic₇₁	—	4	2	no	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	8	2	no	—
42	C₂×Q₈×C₇₁	—	2	1	no	—
43	C₂×Q₈×C₇₁	—	2	2	no	—
44	C₂×Q₈×C₇₁	—	2	2	no	—
45	C₂×Q₈×C₇₁	Q₁₆	1	2	yes	—
46	C₂²×C₄×C₇₁	—	2	1	no	—
47	C₂²×C₄×C₇₁	—	2	1	no	—
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
51	C₂×C₈×C₇₁	—	2	2	no	—
52	C₂×C₈×C₇₁	—	2	2	no	—
53	C₂×C₈×C₇₁	—	2	2	no	—
54	C₂×C₈×C₇₁	—	4	1	no	—
55	C₂×C₈×C₇₁	—	4	2	no	—
56	C₄⋊C₄×C₇₁	—	2	2	no	—
57	C₄⋊C₄×C₇₁	—	4	2	no	—
58	C₄²×C₇₁	—	4	1	no	—
59	C₄²×C₇₁	—	4	1	no	—
60	C₄²×C₇₁	#41	1	2	yes	—
61	C₄²×C₇₁	#41	1	2	yes	—
62	C₂³×C₇₁	#121	1	1	yes	characteristic
63	Q₈×C₇₁	—	2	4	no	—
64	Q₈×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	4	1	no	—
68	C₂×C₄×C₇₁	—	4	1	no	—
69	C₂×C₄×C₇₁	—	4	1	no	—
70	C₂×C₄×C₇₁	—	4	2	no	—
71	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#123	1	2	yes	—
73	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#126	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#135	1	2	yes	—
77	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
78	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₇₁	#137	1	2	yes	—
80	C₂×Dic₇₁	—	8	2	no	—
81	C₂×Dic₇₁	—	8	2	no	—
82	C₂×Dic₇₁	—	8	2	no	—
83	C₈×C₇₁	—	4	2	no	—
84	C₈×C₇₁	—	8	1	no	—
85	C₇₁⋊C₈	—	8	2	no	—
86	C₂²×C₇₁	#397	1	1	yes	characteristic
87	C₂²×C₇₁	#464	1	2	yes	—
88	C₂²×C₇₁	#467	1	1	yes	characteristic
89	C₂²×C₇₁	#469	1	1	yes	characteristic
90	C₂²×C₇₁	#471	1	2	yes	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	4	2	no	—
93	C₄×C₇₁	—	8	1	no	—
94	C₄×C₇₁	#482	1	2	yes	—
95	C₄×C₇₁	#484	1	2	yes	—
96	C₄×C₇₁	#485	1	2	yes	—
97	C₄×C₇₁	#488	1	1	yes	characteristic
98	C₄×C₇₁	#491	1	1	yes	characteristic
99	Dic₇₁	—	16	2	no	—
100	#10624	—	71	1	no	maximal, Sylow
101	C₂×C₇₁	#1684	1	2	yes	—
102	C₂×C₇₁	#1702	1	2	yes	—
103	C₂×C₇₁	#2074	1	1	yes	characteristic
104	C₂×C₇₁	#2076	1	1	yes	characteristic
105	C₂×C₇₁	#2079	1	1	yes	characteristic
106	#1984	D₇₁	1	1	yes	characteristic
107	#1834	—	71	1	no	—
108	#1803	—	71	2	no	—
109	#1296	—	71	2	no	—
110	#1277	—	71	1	no	—
111	C₇₁	#10624	1	1	yes	characteristic, complemented, Sylow
112	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
113	C₂×C₄⋊C₈	—	71	2	no	—
114	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₂×#103	—	142	2	no	—
116	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
117	#384	—	71	2	no	—
118	#340	—	142	2	no	—
119	#340	—	142	2	no	—
120	#336	—	2	2	no	—
121	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	142	2	no	—
124	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
125	C₄×Q₈	—	2	2	no	—
126	C₄×Q₈	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	1	no	—
128	C₂×C₄⋊C₄	—	142	2	no	—
129	C₂×C₄⋊C₄	—	142	2	no	—
130	C₂×C₄⋊C₄	—	142	2	no	—
131	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	142	2	no	—
136	C₄⋊C₈	—	142	2	no	—
137	#103	—	284	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₂₈₄	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	284	2	no	—
146	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
148	C₂²×C₄	#11765865	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	284	2	no	—
155	C₂×C₈	—	284	2	no	—
156	C₂×C₈	—	284	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	284	2	no	—
160	C₄⋊C₄	—	284	2	no	—
161	C₄⋊C₄	—	284	2	no	—
162	C₄⋊C₄	—	284	2	no	—
163	C₄⋊C₄	—	568	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#11765864	1	2	yes	—
167	C₄²	#11765864	1	2	yes	—
168	C₂³	#14556102	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	568	2	no	—
178	C₂×C₄	—	568	2	no	—
179	C₂×C₄	—	568	2	no	—
180	C₂×C₄	#14556109	1	1	yes	characteristic
181	C₂×C₄	#14556112	1	2	yes	—
182	C₂×C₄	#14556115	1	1	yes	characteristic
183	C₂×C₄	#14556121	1	2	yes	—
184	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
185	C₂×C₄	#14556148	1	2	yes	—
186	C₂×C₄	#14556150	1	1	yes	characteristic
187	C₂×C₄	#14556156	1	2	yes	—
188	C₂×C₄	#14556159	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	568	2	no	—
192	C₂²	#17985924	1	1	yes	characteristic
193	C₂²	#17986221	1	2	yes	—
194	C₂²	#17986240	1	1	yes	characteristic
195	C₂²	#17986252	1	1	yes	characteristic
196	C₂²	#17986262	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	1136	2	no	—
201	C₄	#17986324	1	2	yes	—
202	C₄	#17986335	1	2	yes	—
203	C₄	#17986337	1	2	yes	—
204	C₄	#17986353	1	1	yes	characteristic
205	C₄	#17986367	1	1	yes	characteristic
206	C₂	#22587798	1	2	yes	—
207	C₂	#22587908	1	2	yes	—
208	C₂	#22589758	1	1	yes	characteristic
209	C₂	#22589768	1	1	yes	characteristic
210	C₂	#22589785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22588468

C₂

yes

maximal, characteristic

#22588345

C₂

yes

maximal

#22586224

C₂

yes

maximal

#22586175

C₂

yes

maximal, characteristic

C₇₁×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂×#14556052

—

#17985869

C₂²

yes

—

#17985762

—

#17985761

—

C₂×C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#336×C₇₁

—

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556052

—

C₂²×Q₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

D₄

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

Q₁₆

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

#41

yes

—

C₄²×C₇₁

#41

yes

—

C₂³×C₇₁

#121

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#123

yes

characteristic

C₂×C₄×C₇₁

#123

yes

—

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

#126

yes

—

C₂×C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

#135

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

#397

yes

characteristic

C₂²×C₇₁

#464

yes

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#482

yes

—

C₄×C₇₁

#484

yes

—

C₄×C₇₁

#485

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

100

#10624

—

maximal, Sylow

101

C₂×C₇₁

#1684

yes

—

102

C₂×C₇₁

#1702

yes

—

103

C₂×C₇₁

#2074

yes

characteristic

104

C₂×C₇₁

#2076

yes

characteristic

105

C₂×C₇₁

#2079

yes

characteristic

106

#1984

D₇₁

yes

characteristic

107

#1834

—

108

#1803

—

109

#1296

—

110

#1277

—

111

C₇₁

#10624

yes

characteristic, complemented, Sylow

112

C₂×C₄×Q₈

C₂×D₇₁

yes

characteristic

113

C₂×C₄⋊C₈

—

114

C₂×C₄⋊C₈

C₂×D₇₁

yes

characteristic

115

C₂×#103

—

142

—

116

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

117

#384

—

118

#340

—

142

—

119

#340

—

142

—

120

#336

—

121

C₂²×Q₈

D₂₈₄

yes

characteristic

122

C₂²×C₈

—

123

C₂²×C₈

—

142

—

124

C₂²×C₈

C₂²⋊D₇₁

yes

—

125

C₄×Q₈

—

126

C₄×Q₈

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

142

—

129

C₂×C₄⋊C₄

—

142

—

130

C₂×C₄⋊C₄

—

142

—

131

C₂×C₄⋊C₄

D₂₈₄

yes

characteristic

132

C₂×C₄²

—

133

C₂×C₄²

C₂²×D₇₁

yes

characteristic

134

C₄⋊C₈

—

135

C₄⋊C₈

—

142

—

136

C₄⋊C₈

—

142

—

137

#103

—

284

—

138

C₄×C₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₂₈₄

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

284

—

146

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

147

C₂²×C₄

C₂×D₂₈₄

yes

characteristic

148

C₂²×C₄

#11765865

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

284

—

155

C₂×C₈

—

284

—

156

C₂×C₈

—

284

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

284

—

160

C₄⋊C₄

—

284

—

161

C₄⋊C₄

—

284

—

162

C₄⋊C₄

—

284

—

163

C₄⋊C₄

—

568

—

164

C₄²

—

165

C₄²

—

166

C₄²

#11765864

yes

—

167

C₄²

#11765864

yes

—

168

C₂³

#14556102

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

568

—

178

C₂×C₄

—

568

—

179

C₂×C₄

—

568

—

180

C₂×C₄

#14556109

yes

characteristic

181

C₂×C₄

#14556112

yes

—

182

C₂×C₄

#14556115

yes

characteristic

183

C₂×C₄

#14556121

yes

—

184

C₂×C₄

C₂×Dic₂₈₄

yes

characteristic

185

C₂×C₄

#14556148

yes

—

186

C₂×C₄

#14556150

yes

characteristic

187

C₂×C₄

#14556156

yes

—

188

C₂×C₄

#14556159

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

568

—

192

C₂²

#17985924

yes

characteristic

193

C₂²

#17986221

yes

—

194

C₂²

#17986240

yes

characteristic

195

C₂²

#17986252

yes

characteristic

196

C₂²

#17986262

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

1136

—

201

C₄

#17986324

yes

—

202

C₄

#17986335

yes

—

203

C₄

#17986337

yes

—

204

C₄

#17986353

yes

characteristic

205

C₄

#17986367

yes

characteristic

206

C₂

#22587798

yes

—

207

C₂

#22587908

yes

—

208

C₂

#22589758

yes

characteristic

209

C₂

#22589768

yes

characteristic

210

C₂

#22589785

yes

characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1984	C₂
socle, prime radical	C₂³×C₇₁	#121
hypercenter	#1984	D₇₁
hypocenter	C₇₁	#10624
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#14556102

commutator, cosocle

C₂×C₄²×C₇₁

C₂³

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₇₁×#1984

C₂

C₂³×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
111. C₇₁	#10624	semidirect

H ≅ G/N

type

111. C₇₁

#10624

semidirect

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

subgroup	index
#22588468	2
#22588345	2
#22586224	2
#22586175	2
C₇₁×#1984	2
#10624	71

subgroup

index

C₇₁×#1984

Subgroups — #31902201

Conjugacy classes of subgroups

Complements

Maximal subgroups