Group #31897137 — subgroups

Subgroups — #31897137

Order 18176.

402 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 135 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	#22589330	C₂	1	2	yes	maximal
2	#22587096	C₂	1	1	yes	maximal, characteristic
3	#22587025	C₂	1	2	yes	maximal
4	#22586629	C₂	1	1	yes	maximal, characteristic
5	C₂×#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	1	yes	characteristic
7	#17986023	C₂²	1	2	yes	—
8	C₄×#11765849	C₂²	1	1	yes	characteristic
9	#17985768	—	2	4	no	—
10	#17985750	—	2	2	no	—
11	#17985745	—	2	4	no	—
12	#17985721	—	2	1	no	—
13	#17985720	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	#500×C₇₁	—	2	1	no	—
16	#500×C₇₁	—	2	1	no	—
17	#500×C₇₁	—	2	2	no	—
18	C₂×#107×C₇₁	C₄	1	4	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	C₂×#11765849	D₄	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
26	#14556039	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₂²×Q₈×C₇₁	—	2	2	no	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄⋊Q₈×C₇₁	—	2	2	no	—
32	C₄⋊Q₈×C₇₁	—	2	2	no	—
33	C₄⋊Q₈×C₇₁	—	2	2	no	—
34	C₄⋊Q₈×C₇₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	#107×C₇₁	—	2	4	no	—
39	#107×C₇₁	—	2	4	no	—
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₄×C₈×C₇₁	D₄	1	2	yes	—
43	#11765849	—	2	4	no	—
44	#11765849	—	4	4	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₂×C₇₁⋊C₈	—	2	2	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×Q₈×C₇₁	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	8	4	no	—
51	C₂²×C₄×C₇₁	—	2	4	no	—
52	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
58	C₄⋊C₄×C₇₁	—	2	4	no	—
59	C₄⋊C₄×C₇₁	—	2	4	no	—
60	C₄⋊C₄×C₇₁	—	4	2	no	—
61	C₄⋊C₄×C₇₁	—	4	2	no	—
62	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
67	Q₈×C₇₁	—	8	4	no	—
68	Q₈×C₇₁	—	8	4	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	4	4	no	—
72	C₂×C₄×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
77	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
78	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#137	1	4	yes	—
80	C₈×C₇₁	—	2	2	no	—
81	C₈×C₇₁	#101	1	4	yes	—
82	C₇₁⋊C₈	—	4	2	no	—
83	C₇₁⋊C₈	—	4	2	no	—
84	C₇₁⋊C₈	—	8	2	no	—
85	C₂²×C₇₁	—	2	1	no	—
86	C₂²×C₇₁	—	2	1	no	—
87	C₂²×C₇₁	#390	1	1	yes	characteristic
88	C₂²×C₇₁	#441	1	2	yes	—
89	C₄×C₇₁	—	2	1	no	—
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	—	8	4	no	—
92	C₄×C₇₁	C₂×#101	1	2	yes	—
93	C₄×C₇₁	#470	1	2	yes	—
94	C₄×C₇₁	#472	1	2	yes	—
95	C₄×C₇₁	#497	1	2	yes	—
96	#9810	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	—	2	2	no	—
98	C₂×C₇₁	#1966	1	1	yes	characteristic
99	C₂×C₇₁	#1971	1	1	yes	characteristic
100	C₂×C₇₁	#1978	1	1	yes	characteristic
101	C₂×#500	D₇₁	1	1	yes	characteristic
102	#1997	—	71	2	no	—
103	#1566	—	71	1	no	—
104	#1549	—	71	2	no	—
105	#1451	—	71	1	no	—
106	C₇₁	#9810	1	1	yes	characteristic, complemented, Sylow
107	C₂×C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
108	#500	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	2	no	—
111	#431	—	71	1	no	—
112	C₂×#107	Dic₇₁	1	4	yes	—
113	#423	—	71	2	no	—
114	C₄×M₁₆	—	71	1	no	—
115	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
116	#343	—	142	4	no	—
117	#334	—	142	2	no	—
118	#332	—	142	4	no	—
119	#322	—	142	1	no	—
120	#322	—	142	1	no	—
121	C₂²×Q₈	—	2	2	no	—
122	C₂×M₁₆	—	71	2	no	—
123	C₂×M₁₆	—	142	2	no	—
124	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	—	2	2	no	—
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
131	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
132	#107	—	2	4	no	—
133	#107	—	2	4	no	—
134	C₄⋊C₈	—	71	2	no	—
135	C₄⋊C₈	—	142	2	no	—
136	C₄⋊C₈	—	142	2	no	—
137	C₂²⋊C₈	—	142	1	no	—
138	#97	—	142	1	no	—
139	C₄×C₈	—	142	1	no	—
140	C₄×C₈	—	142	1	no	—
141	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	8	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
150	C₂²×C₄	#11765864	1	1	yes	characteristic
151	M₁₆	—	284	4	no	—
152	M₁₆	—	142	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	284	2	no	—
155	C₂×C₈	—	142	2	no	—
156	C₂×C₈	—	142	2	no	—
157	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄²	D₄×D₇₁	1	1	yes	characteristic
164	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
166	C₄²	#11765864	1	1	yes	characteristic
167	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
168	Q₈	—	8	4	no	—
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	4	no	—
174	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
176	C₂×C₄	#14556099	1	1	yes	characteristic
177	C₂×C₄	#14556099	1	1	yes	characteristic
178	C₂×C₄	#14556115	1	1	yes	characteristic
179	C₂×C₄	#14556124	1	1	yes	characteristic
180	C₂×C₄	#14556160	1	4	yes	—
181	C₈	—	2	2	no	—
182	C₈	—	284	2	no	—
183	C₈	—	284	2	no	—
184	C₈	—	568	2	no	—
185	C₈	#14556047	1	4	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	#17985902	1	1	yes	characteristic
189	C₂²	#17986106	1	2	yes	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	8	4	no	—
193	C₄	C₂×#14556047	1	2	yes	—
194	C₄	#17986254	1	2	yes	—
195	C₄	#17986266	1	2	yes	—
196	C₄	#17986392	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	#22589185	1	1	yes	characteristic
199	C₂	#22589212	1	1	yes	characteristic
200	C₂	#22589251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22589330

C₂

yes

maximal

#22587096

C₂

yes

maximal, characteristic

#22587025

C₂

yes

maximal

#22586629

C₂

yes

maximal, characteristic

C₂×#500×C₇₁

C₂

yes

maximal, characteristic

#17986052

C₂²

yes

characteristic

#17986023

C₂²

yes

—

C₄×#11765849

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈×C₇₁

C₂²

yes

—

#500×C₇₁

—

#500×C₇₁

—

#500×C₇₁

—

C₂×#107×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×#11765849

—

C₂×#11765849

D₄

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

D₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

characteristic

C₄×C₈×C₇₁

D₄

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₁

#41

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₄×D₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

—

C₈×C₇₁

—

C₈×C₇₁

#101

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#390

yes

characteristic

C₂²×C₇₁

#441

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×#101

yes

—

C₄×C₇₁

#470

yes

—

C₄×C₇₁

#472

yes

—

C₄×C₇₁

#497

yes

—

#9810

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

100

C₂×C₇₁

#1978

yes

characteristic

101

C₂×#500

D₇₁

yes

characteristic

102

#1997

—

103

#1566

—

104

#1549

—

105

#1451

—

106

C₇₁

#9810

yes

characteristic, complemented, Sylow

107

C₂×C₄⋊Q₈

C₂×D₇₁

yes

—

108

#500

—

109

#500

—

110

#500

—

111

#431

—

112

C₂×#107

Dic₇₁

yes

—

113

#423

—

114

C₄×M₁₆

—

115

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

116

#343

—

142

—

117

#334

—

142

—

118

#332

—

142

—

119

#322

—

142

—

120

#322

—

142

—

121

C₂²×Q₈

—

122

C₂×M₁₆

—

123

C₂×M₁₆

—

142

—

124

C₂²×C₈

C₂×Dic₇₁

yes

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

C₂×Dic₇₁

yes

—

131

C₂×C₄²

C₂²×D₇₁

yes

characteristic

132

#107

—

133

#107

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

142

—

136

C₄⋊C₈

—

142

—

137

C₂²⋊C₈

—

142

—

138

#97

—

142

—

139

C₄×C₈

—

142

—

140

C₄×C₈

—

142

—

141

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

142

C₄×C₈

C₂²⋊D₇₁

yes

characteristic

143

C₄×C₈

C₂²⋊D₇₁

yes

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

149

C₂²×C₄

D₄×D₇₁

yes

characteristic

150

C₂²×C₄

#11765864

yes

characteristic

151

M₁₆

—

284

—

152

M₁₆

—

142

—

153

C₂×C₈

—

154

C₂×C₈

—

284

—

155

C₂×C₈

—

142

—

156

C₂×C₈

—

142

—

157

C₂×C₈

C₂²⋊Dic₇₁

yes

—

158

C₂×C₈

C₂²⋊Dic₇₁

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄²

D₄×D₇₁

yes

characteristic

164

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

165

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

166

C₄²

#11765864

yes

characteristic

167

C₂³

D₄×Dic₇₁

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

175

C₂×C₄

D₄×Dic₇₁

yes

characteristic

176

C₂×C₄

#14556099

yes

characteristic

177

C₂×C₄

#14556099

yes

characteristic

178

C₂×C₄

#14556115

yes

characteristic

179

C₂×C₄

#14556124

yes

characteristic

180

C₂×C₄

#14556160

yes

—

181

C₈

—

182

C₈

—

284

—

183

C₈

—

284

—

184

C₈

—

568

—

185

C₈

#14556047

yes

—

186

C₂²

—

187

C₂²

—

188

C₂²

#17985902

yes

characteristic

189

C₂²

#17986106

yes

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

C₂×#14556047

yes

—

194

C₄

#17986254

yes

—

195

C₄

#17986266

yes

—

196

C₄

#17986392

yes

—

197

C₂

—

198

C₂

#22589185

yes

characteristic

199

C₂

#22589212

yes

characteristic

200

C₂

#22589251

yes

characteristic

	subgroup	G/H
center	C₂²	#17985902
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#500×C₇₁	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#500	D₇₁
hypocenter	C₇₁	#9810
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

subgroup

G/H

center

C₂²

#17985902

commutator

C₂²×C₄×C₇₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₁

Fitting

C₂×#500×C₇₁

C₂

socle

C₂²×C₇₁

#390

prime radical

C₂³×C₇₁

C₄×D₄

cosocle

C₂×C₄²×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

N	H ≅ G/N	type
106. C₇₁	#9810	semidirect

H ≅ G/N

type

106. C₇₁

#9810

semidirect

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

subgroup	index
#22589330	2
#22587096	2
#22587025	2
#22586629	2
C₂×#500×C₇₁	2
#9810	71

subgroup

index

C₂×#500×C₇₁

Subgroups — #31897137

Conjugacy classes of subgroups

Complements

Maximal subgroups