Group #31916541 — subgroups

Subgroups — #31916541

Order 18176.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22598864	C₂	1	1	yes	maximal, characteristic
2	#22590060	C₂	1	2	yes	maximal
3	#22586447	C₂	1	2	yes	maximal
4	#22586424	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1971	C₂	1	1	yes	maximal, characteristic
6	#17986556	—	2	1	no	—
7	C₂×#14556068	C₂²	1	2	yes	—
8	#17986408	—	2	4	no	—
9	#17986037	—	2	1	no	—
10	#17986037	C₂²	1	2	yes	—
11	#17985803	—	2	4	no	—
12	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₁	—	2	1	no	—
15	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
16	#391×C₇₁	—	2	1	no	—
17	#14556128	—	2	2	no	—
18	#14556120	—	2	2	no	—
19	#14556107	—	4	2	no	—
20	C₄×Dic₁₄₂	—	4	2	no	—
21	#14556087	—	2	2	no	—
22	#14556068	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	#14556062	—	2	2	no	—
26	#14556062	D₄	1	4	yes	—
27	#14556060	—	2	2	no	—
28	#14556060	—	2	2	no	—
29	C₂²×Q₈×C₇₁	—	2	1	no	—
30	C₂×Q₁₆×C₇₁	—	2	2	no	—
31	C₂×Q₁₆×C₇₁	—	2	2	no	—
32	C₂×Q₁₆×C₇₁	—	2	1	no	—
33	C₂×Q₁₆×C₇₁	Q₈	1	2	yes	complemented
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	#106×C₇₁	—	2	2	no	—
39	#103×C₇₁	—	4	2	no	—
40	#97×C₇₁	—	2	2	no	—
41	#95×C₇₁	—	4	2	no	—
42	C₂×Dic₁₄₂	—	4	2	no	—
43	C₄⋊Dic₇₁	—	2	4	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	Dic₇₁⋊C₄	—	4	4	no	—
46	C₂²⋊Dic₇₁	—	4	2	no	—
47	C₄×Dic₇₁	—	4	2	no	—
48	C₇₁⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	—	4	1	no	—
51	C₂×Q₈×C₇₁	—	4	1	no	—
52	C₂×Q₈×C₇₁	—	4	1	no	—
53	C₂×Q₈×C₇₁	—	4	1	no	—
54	C₂×Q₈×C₇₁	—	4	1	no	—
55	C₂²×C₄×C₇₁	—	4	1	no	—
56	C₂²×C₄×C₇₁	—	4	1	no	—
57	C₂²×C₄×C₇₁	D₈	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₇₁	—	2	4	no	—
60	Q₁₆×C₇₁	—	4	2	no	—
61	C₂×C₈×C₇₁	—	2	1	no	—
62	C₂×C₈×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
64	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₇₁	#41	1	2	yes	—
67	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
68	C₄⋊C₄×C₇₁	—	4	1	no	—
69	C₄⋊C₄×C₇₁	—	4	1	no	—
70	C₄²×C₇₁	—	2	1	no	—
71	C₄²×C₇₁	—	2	1	no	—
72	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₇₁	—	4	2	no	—
74	Q₈×C₇₁	—	4	2	no	—
75	Q₈×C₇₁	—	8	1	no	—
76	Q₈×C₇₁	—	8	1	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	4	2	no	—
79	C₂×C₄×C₇₁	—	4	2	no	—
80	C₂×C₄×C₇₁	—	8	1	no	—
81	C₂×C₄×C₇₁	—	8	1	no	—
82	C₂×C₄×C₇₁	—	8	2	no	—
83	C₂×C₄×C₇₁	—	8	1	no	—
84	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
90	C₂×Dic₇₁	—	4	4	no	—
91	Dic₁₄₂	—	8	4	no	—
92	C₈×C₇₁	—	2	2	no	—
93	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₇₁	#124	1	2	yes	—
95	C₂²×C₇₁	—	2	2	no	—
96	C₂²×C₇₁	#447	1	1	yes	characteristic
97	C₂²×C₇₁	#474	1	1	yes	characteristic
98	C₂²×C₇₁	#486	1	1	yes	characteristic
99	C₄×C₇₁	—	4	2	no	—
100	C₄×C₇₁	—	8	2	no	—
101	C₄×C₇₁	—	16	1	no	—
102	C₄×C₇₁	#393	1	1	yes	characteristic
103	C₄×C₇₁	#448	1	1	yes	characteristic
104	C₄×C₇₁	#476	1	1	yes	characteristic
105	C₄×C₇₁	#489	1	1	yes	characteristic
106	Dic₇₁	—	8	4	no	—
107	#13368	—	71	1	no	maximal, Sylow
108	C₂×C₇₁	—	2	1	no	—
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	#2023	1	1	yes	characteristic
111	C₂×C₇₁	#2191	1	2	yes	—
112	#3162	—	71	1	no	—
113	#2141	—	71	2	no	—
114	#1971	D₇₁	1	1	yes	characteristic
115	#1386	—	71	2	no	—
116	#1377	—	71	1	no	—
117	C₇₁	#13368	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
119	#533	—	142	1	no	—
120	C₂×M₃₂	—	71	2	no	—
121	#501	—	142	4	no	—
122	#427	—	71	2	no	—
123	#427	—	142	1	no	—
124	C₂×#106	C₂×D₇₁	1	1	yes	characteristic
125	C₂×#103	—	2	1	no	—
126	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
127	#391	—	2	1	no	—
128	#358	—	142	4	no	—
129	C₂²×Q₈	—	2	1	no	—
130	C₂×Q₁₆	—	2	2	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	—	2	1	no	—
133	C₂×Q₁₆	Dic₁₄₂	1	2	yes	complemented
134	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	142	2	no	—
137	#125	—	142	2	no	—
138	C₂²⋊Q₈	—	284	2	no	—
139	C₄×Q₈	—	284	2	no	—
140	#117	—	142	2	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
143	M₃₂	—	142	2	no	—
144	M₃₂	—	142	2	no	—
145	C₂×C₁₆	—	142	2	no	—
146	#107	—	71	4	no	—
147	#107	—	142	2	no	—
148	#106	—	2	2	no	—
149	#106	—	142	2	no	—
150	#106	—	142	2	no	—
151	#103	—	4	2	no	—
152	#97	—	2	2	no	—
153	#95	—	4	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	—	4	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	C₂×Q₈	—	284	2	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	#11765850	1	2	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	4	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	#11765853	1	2	yes	—
170	C₂×C₈	D₄×D₇₁	1	2	yes	—
171	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
172	C₂×C₈	#11765864	1	2	yes	—
173	C₂×C₈	#11765865	1	1	yes	characteristic
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	284	4	no	—
177	C₄⋊C₄	—	284	2	no	—
178	C₄⋊C₄	—	142	4	no	—
179	C₂²⋊C₄	—	284	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	284	2	no	—
183	C₁₆	—	284	2	no	—
184	C₂³	C₂×#11765850	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	Q₈	—	8	1	no	—
188	Q₈	—	8	1	no	—
189	Q₈	—	568	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	284	4	no	—
198	C₂×C₄	#14556096	1	1	yes	characteristic
199	C₂×C₄	#14556107	1	1	yes	characteristic
200	C₂×C₄	#14556115	1	1	yes	characteristic
201	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
202	C₂×C₄	#14556153	1	1	yes	characteristic
203	C₂×C₄	#14556158	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	#14556105	1	2	yes	—
206	C₈	#14556114	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	#17986132	1	1	yes	characteristic
209	C₂²	#17986279	1	1	yes	characteristic
210	C₂²	#17986345	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	8	2	no	—
213	C₄	—	16	1	no	—
214	C₄	—	568	4	no	—
215	C₄	#17985911	1	1	yes	characteristic
216	C₄	#17986137	1	1	yes	characteristic
217	C₄	#17986289	1	1	yes	characteristic
218	C₄	#17986358	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#22589476	1	1	yes	characteristic
222	C₂	#22590290	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22598864

C₂

yes

maximal, characteristic

#22590060

C₂

yes

maximal

#22586447

C₂

yes

maximal

#22586424

C₂

yes

maximal, characteristic

C₇₁×#1971

C₂

yes

maximal, characteristic

#17986556

—

C₂×#14556068

C₂²

yes

—

#17986408

—

#17986037

—

#17986037

C₂²

yes

—

#17985803

—

C₂²×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×#106×C₇₁

C₂²

yes

characteristic

C₂×#103×C₇₁

—

C₂×#97×C₇₁

C₂²

yes

characteristic

#391×C₇₁

—

#14556128

—

#14556120

—

#14556107

—

C₄×Dic₁₄₂

—

#14556087

—

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556062

—

#14556062

D₄

yes

—

#14556060

—

#14556060

—

C₂²×Q₈×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

Q₈

yes

complemented

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#106×C₇₁

—

#103×C₇₁

—

#97×C₇₁

—

#95×C₇₁

—

C₂×Dic₁₄₂

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₈

yes

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₂×C₈×C₇₁

#41

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

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

—

C₂×C₄×C₇₁

#120

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₁

#124

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#124

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

100

C₄×C₇₁

—

101

C₄×C₇₁

—

102

C₄×C₇₁

#393

yes

characteristic

103

C₄×C₇₁

#448

yes

characteristic

104

C₄×C₇₁

#476

yes

characteristic

105

C₄×C₇₁

#489

yes

characteristic

106

Dic₇₁

—

107

#13368

—

maximal, Sylow

108

C₂×C₇₁

—

109

C₂×C₇₁

—

110

C₂×C₇₁

#2023

yes

characteristic

111

C₂×C₇₁

#2191

yes

—

112

#3162

—

113

#2141

—

114

#1971

D₇₁

yes

characteristic

115

#1386

—

116

#1377

—

117

C₇₁

#13368

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

C₂×D₇₁

yes

characteristic

119

#533

—

142

—

120

C₂×M₃₂

—

121

#501

—

142

—

122

#427

—

123

#427

—

142

—

124

C₂×#106

C₂×D₇₁

yes

characteristic

125

C₂×#103

—

126

C₂×#97

C₂×D₇₁

yes

characteristic

127

#391

—

128

#358

—

142

—

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

Dic₁₄₂

yes

complemented

134

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

135

C₂²×C₈

C₂²×D₇₁

yes

characteristic

136

C₄⋊Q₈

—

142

—

137

#125

—

142

—

138

C₂²⋊Q₈

—

284

—

139

C₄×Q₈

—

284

—

140

#117

—

142

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

143

M₃₂

—

142

—

144

M₃₂

—

142

—

145

C₂×C₁₆

—

142

—

146

#107

—

147

#107

—

142

—

148

#106

—

149

#106

—

142

—

150

#106

—

142

—

151

#103

—

152

#97

—

153

#95

—

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂×Q₈

—

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

284

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

#11765850

yes

—

164

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

#11765853

yes

—

170

C₂×C₈

D₄×D₇₁

yes

—

171

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

172

C₂×C₈

#11765864

yes

—

173

C₂×C₈

#11765865

yes

characteristic

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

284

—

177

C₄⋊C₄

—

284

—

178

C₄⋊C₄

—

142

—

179

C₂²⋊C₄

—

284

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

284

—

183

C₁₆

—

284

—

184

C₂³

C₂×#11765850

yes

characteristic

185

Q₈

—

186

Q₈

—

187

Q₈

—

188

Q₈

—

189

Q₈

—

568

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

284

—

198

C₂×C₄

#14556096

yes

characteristic

199

C₂×C₄

#14556107

yes

characteristic

200

C₂×C₄

#14556115

yes

characteristic

201

C₂×C₄

C₂×#11765853

yes

characteristic

202

C₂×C₄

#14556153

yes

characteristic

203

C₂×C₄

#14556158

yes

characteristic

204

C₈

—

205

C₈

#14556105

yes

—

206

C₈

#14556114

yes

—

207

C₂²

—

208

C₂²

#17986132

yes

characteristic

209

C₂²

#17986279

yes

characteristic

210

C₂²

#17986345

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

568

—

215

C₄

#17985911

yes

characteristic

216

C₄

#17986137

yes

characteristic

217

C₄

#17986289

yes

characteristic

218

C₄

#17986358

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#22589476

yes

characteristic

222

C₂

#22590290

yes

—

	subgroup	G/H
center	C₂²	#17986132
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1971	C₂
socle	C₂²×C₇₁	#447
prime radical	C₂³×C₇₁	C₂×D₈
hypercenter	#1971	D₇₁
hypocenter	C₇₁	#13368
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#17986132

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1971

C₂

socle

C₂²×C₇₁

#447

prime radical

C₂³×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

N	H ≅ G/N	type
33. C₂×Q₁₆×C₇₁	Q₈	semidirect
117. C₇₁	#13368	semidirect
133. C₂×Q₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

33. C₂×Q₁₆×C₇₁

Q₈

semidirect

117. C₇₁

#13368

semidirect

133. C₂×Q₁₆

Dic₁₄₂

semidirect

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

subgroup	index
#22598864	2
#22590060	2
#22586447	2
#22586424	2
C₇₁×#1971	2
#13368	71

subgroup

index

C₇₁×#1971

Subgroups — #31916541

Conjugacy classes of subgroups

Complements

Maximal subgroups