Group #31917558 — subgroups

Subgroups — #31917558

Order 18176.

330 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×#17985821	C₂	1	2	yes	maximal, complemented
2	C₂×#17985808	C₂	1	2	yes	maximal
3	#22589228	C₂	1	1	yes	maximal, characteristic
4	#22586422	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14556150	C₂²	1	1	yes	characteristic
7	C₂×#14556068	C₂²	1	2	yes	—
8	C₂×#14556064	C₂²	1	1	yes	characteristic
9	C₂×#14556056	—	2	1	no	—
10	#17985965	—	2	1	no	—
11	#17985821	—	2	4	no	—
12	#17985808	—	2	4	no	—
13	C₂×#107×C₇₁	—	2	1	no	—
14	C₂×#106×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#382×C₇₁	—	2	1	no	—
17	C₂×#11765865	—	2	1	no	—
18	#14556150	—	2	2	no	—
19	#14556150	—	2	2	no	—
20	C₂×Dic₂₈₄	Q₈	1	1	yes	characteristic
21	C₂×#11765854	—	2	1	no	—
22	C₂×D₅₆₈	Q₈	1	1	yes	characteristic
23	C₂×#11765849	—	2	1	no	—
24	#14556068	—	2	2	no	—
25	#14556068	D₄	1	4	yes	—
26	C₂×C₇₁⋊C₁₆	—	2	2	no	—
27	#14556064	—	2	2	no	—
28	#14556056	—	4	2	no	—
29	(C₂×D₇₁)⋊C₈	—	4	2	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
34	#107×C₇₁	—	4	2	no	—
35	#106×C₇₁	—	2	2	no	—
36	#106×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	#95×C₇₁	—	4	2	no	—
39	#11765865	—	4	1	no	—
40	#11765865	—	4	1	no	—
41	#11765865	—	4	1	no	—
42	#11765865	—	4	1	no	—
43	C₂×Dic₁₄₂	—	2	1	no	—
44	C₂×D₂₈₄	—	2	1	no	—
45	C₂×C₂²⋊D₇₁	—	4	1	no	—
46	C₂×C₄×D₇₁	—	4	1	no	—
47	Dic₂₈₄	—	2	2	no	—
48	#11765854	—	4	2	no	—
49	D₅₆₈	—	2	2	no	—
50	#11765849	—	4	1	no	—
51	#11765849	—	4	1	no	—
52	C₂×C₇₁⋊C₈	—	4	1	no	—
53	C₇₁⋊C₁₆	—	2	4	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₇₁	—	4	2	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄²×C₇₁	—	2	1	no	—
67	C₄²×C₇₁	SD₁₆	1	2	yes	—
68	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₂²×D₇₁	—	4	1	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	—	4	2	no	—
72	C₂×C₄×C₇₁	—	8	4	no	—
73	C₂×C₄×C₇₁	—	8	2	no	—
74	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
76	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂²⋊D₇₁	—	8	1	no	—
81	C₂²⋊D₇₁	—	8	1	no	—
82	C₂×Dic₇₁	—	4	1	no	—
83	D₂₈₄	—	4	1	no	—
84	D₂₈₄	—	4	1	no	—
85	C₄×D₇₁	—	8	1	no	—
86	C₄×D₇₁	—	8	1	no	—
87	Dic₁₄₂	—	4	1	no	—
88	Dic₁₄₂	—	4	1	no	—
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
91	C₈×C₇₁	#123	1	2	yes	—
92	C₇₁⋊C₈	—	8	1	no	—
93	C₂²×C₇₁	—	2	2	no	—
94	C₂²×C₇₁	#476	1	1	yes	characteristic
95	C₂²×C₇₁	#481	1	1	yes	characteristic
96	C₂²×C₇₁	#496	1	1	yes	characteristic
97	C₂×D₇₁	—	8	1	no	—
98	C₂×D₇₁	—	8	2	no	—
99	C₄×C₇₁	—	4	2	no	—
100	C₄×C₇₁	—	16	2	no	—
101	C₄×C₇₁	#399	1	1	yes	characteristic
102	C₄×C₇₁	#476	1	1	yes	characteristic
103	C₄×C₇₁	#481	1	1	yes	characteristic
104	C₄×C₇₁	#492	1	1	yes	characteristic
105	Dic₇₁	—	8	1	no	—
106	#13534	—	71	1	no	maximal, Sylow
107	C₂×C₇₁	—	2	1	no	—
108	C₂×C₇₁	—	2	1	no	—
109	C₂×C₇₁	#2094	1	1	yes	characteristic
110	C₂×C₇₁	#2231	1	2	yes	—
111	D₇₁	—	16	1	no	—
112	C₂×#365	—	71	2	no	—
113	C₂×#360	—	71	2	no	—
114	#1974	—	71	1	no	—
115	#1846	D₇₁	1	1	yes	characteristic, complemented
116	#1376	—	71	1	no	—
117	C₇₁	#13534	1	1	yes	characteristic, complemented, Sylow
118	C₂×#135	—	71	1	no	—
119	C₂×M₃₂	—	71	2	no	—
120	C₂×#108	—	71	1	no	—
121	C₂×#107	—	2	1	no	—
122	C₂×#106	C₂×D₇₁	1	2	yes	—
123	C₂×#104	—	142	1	no	—
124	#408	—	142	1	no	—
125	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
126	#382	—	2	1	no	—
127	#365	—	142	4	no	—
128	#360	—	142	4	no	—
129	C₂×#41	—	142	1	no	—
130	#135	—	142	2	no	—
131	#135	—	142	2	no	—
132	C₂×Q₁₆	—	71	1	no	—
133	C₂×SD₁₆	—	142	1	no	—
134	C₂×D₈	—	71	1	no	—
135	C₂×M₁₆	—	142	1	no	—
136	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
140	M₃₂	—	71	4	no	—
141	M₃₂	—	142	2	no	—
142	C₂×C₁₆	—	142	2	no	—
143	#108	—	142	2	no	—
144	#107	—	4	2	no	—
145	#106	—	2	2	no	—
146	#106	—	2	2	no	—
147	#104	—	284	2	no	—
148	C₂²⋊C₈	—	284	2	no	—
149	C₄×C₈	—	2	2	no	—
150	#95	—	4	2	no	—
151	#41	—	284	1	no	—
152	#41	—	284	1	no	—
153	#41	—	284	1	no	—
154	#41	—	284	1	no	—
155	C₂×Q₈	—	142	1	no	—
156	C₂×D₄	—	284	1	no	—
157	C₂×D₄	—	142	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	284	1	no	—
161	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
162	Q₁₆	—	142	2	no	—
163	SD₁₆	—	284	2	no	—
164	D₈	—	142	2	no	—
165	M₁₆	—	284	1	no	—
166	M₁₆	—	284	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	284	1	no	—
170	C₂×C₈	#11765852	1	2	yes	—
171	C₂×C₈	D₄×D₇₁	1	2	yes	—
172	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
173	C₂×C₈	#11765865	1	2	yes	—
174	C₂×C₈	#11765866	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄²	—	2	1	no	—
178	C₄²	#11765852	1	2	yes	—
179	C₁₆	—	142	4	no	—
180	C₂³	—	284	1	no	—
181	C₂³	#14556154	1	1	yes	characteristic
182	Q₈	—	284	1	no	—
183	Q₈	—	284	1	no	—
184	D₄	—	284	1	no	—
185	D₄	—	284	1	no	—
186	D₄	—	568	1	no	—
187	D₄	—	568	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	8	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	284	1	no	—
193	C₂×C₄	—	568	1	no	—
194	C₂×C₄	—	568	1	no	—
195	C₂×C₄	#14556104	1	1	yes	characteristic
196	C₂×C₄	#14556111	1	1	yes	characteristic
197	C₂×C₄	#14556124	1	1	yes	characteristic
198	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
199	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
200	C₂×C₄	#14556154	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	568	1	no	—
203	C₈	#14556106	1	2	yes	—
204	C₈	#14556113	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	568	2	no	—
207	C₂²	—	568	1	no	—
208	C₂²	#17986286	1	1	yes	characteristic
209	C₂²	#17986317	1	1	yes	characteristic
210	C₂²	#17986386	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	16	2	no	—
213	C₄	—	568	1	no	—
214	C₄	#17985934	1	1	yes	characteristic
215	C₄	#17986286	1	1	yes	characteristic
216	C₄	#17986317	1	1	yes	characteristic
217	C₄	#17986373	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	—	1136	1	no	—
221	C₂	#22589874	1	1	yes	characteristic
222	C₂	#22590513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985821

C₂

yes

maximal, complemented

C₂×#17985808

C₂

yes

maximal

#22589228

C₂

yes

maximal, characteristic

#22586422

C₂

yes

maximal, characteristic, complemented

C₇₁×#1846

C₂

yes

maximal, characteristic, complemented

C₂×#14556150

C₂²

yes

characteristic

C₂×#14556068

C₂²

yes

—

C₂×#14556064

C₂²

yes

—

—

—

—

C₂×#107×C₇₁

—

C₂×#106×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#382×C₇₁

—

C₂×#11765865

—

#14556150

—

#14556150

—

C₂×Dic₂₈₄

Q₈

yes

characteristic

C₂×#11765854

—

C₂×D₅₆₈

Q₈

yes

—

—

D₄

yes

—

C₂×C₇₁⋊C₁₆

—

#14556064

—

#14556056

—

(C₂×D₇₁)⋊C₈

—

C₂²×C₈×C₇₁

D₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

D₄

yes

characteristic

#107×C₇₁

—

#106×C₇₁

—

#106×C₇₁

—

C₄×C₈×C₇₁

—

#95×C₇₁

—

—

—

—

—

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

C₂×C₄×D₇₁

—

Dic₂₈₄

—

#11765854

—

D₅₆₈

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

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

SD₁₆

yes

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂²⋊D₇₁

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊Q₈

yes

—

C₈×C₇₁

#123

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

100

C₄×C₇₁

—

101

C₄×C₇₁

#399

yes

characteristic

102

C₄×C₇₁

#476

yes

characteristic

103

C₄×C₇₁

#481

yes

characteristic

104

C₄×C₇₁

#492

yes

characteristic

105

Dic₇₁

—

106

#13534

—

maximal, Sylow

107

C₂×C₇₁

—

108

C₂×C₇₁

—

109

C₂×C₇₁

#2094

yes

characteristic

110

C₂×C₇₁

#2231

yes

—

111

D₇₁

—

112

C₂×#365

—

113

C₂×#360

—

114

#1974

—

115

#1846

D₇₁

yes

characteristic, complemented

116

#1376

—

117

C₇₁

#13534

yes

characteristic, complemented, Sylow

118

C₂×#135

—

119

C₂×M₃₂

—

120

C₂×#108

—

121

C₂×#107

—

122

C₂×#106

C₂×D₇₁

yes

—

123

C₂×#104

—

142

—

124

#408

—

142

—

125

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

126

#382

—

127

#365

—

142

—

128

#360

—

142

—

129

C₂×#41

—

142

—

130

#135

—

142

—

131

#135

—

142

—

132

C₂×Q₁₆

—

133

C₂×SD₁₆

—

142

—

134

C₂×D₈

—

135

C₂×M₁₆

—

142

—

136

C₂²×C₈

C₂²⋊D₇₁

yes

characteristic

137

C₂²×C₈

C₂²×D₇₁

yes

characteristic

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

C₂²⋊D₇₁

yes

characteristic

140

M₃₂

—

141

M₃₂

—

142

—

142

C₂×C₁₆

—

142

—

143

#108

—

142

—

144

#107

—

145

#106

—

146

#106

—

147

#104

—

284

—

148

C₂²⋊C₈

—

284

—

149

C₄×C₈

—

150

#95

—

151

#41

—

284

—

152

#41

—

284

—

153

#41

—

284

—

154

#41

—

284

—

155

C₂×Q₈

—

142

—

156

C₂×D₄

—

284

—

157

C₂×D₄

—

142

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

284

—

161

C₂²×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

162

Q₁₆

—

142

—

163

SD₁₆

—

284

—

164

D₈

—

142

—

165

M₁₆

—

284

—

166

M₁₆

—

284

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

284

—

170

C₂×C₈

#11765852

yes

—

171

C₂×C₈

D₄×D₇₁

yes

—

172

C₂×C₈

Q₈×D₇₁

yes

characteristic

173

C₂×C₈

#11765865

yes

—

174

C₂×C₈

#11765866

yes

characteristic

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄²

—

178

C₄²

#11765852

yes

—

179

C₁₆

—

142

—

180

C₂³

—

284

—

181

C₂³

#14556154

yes

characteristic

182

Q₈

—

284

—

183

Q₈

—

284

—

184

D₄

—

284

—

185

D₄

—

284

—

186

D₄

—

568

—

187

D₄

—

568

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

284

—

193

C₂×C₄

—

568

—

194

C₂×C₄

—

568

—

195

C₂×C₄

#14556104

yes

characteristic

196

C₂×C₄

#14556111

yes

characteristic

197

C₂×C₄

#14556124

yes

characteristic

198

C₂×C₄

C₂×#11765852

yes

characteristic

199

C₂×C₄

C₂×#11765852

yes

characteristic

200

C₂×C₄

#14556154

yes

characteristic

201

C₈

—

202

C₈

—

568

—

203

C₈

#14556106

yes

—

204

C₈

#14556113

yes

—

205

C₂²

—

206

C₂²

—

568

—

207

C₂²

—

568

—

208

C₂²

#17986286

yes

characteristic

209

C₂²

#17986317

yes

characteristic

210

C₂²

#17986386

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

568

—

214

C₄

#17985934

yes

characteristic

215

C₄

#17986286

yes

characteristic

216

C₄

#17986317

yes

characteristic

217

C₄

#17986373

yes

characteristic

218

C₂

—

219

C₂

—

220

C₂

—

1136

—

221

C₂

#22589874

yes

characteristic

222

C₂

#22590513

yes

—

	subgroup	G/H
center	C₂²	#17986286
commutator, cosocle	C₂²×C₈×C₇₁	C₂³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1846	C₂
socle	C₂²×C₇₁	#476
prime radical	C₂×#14556150	C₂²
hypercenter	#1846	D₇₁
hypocenter	C₇₁	#13534
LCS	G ▷ C₂²×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂²

#17986286

commutator, cosocle

C₂²×C₈×C₇₁

C₂³

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#1846

C₂

C₂²×C₇₁

C₂²

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
1. C₂×#17985821	C₂	semidirect
4. #22586422	C₂	semidirect
5. C₇₁×#1846	C₂	semidirect
115. #1846	D₇₁	semidirect
117. C₇₁	#13534	semidirect

H ≅ G/N

type

1. C₂×#17985821

C₂

semidirect

4. #22586422

C₂

semidirect

5. C₇₁×#1846

C₂

semidirect

115. #1846

D₇₁

semidirect

117. C₇₁

#13534

semidirect

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

subgroup	index
C₂×#17985821	2
C₂×#17985808	2
#22589228	2
#22586422	2
C₇₁×#1846	2
#13534	71

subgroup

index

C₇₁×#1846

Subgroups — #31917558

Conjugacy classes of subgroups

Complements

Maximal subgroups