Group #34208380 — subgroups

Subgroups — #34208380

Order 20224.

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

normal subgroups only

This group has 354 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×#19098387	C₂	1	2	yes	maximal, complemented
2	C₂×#19098375	C₂	1	2	yes	maximal
3	#23764276	C₂	1	1	yes	maximal, characteristic
4	#23761484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
7	C₂×#14579101	C₂²	1	2	yes	—
8	C₂×#14579093	C₂²	1	1	yes	characteristic
9	C₂×#14579083	—	2	1	no	—
10	#19098465	—	2	1	no	—
11	#19098387	—	2	4	no	—
12	#19098375	—	2	4	no	—
13	C₂×#108×C₇₉	—	2	1	no	—
14	C₂×#108×C₇₉	C₂²	1	2	yes	—
15	#423×C₇₉	—	2	1	no	—
16	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×D₃₁₆	—	2	1	no	—
18	C₂×D₆₃₂	—	2	2	no	—
19	C₂×D₆₃₂	—	2	2	no	—
20	C₂×D₆₃₂	—	2	1	no	—
21	C₂×D₆₃₂	Q₈	1	2	yes	—
22	C₂×C₄⋊Dic₇₉	—	2	1	no	—
23	C₂×(C₂×D₇₉)⋊C₄	—	4	2	no	—
24	#14579101	—	2	2	no	—
25	#14579101	D₄	1	4	yes	—
26	C₂×C₇₉⋊C₁₆	—	2	2	no	—
27	#14579093	—	2	2	no	—
28	#14579083	—	4	2	no	—
29	C₂×M₁₆×C₇₉	—	2	2	no	—
30	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
33	#108×C₇₉	—	2	2	no	—
34	#108×C₇₉	—	2	2	no	—
35	#108×C₇₉	—	4	2	no	—
36	C₄⋊C₈×C₇₉	—	4	2	no	—
37	#97×C₇₉	—	2	2	no	—
38	C₂³×D₇₉	—	4	1	no	—
39	C₂×D₃₁₆	—	2	2	no	—
40	C₂×D₃₁₆	—	4	1	no	—
41	C₂×D₃₁₆	—	4	1	no	—
42	C₂×D₃₁₆	—	4	1	no	—
43	C₂×D₃₁₆	—	4	1	no	—
44	C₂×D₃₁₆	—	4	1	no	—
45	C₂²×Dic₇₉	—	4	1	no	—
46	D₆₃₂	—	2	4	no	—
47	D₆₃₂	—	4	2	no	—
48	C₄⋊Dic₇₉	—	4	1	no	—
49	C₄⋊Dic₇₉	—	4	1	no	—
50	(C₂×D₇₉)⋊C₄	—	8	4	no	—
51	C₇₉⋊C₁₆	—	2	4	no	—
52	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₇₉	—	4	2	no	—
55	M₁₆×C₇₉	—	4	2	no	—
56	C₂×C₈×C₇₉	—	2	1	no	—
57	C₂×C₈×C₇₉	—	2	1	no	—
58	C₂×C₈×C₇₉	—	4	2	no	—
59	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
60	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
63	C₂×C₈×C₇₉	#41	1	2	yes	—
64	C₄²×C₇₉	—	2	1	no	—
65	C₄²×C₇₉	—	2	1	no	—
66	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
67	C₂²×D₇₉	—	4	2	no	—
68	C₂²×D₇₉	—	8	2	no	—
69	C₂²×D₇₉	—	8	2	no	—
70	C₂²×D₇₉	—	8	1	no	—
71	C₂²×D₇₉	—	8	1	no	—
72	C₂×C₄×C₇₉	—	2	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×Dic₇₉	—	8	2	no	—
81	C₂×Dic₇₉	—	8	1	no	—
82	D₃₁₆	—	4	2	no	—
83	D₃₁₆	—	4	2	no	—
84	D₃₁₆	—	8	1	no	—
85	D₃₁₆	—	8	1	no	—
86	C₈×C₇₉	—	2	2	no	—
87	C₈×C₇₉	—	8	2	no	—
88	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
89	C₈×C₇₉	#123	1	2	yes	—
90	C₂²×C₇₉	—	2	2	no	—
91	C₂²×C₇₉	#476	1	1	yes	characteristic
92	C₂²×C₇₉	#481	1	1	yes	characteristic
93	C₂²×C₇₉	#492	1	1	yes	characteristic
94	C₂×D₇₉	—	8	4	no	—
95	C₂×D₇₉	—	8	2	no	—
96	C₂×D₇₉	—	16	1	no	—
97	C₂×D₇₉	—	16	1	no	—
98	C₂×D₇₉	—	16	2	no	—
99	C₄×C₇₉	—	4	2	no	—
100	C₄×C₇₉	#399	1	1	yes	characteristic
101	C₄×C₇₉	#476	1	1	yes	characteristic
102	C₄×C₇₉	#481	1	1	yes	characteristic
103	C₄×C₇₉	#496	1	1	yes	characteristic
104	Dic₇₉	—	16	1	no	—
105	#13535	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	2	1	no	—
107	C₂×C₇₉	—	2	1	no	—
108	C₂×C₇₉	#2094	1	1	yes	characteristic
109	C₂×C₇₉	#2238	1	2	yes	—
110	D₇₉	—	16	2	no	—
111	C₂×#365	—	79	2	no	—
112	C₂×#361	—	79	2	no	—
113	#1972	—	79	1	no	—
114	#1849	D₇₉	1	1	yes	characteristic, complemented
115	#1377	—	79	1	no	—
116	C₇₉	#13535	1	1	yes	characteristic, complemented, Sylow
117	C₂²×D₈	—	79	1	no	—
118	C₂×M₃₂	—	79	2	no	—
119	C₂×#108	—	2	1	no	—
120	C₂×#108	C₂×D₇₉	1	2	yes	—
121	C₂×#106	—	79	1	no	—
122	#423	—	2	1	no	—
123	C₂×#102	—	158	1	no	—
124	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
125	#390	—	158	1	no	—
126	#365	—	158	4	no	—
127	#361	—	158	4	no	—
128	C₂²×D₄	—	158	1	no	—
129	C₂×D₈	—	79	2	no	—
130	C₂×D₈	—	158	2	no	—
131	C₂×D₈	—	158	2	no	—
132	C₂×D₈	—	158	1	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
136	C₂×C₄⋊C₄	—	158	1	no	—
137	C₂×C₂²⋊C₄	—	316	2	no	—
138	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
139	M₃₂	—	79	4	no	—
140	M₃₂	—	158	2	no	—
141	C₂×C₁₆	—	158	2	no	—
142	#108	—	2	2	no	—
143	#108	—	2	2	no	—
144	#108	—	4	2	no	—
145	#106	—	158	2	no	—
146	C₄⋊C₈	—	4	2	no	—
147	#102	—	316	2	no	—
148	#97	—	2	2	no	—
149	C₂⁴	—	316	1	no	—
150	C₂×D₄	—	316	1	no	—
151	C₂×D₄	—	316	1	no	—
152	C₂×D₄	—	316	1	no	—
153	C₂×D₄	—	316	1	no	—
154	C₂×D₄	—	316	1	no	—
155	C₂×D₄	—	158	2	no	—
156	C₂²×C₄	—	316	1	no	—
157	C₂²×C₄	#11929611	1	2	yes	—
158	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
159	D₈	—	316	2	no	—
160	D₈	—	158	4	no	—
161	M₁₆	—	4	2	no	—
162	M₁₆	—	4	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	#11929611	1	2	yes	—
167	C₂×C₈	D₄×D₇₉	1	2	yes	—
168	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
169	C₂×C₈	#11929624	1	2	yes	—
170	C₂×C₈	#11929625	1	1	yes	characteristic
171	C₄⋊C₄	—	316	1	no	—
172	C₄⋊C₄	—	316	1	no	—
173	C₂²⋊C₄	—	632	4	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₁₆	—	158	4	no	—
177	C₂³	—	316	2	no	—
178	C₂³	—	632	2	no	—
179	C₂³	—	632	2	no	—
180	C₂³	—	632	1	no	—
181	C₂³	—	632	1	no	—
182	C₂³	C₂×#11929611	1	1	yes	characteristic
183	D₄	—	316	2	no	—
184	D₄	—	316	2	no	—
185	D₄	—	632	1	no	—
186	D₄	—	632	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	632	2	no	—
190	C₂×C₄	—	632	1	no	—
191	C₂×C₄	#14579137	1	1	yes	characteristic
192	C₂×C₄	#14579144	1	1	yes	characteristic
193	C₂×C₄	#14579157	1	1	yes	characteristic
194	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
195	C₂×C₄	#14579187	1	1	yes	characteristic
196	C₂×C₄	#14579187	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	8	2	no	—
199	C₈	#14579139	1	2	yes	—
200	C₈	#14579146	1	2	yes	—
201	C₂²	—	2	2	no	—
202	C₂²	—	632	4	no	—
203	C₂²	—	632	2	no	—
204	C₂²	—	1264	1	no	—
205	C₂²	—	1264	1	no	—
206	C₂²	—	1264	2	no	—
207	C₂²	#19098850	1	1	yes	characteristic
208	C₂²	#19098881	1	1	yes	characteristic
209	C₂²	#19098937	1	1	yes	characteristic
210	C₄	—	4	2	no	—
211	C₄	—	1264	1	no	—
212	C₄	#19098498	1	1	yes	characteristic
213	C₄	#19098850	1	1	yes	characteristic
214	C₄	#19098881	1	1	yes	characteristic
215	C₄	#19098950	1	1	yes	characteristic
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	1264	2	no	—
219	C₂	#23764934	1	1	yes	characteristic
220	C₂	#23765614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098387

C₂

yes

maximal, complemented

C₂×#19098375

C₂

yes

maximal

#23764276

C₂

yes

maximal, characteristic

#23761484

C₂

yes

maximal, characteristic, complemented

C₇₉×#1849

C₂

yes

maximal, characteristic, complemented

C₂²×D₆₃₂

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579093

C₂²

yes

—

—

—

—

C₂×#108×C₇₉

—

C₂×#108×C₇₉

C₂²

yes

—

#423×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

C₂²×D₃₁₆

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

—

C₂×D₆₃₂

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×(C₂×D₇₉)⋊C₄

—

#14579101

—

#14579101

D₄

yes

—

C₂×C₇₉⋊C₁₆

—

#14579093

—

#14579083

—

C₂×M₁₆×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

#108×C₇₉

—

#108×C₇₉

—

#108×C₇₉

—

C₄⋊C₈×C₇₉

—

#97×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

SD₁₆

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

C₂×C₈×C₇₉

yes

C₂×C₈×C₇₉

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×SD₁₆

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

C₂×SD₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#123

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

100

C₄×C₇₉

#399

yes

characteristic

101

C₄×C₇₉

#476

yes

characteristic

102

C₄×C₇₉

#481

yes

characteristic

103

C₄×C₇₉

#496

yes

characteristic

104

Dic₇₉

—

105

#13535

—

maximal, Sylow

106

C₂×C₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

#2094

yes

characteristic

109

C₂×C₇₉

#2238

yes

—

110

D₇₉

—

111

C₂×#365

—

112

C₂×#361

—

113

#1972

—

114

#1849

D₇₉

yes

characteristic, complemented

115

#1377

—

116

C₇₉

#13535

yes

characteristic, complemented, Sylow

117

C₂²×D₈

—

118

C₂×M₃₂

—

119

C₂×#108

—

120

C₂×#108

C₂×D₇₉

yes

—

121

C₂×#106

—

122

#423

—

123

C₂×#102

—

158

—

124

C₂×#97

C₂×D₇₉

yes

characteristic

125

#390

—

158

—

126

#365

—

158

—

127

#361

—

158

—

128

C₂²×D₄

—

158

—

129

C₂×D₈

—

130

C₂×D₈

—

158

—

131

C₂×D₈

—

158

—

132

C₂×D₈

—

158

—

133

C₂×M₁₆

—

134

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

135

C₂²×C₈

C₂²×D₇₉

yes

characteristic

136

C₂×C₄⋊C₄

—

158

—

137

C₂×C₂²⋊C₄

—

316

—

138

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

139

M₃₂

—

140

M₃₂

—

158

—

141

C₂×C₁₆

—

158

—

142

#108

—

143

#108

—

144

#108

—

145

#106

—

158

—

146

C₄⋊C₈

—

147

#102

—

316

—

148

#97

—

149

C₂⁴

—

316

—

150

C₂×D₄

—

316

—

151

C₂×D₄

—

316

—

152

C₂×D₄

—

316

—

153

C₂×D₄

—

316

—

154

C₂×D₄

—

316

—

155

C₂×D₄

—

158

—

156

C₂²×C₄

—

316

—

157

C₂²×C₄

#11929611

yes

—

158

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

159

D₈

—

316

—

160

D₈

—

158

—

161

M₁₆

—

162

M₁₆

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

#11929611

yes

—

167

C₂×C₈

D₄×D₇₉

yes

—

168

C₂×C₈

Q₈×D₇₉

yes

characteristic

169

C₂×C₈

#11929624

yes

—

170

C₂×C₈

#11929625

yes

characteristic

171

C₄⋊C₄

—

316

—

172

C₄⋊C₄

—

316

—

173

C₂²⋊C₄

—

632

—

174

C₄²

—

175

C₄²

—

176

C₁₆

—

158

—

177

C₂³

—

316

—

178

C₂³

—

632

—

179

C₂³

—

632

—

180

C₂³

—

632

—

181

C₂³

—

632

—

182

C₂³

C₂×#11929611

yes

characteristic

183

D₄

—

316

—

184

D₄

—

316

—

185

D₄

—

632

—

186

D₄

—

632

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

632

—

190

C₂×C₄

—

632

—

191

C₂×C₄

#14579137

yes

characteristic

192

C₂×C₄

#14579144

yes

characteristic

193

C₂×C₄

#14579157

yes

characteristic

194

C₂×C₄

C₂×#11929611

yes

characteristic

195

C₂×C₄

#14579187

yes

characteristic

196

C₂×C₄

#14579187

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

#14579139

yes

—

200

C₈

#14579146

yes

—

201

C₂²

—

202

C₂²

—

632

—

203

C₂²

—

632

—

204

C₂²

—

1264

—

205

C₂²

—

1264

—

206

C₂²

—

1264

—

207

C₂²

#19098850

yes

characteristic

208

C₂²

#19098881

yes

characteristic

209

C₂²

#19098937

yes

characteristic

210

C₄

—

211

C₄

—

1264

—

212

C₄

#19098498

yes

characteristic

213

C₄

#19098850

yes

characteristic

214

C₄

#19098881

yes

characteristic

215

C₄

#19098950

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

—

1264

—

219

C₂

#23764934

yes

characteristic

220

C₂

#23765614

yes

—

	subgroup	G/H
center	C₂²	#19098881
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1849	C₂
socle	C₂²×C₇₉	#481
prime radical	C₂²×D₆₃₂	C₂²
hypercenter	#1849	D₇₉
hypocenter	C₇₉	#13535
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

subgroup

G/H

center

C₂²

#19098881

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1849

C₂

socle

C₂²×C₇₉

#481

prime radical

C₂²×D₆₃₂

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1849

N	H ≅ G/N	type
1. C₂×#19098387	C₂	semidirect
4. #23761484	C₂	semidirect
5. C₇₉×#1849	C₂	semidirect
114. #1849	D₇₉	semidirect
116. C₇₉	#13535	semidirect

H ≅ G/N

type

1. C₂×#19098387

C₂

semidirect

4. #23761484

C₂

semidirect

5. C₇₉×#1849

C₂

semidirect

114. #1849

D₇₉

semidirect

116. C₇₉

#13535

semidirect

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

subgroup	index
C₂×#19098387	2
C₂×#19098375	2
#23764276	2
#23761484	2
C₇₉×#1849	2
#13535	79

subgroup

index

C₇₉×#1849

Subgroups — #34208380

Conjugacy classes of subgroups

Complements

Maximal subgroups