Group #34190560 — subgroups

Subgroups — #34190560

Order 20224.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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	#23774544	C₂	1	2	yes	maximal
2	#23773924	C₂	1	1	yes	maximal, characteristic
3	#23761963	C₂	1	2	yes	maximal
4	#23761245	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1960	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	2	yes	—
7	#19098972	—	2	2	no	—
8	#19098972	—	2	2	no	—
9	#19098959	—	2	2	no	—
10	#19098915	—	2	2	no	—
11	#19098893	—	2	4	no	—
12	#19098845	—	2	4	no	—
13	#19098616	C₂²	1	2	yes	—
14	#19098601	—	2	2	no	—
15	#19098299	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#103×C₇₉	—	2	2	no	—
18	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
19	#382×C₇₉	C₂²	1	1	yes	characteristic
20	#14579161	D₄	1	2	yes	—
21	#14579161	D₄	1	2	yes	—
22	#14579153	—	2	2	no	—
23	#14579140	—	2	2	no	—
24	#14579140	—	2	2	no	—
25	C₄×Dic₁₅₈	—	2	2	no	—
26	C₄×Dic₁₅₈	—	2	2	no	—
27	#14579120	—	2	2	no	—
28	#14579095	—	2	4	no	—
29	#14579093	—	2	2	no	—
30	#14579093	—	2	2	no	—
31	#14579092	—	2	8	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
33	#14579084	—	2	8	no	—
34	C₂²⋊(C₇₉⋊C₈)	D₄	1	4	yes	—
35	C₄×C₇₉⋊C₈	—	2	2	no	—
36	C₂²×Q₈×C₇₉	—	2	1	no	—
37	C₂²×C₈×C₇₉	D₄	1	2	yes	—
38	C₄⋊Q₈×C₇₉	—	2	2	no	—
39	C₄⋊Q₈×C₇₉	—	2	1	no	—
40	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
43	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
44	#103×C₇₉	—	4	4	no	—
45	#97×C₇₉	—	2	2	no	—
46	#95×C₇₉	—	2	2	no	—
47	C₂×Dic₁₅₈	—	2	2	no	—
48	C₂×Dic₁₅₈	—	2	2	no	—
49	C₂×C₇₉⋊C₈	—	2	4	no	—
50	C₄⋊Dic₇₉	—	2	2	no	—
51	C₄⋊Dic₇₉	—	2	2	no	—
52	C₄⋊Dic₇₉	—	2	2	no	—
53	C₄⋊Dic₇₉	—	2	2	no	—
54	Dic₇₉⋊C₄	—	4	2	no	—
55	Dic₇₉⋊C₄	—	4	2	no	—
56	C₂²⋊Dic₇₉	—	4	2	no	—
57	C₄×Dic₇₉	—	4	2	no	—
58	C₂×Q₈×C₇₉	—	2	2	no	—
59	C₂×Q₈×C₇₉	—	4	1	no	—
60	C₂×Q₈×C₇₉	—	4	2	no	—
61	C₂×Q₈×C₇₉	—	4	2	no	—
62	C₂²×C₄×C₇₉	—	4	1	no	—
63	C₂²×C₄×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₉	—	2	2	no	—
67	C₂×C₈×C₇₉	—	2	2	no	—
68	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
69	C₄⋊C₄×C₇₉	—	2	4	no	—
70	C₄⋊C₄×C₇₉	—	4	2	no	—
71	C₄⋊C₄×C₇₉	—	4	2	no	—
72	C₄²×C₇₉	C₂×D₄	1	2	yes	—
73	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₇₉	#41	1	1	yes	characteristic
75	C₂³×C₇₉	#127	1	1	yes	characteristic
76	Q₈×C₇₉	—	4	4	no	—
77	Q₈×C₇₉	—	8	2	no	—
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	—	4	2	no	—
80	C₂×C₄×C₇₉	—	8	1	no	—
81	C₂×C₄×C₇₉	—	8	1	no	—
82	C₂×C₄×C₇₉	—	8	2	no	—
83	C₂×C₄×C₇₉	—	8	1	no	—
84	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
85	C₂×C₄×C₇₉	#120	1	2	yes	—
86	C₂×C₄×C₇₉	#121	1	2	yes	—
87	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
88	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₇₉	#137	1	2	yes	—
90	C₂×Dic₇₉	—	4	2	no	—
91	C₂×Dic₇₉	—	4	2	no	—
92	Dic₁₅₈	—	4	4	no	—
93	Dic₁₅₈	—	8	2	no	—
94	C₈×C₇₉	—	2	4	no	—
95	C₇₉⋊C₈	—	4	4	no	—
96	C₂²×C₇₉	—	2	2	no	—
97	C₂²×C₇₉	#392	1	1	yes	characteristic
98	C₂²×C₇₉	#492	1	1	yes	characteristic
99	C₂²×C₇₉	#497	1	1	yes	characteristic
100	C₄×C₇₉	—	2	2	no	—
101	C₄×C₇₉	—	2	2	no	—
102	C₄×C₇₉	—	8	2	no	—
103	C₄×C₇₉	—	16	1	no	—
104	C₄×C₇₉	#448	1	2	yes	—
105	C₄×C₇₉	#461	1	2	yes	—
106	Dic₇₉	—	8	2	no	—
107	Dic₇₉	—	8	2	no	—
108	#10227	—	79	1	no	maximal, Sylow
109	C₂×C₇₉	—	2	1	no	—
110	C₂×C₇₉	—	2	1	no	—
111	C₂×C₇₉	#2009	1	2	yes	—
112	C₂×C₇₉	#2013	1	1	yes	characteristic
113	#3229	—	79	2	no	—
114	#3162	—	79	1	no	—
115	#1960	D₇₉	1	1	yes	characteristic
116	#1518	—	79	2	no	—
117	#1281	—	79	1	no	—
118	C₇₉	#10227	1	1	yes	characteristic, complemented, Sylow
119	#533	—	79	2	no	—
120	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
121	#501	—	158	2	no	—
122	#501	—	158	2	no	—
123	#498	—	158	2	no	—
124	#487	—	158	2	no	—
125	#483	—	158	4	no	—
126	#475	—	158	4	no	—
127	#431	—	79	2	no	—
128	#427	—	158	2	no	—
129	C₂×#103	—	2	2	no	—
130	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
131	#382	C₂×D₇₉	1	1	yes	characteristic
132	#328	—	158	4	no	—
133	C₂²×Q₈	—	2	1	no	—
134	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	1	no	—
137	C₄⋊Q₈	—	79	2	no	—
138	C₄⋊Q₈	—	79	2	no	—
139	C₄⋊Q₈	D₃₁₆	1	2	yes	—
140	#125	—	158	2	no	—
141	C₂²⋊Q₈	—	158	2	no	—
142	C₂²⋊Q₈	—	158	2	no	—
143	C₄×Q₈	—	158	2	no	—
144	C₄×Q₈	—	158	2	no	—
145	#117	—	158	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
149	#107	—	158	4	no	—
150	#106	—	158	8	no	—
151	#106	—	158	2	no	—
152	#106	—	158	2	no	—
153	C₄⋊C₈	—	158	2	no	—
154	#103	—	4	4	no	—
155	#103	—	158	8	no	—
156	C₂²⋊C₈	—	79	4	no	—
157	#97	—	2	2	no	—
158	C₄×C₈	—	158	2	no	—
159	#95	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	4	1	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	158	2	no	—
165	C₂×Q₈	—	158	2	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	D₄×D₇₉	1	2	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	158	4	no	—
173	C₂×C₈	#11929612	1	4	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	316	2	no	—
178	C₄⋊C₄	—	316	2	no	—
179	C₄⋊C₄	—	158	2	no	—
180	C₄⋊C₄	—	158	2	no	—
181	C₄⋊C₄	—	158	2	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₂²⋊C₄	—	316	2	no	—
184	C₄²	—	316	2	no	—
185	C₄²	D₄×D₇₉	1	2	yes	—
186	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
187	C₄²	#11929624	1	1	yes	characteristic
188	C₂³	#14579157	1	1	yes	characteristic
189	Q₈	—	4	4	no	—
190	Q₈	—	8	2	no	—
191	Q₈	—	316	4	no	—
192	Q₈	—	632	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	1	no	—
199	C₂×C₄	—	316	2	no	—
200	C₂×C₄	—	316	2	no	—
201	C₂×C₄	#14579129	1	1	yes	characteristic
202	C₂×C₄	#14579131	1	2	yes	—
203	C₂×C₄	#14579134	1	2	yes	—
204	C₂×C₄	#14579135	1	1	yes	characteristic
205	C₂×C₄	C₂×#11929612	1	2	yes	—
206	C₂×C₄	#14579191	1	2	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	316	4	no	—
209	C₂²	—	2	2	no	—
210	C₂²	#19098472	1	1	yes	characteristic
211	C₂²	#19098935	1	1	yes	characteristic
212	C₂²	#19098958	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	8	2	no	—
216	C₄	—	16	1	no	—
217	C₄	—	632	2	no	—
218	C₄	—	632	2	no	—
219	C₄	#19098701	1	2	yes	—
220	C₄	#19098768	1	2	yes	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	#23764462	1	2	yes	—
224	C₂	#23764482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23774544

C₂

yes

maximal

#23773924

C₂

yes

maximal, characteristic

#23761963

C₂

yes

maximal

#23761245

C₂

yes

maximal, characteristic

C₇₉×#1960

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

—

#19098601

—

#19098299

—

C₂×C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

C₂×#103×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

#382×C₇₉

C₂²

yes

characteristic

#14579161

D₄

yes

—

#14579161

D₄

yes

—

#14579153

—

#14579140

—

#14579140

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

—

—

—

—

—

C₄⋊(C₇₉⋊C₈)

—

#14579084

—

C₂²⋊(C₇₉⋊C₈)

D₄

yes

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#103×C₇₉

—

#97×C₇₉

—

#95×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

C₄²×C₇₉

yes

C₂³×C₇₉

yes

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#120

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#121

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

—

103

C₄×C₇₉

—

104

C₄×C₇₉

#448

yes

—

105

C₄×C₇₉

#461

yes

—

106

Dic₇₉

—

107

Dic₇₉

—

108

#10227

—

maximal, Sylow

109

C₂×C₇₉

—

110

C₂×C₇₉

—

111

C₂×C₇₉

#2009

yes

—

112

C₂×C₇₉

#2013

yes

characteristic

113

#3229

—

114

#3162

—

115

#1960

D₇₉

yes

characteristic

116

#1518

—

117

#1281

—

118

C₇₉

#10227

yes

characteristic, complemented, Sylow

119

#533

—

120

C₂×C₄⋊Q₈

C₂×D₇₉

yes

characteristic

121

#501

—

158

—

122

#501

—

158

—

123

#498

—

158

—

124

#487

—

158

—

125

#483

—

158

—

126

#475

—

158

—

127

#431

—

128

#427

—

158

—

129

C₂×#103

—

130

C₂×#97

C₂×D₇₉

yes

characteristic

131

#382

C₂×D₇₉

yes

characteristic

132

#328

—

158

—

133

C₂²×Q₈

—

134

C₂²×C₈

C₂²⋊D₇₉

yes

—

135

C₄⋊Q₈

—

136

C₄⋊Q₈

—

137

C₄⋊Q₈

—

138

C₄⋊Q₈

—

139

C₄⋊Q₈

D₃₁₆

yes

—

140

#125

—

158

—

141

C₂²⋊Q₈

—

158

—

142

C₂²⋊Q₈

—

158

—

143

C₄×Q₈

—

158

—

144

C₄×Q₈

—

158

—

145

#117

—

158

—

146

C₂×C₄⋊C₄

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄²

C₂²×D₇₉

yes

characteristic

149

#107

—

158

—

150

#106

—

158

—

151

#106

—

158

—

152

#106

—

158

—

153

C₄⋊C₈

—

158

—

154

#103

—

155

#103

—

158

—

156

C₂²⋊C₈

—

157

#97

—

158

C₄×C₈

—

158

—

159

#95

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

158

—

165

C₂×Q₈

—

158

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

D₄×D₇₉

yes

—

169

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

158

—

173

C₂×C₈

#11929612

yes

—

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

177

C₄⋊C₄

—

316

—

178

C₄⋊C₄

—

316

—

179

C₄⋊C₄

—

158

—

180

C₄⋊C₄

—

158

—

181

C₄⋊C₄

—

158

—

182

C₄⋊C₄

—

158

—

183

C₂²⋊C₄

—

316

—

184

C₄²

—

316

—

185

C₄²

D₄×D₇₉

yes

—

186

C₄²

C₂×D₃₁₆

yes

characteristic

187

C₄²

#11929624

yes

characteristic

188

C₂³

#14579157

yes

characteristic

189

Q₈

—

190

Q₈

—

191

Q₈

—

316

—

192

Q₈

—

632

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

316

—

200

C₂×C₄

—

316

—

201

C₂×C₄

#14579129

yes

characteristic

202

C₂×C₄

#14579131

yes

—

203

C₂×C₄

#14579134

yes

—

204

C₂×C₄

#14579135

yes

characteristic

205

C₂×C₄

C₂×#11929612

yes

—

206

C₂×C₄

#14579191

yes

—

207

C₈

—

208

C₈

—

316

—

209

C₂²

—

210

C₂²

#19098472

yes

characteristic

211

C₂²

#19098935

yes

characteristic

212

C₂²

#19098958

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

632

—

218

C₄

—

632

—

219

C₄

#19098701

yes

—

220

C₄

#19098768

yes

—

221

C₂

—

222

C₂

—

223

C₂

#23764462

yes

—

224

C₂

#23764482

yes

characteristic

	subgroup	G/H
center	C₂²	#19098472
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1960	C₂
socle	C₂²×C₇₉	#392
prime radical	C₂³×C₇₉	#127
hypercenter	#1960	D₇₉
hypocenter	C₇₉	#10227
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

subgroup

G/H

center

C₂²

#19098472

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1960

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1960

N	H ≅ G/N	type
118. C₇₉	#10227	semidirect

H ≅ G/N

type

118. C₇₉

#10227

semidirect

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

subgroup	index
#23774544	2
#23773924	2
#23761963	2
#23761245	2
C₇₉×#1960	2
#10227	79

subgroup

index

C₇₉×#1960

Subgroups — #34190560

Conjugacy classes of subgroups

Complements

Maximal subgroups