Group #34207358 — subgroups

Subgroups — #34207358

Order 20224.

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	#23773924	C₂	1	1	yes	maximal, characteristic
2	#23765120	C₂	1	2	yes	maximal
3	#23761507	C₂	1	2	yes	maximal
4	#23761484	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1971	C₂	1	1	yes	maximal, characteristic
6	#19099120	—	2	1	no	—
7	C₂×#14579101	C₂²	1	2	yes	—
8	#19098972	—	2	4	no	—
9	#19098601	—	2	1	no	—
10	#19098601	C₂²	1	2	yes	—
11	#19098367	—	2	4	no	—
12	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₉	—	2	1	no	—
15	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
16	#391×C₇₉	—	2	1	no	—
17	#14579161	—	2	2	no	—
18	#14579153	—	2	2	no	—
19	#14579140	—	4	2	no	—
20	C₄×Dic₁₅₈	—	4	2	no	—
21	#14579120	—	2	2	no	—
22	#14579101	—	2	2	no	—
23	#14579101	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	#14579095	—	2	2	no	—
26	#14579095	D₄	1	4	yes	—
27	#14579093	—	2	2	no	—
28	#14579093	—	2	2	no	—
29	C₂²×Q₈×C₇₉	—	2	1	no	—
30	C₂×Q₁₆×C₇₉	—	2	2	no	—
31	C₂×Q₁₆×C₇₉	—	2	2	no	—
32	C₂×Q₁₆×C₇₉	—	2	1	no	—
33	C₂×Q₁₆×C₇₉	Q₈	1	2	yes	complemented
34	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
37	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
38	#106×C₇₉	—	2	2	no	—
39	#103×C₇₉	—	4	2	no	—
40	#97×C₇₉	—	2	2	no	—
41	#95×C₇₉	—	4	2	no	—
42	C₂×Dic₁₅₈	—	4	2	no	—
43	C₄⋊Dic₇₉	—	2	4	no	—
44	C₄⋊Dic₇₉	—	4	2	no	—
45	Dic₇₉⋊C₄	—	4	4	no	—
46	C₂²⋊Dic₇₉	—	4	2	no	—
47	C₄×Dic₇₉	—	4	2	no	—
48	C₇₉⋊C₁₆	—	4	2	no	—
49	C₂×Q₈×C₇₉	—	2	2	no	—
50	C₂×Q₈×C₇₉	—	4	1	no	—
51	C₂×Q₈×C₇₉	—	4	1	no	—
52	C₂×Q₈×C₇₉	—	4	1	no	—
53	C₂×Q₈×C₇₉	—	4	1	no	—
54	C₂×Q₈×C₇₉	—	4	1	no	—
55	C₂²×C₄×C₇₉	—	4	1	no	—
56	C₂²×C₄×C₇₉	—	4	1	no	—
57	C₂²×C₄×C₇₉	D₈	1	2	yes	—
58	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₇₉	—	2	4	no	—
60	Q₁₆×C₇₉	—	4	2	no	—
61	C₂×C₈×C₇₉	—	2	1	no	—
62	C₂×C₈×C₇₉	—	2	1	no	—
63	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
64	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₉	#41	1	2	yes	—
68	C₄⋊C₄×C₇₉	—	4	1	no	—
69	C₄⋊C₄×C₇₉	—	4	1	no	—
70	C₄²×C₇₉	—	2	1	no	—
71	C₄²×C₇₉	—	2	1	no	—
72	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
73	Q₈×C₇₉	—	4	2	no	—
74	Q₈×C₇₉	—	4	2	no	—
75	Q₈×C₇₉	—	8	1	no	—
76	Q₈×C₇₉	—	8	1	no	—
77	C₂×C₄×C₇₉	—	2	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	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
90	C₂×Dic₇₉	—	4	4	no	—
91	Dic₁₅₈	—	8	4	no	—
92	C₈×C₇₉	—	2	2	no	—
93	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₇₉	#124	1	2	yes	—
95	C₂²×C₇₉	—	2	2	no	—
96	C₂²×C₇₉	#447	1	1	yes	characteristic
97	C₂²×C₇₉	#474	1	1	yes	characteristic
98	C₂²×C₇₉	#486	1	1	yes	characteristic
99	C₄×C₇₉	—	4	2	no	—
100	C₄×C₇₉	—	8	2	no	—
101	C₄×C₇₉	—	16	1	no	—
102	C₄×C₇₉	#393	1	1	yes	characteristic
103	C₄×C₇₉	#448	1	1	yes	characteristic
104	C₄×C₇₉	#476	1	1	yes	characteristic
105	C₄×C₇₉	#489	1	1	yes	characteristic
106	Dic₇₉	—	8	4	no	—
107	#13368	—	79	1	no	maximal, Sylow
108	C₂×C₇₉	—	2	1	no	—
109	C₂×C₇₉	—	2	1	no	—
110	C₂×C₇₉	#2023	1	1	yes	characteristic
111	C₂×C₇₉	#2191	1	2	yes	—
112	#3162	—	79	1	no	—
113	#2141	—	79	2	no	—
114	#1971	D₇₉	1	1	yes	characteristic
115	#1386	—	79	2	no	—
116	#1377	—	79	1	no	—
117	C₇₉	#13368	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
119	#533	—	158	1	no	—
120	C₂×M₃₂	—	79	2	no	—
121	#501	—	158	4	no	—
122	#427	—	79	2	no	—
123	#427	—	158	1	no	—
124	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
125	C₂×#103	—	2	1	no	—
126	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
127	#391	—	2	1	no	—
128	#358	—	158	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	Dic₁₅₈	1	2	yes	complemented
134	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
136	C₄⋊Q₈	—	158	2	no	—
137	#125	—	158	2	no	—
138	C₂²⋊Q₈	—	316	2	no	—
139	C₄×Q₈	—	316	2	no	—
140	#117	—	158	2	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
143	M₃₂	—	158	2	no	—
144	M₃₂	—	158	2	no	—
145	C₂×C₁₆	—	158	2	no	—
146	#107	—	79	4	no	—
147	#107	—	158	2	no	—
148	#106	—	2	2	no	—
149	#106	—	158	2	no	—
150	#106	—	158	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	—	316	2	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	1	no	—
163	C₂²×C₄	#11929609	1	2	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₇₉	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	#11929612	1	2	yes	—
170	C₂×C₈	D₄×D₇₉	1	2	yes	—
171	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
172	C₂×C₈	#11929623	1	2	yes	—
173	C₂×C₈	#11929624	1	1	yes	characteristic
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	316	4	no	—
177	C₄⋊C₄	—	316	2	no	—
178	C₄⋊C₄	—	158	4	no	—
179	C₂²⋊C₄	—	316	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	316	2	no	—
183	C₁₆	—	316	2	no	—
184	C₂³	C₂×#11929609	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	—	632	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	—	316	4	no	—
198	C₂×C₄	#14579129	1	1	yes	characteristic
199	C₂×C₄	#14579140	1	1	yes	characteristic
200	C₂×C₄	#14579148	1	1	yes	characteristic
201	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
202	C₂×C₄	#14579186	1	1	yes	characteristic
203	C₂×C₄	#14579191	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	#14579138	1	2	yes	—
206	C₈	#14579147	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	#19098696	1	1	yes	characteristic
209	C₂²	#19098843	1	1	yes	characteristic
210	C₂²	#19098909	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	8	2	no	—
213	C₄	—	16	1	no	—
214	C₄	—	632	4	no	—
215	C₄	#19098475	1	1	yes	characteristic
216	C₄	#19098701	1	1	yes	characteristic
217	C₄	#19098853	1	1	yes	characteristic
218	C₄	#19098922	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#23764536	1	1	yes	characteristic
222	C₂	#23765350	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23773924

C₂

yes

maximal, characteristic

#23765120

C₂

yes

maximal

#23761507

C₂

yes

maximal

#23761484

C₂

yes

maximal, characteristic

C₇₉×#1971

C₂

yes

maximal, characteristic

#19099120

—

C₂×#14579101

C₂²

yes

—

#19098972

—

#19098601

—

#19098601

C₂²

yes

—

#19098367

—

C₂²×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂×#106×C₇₉

C₂²

yes

characteristic

C₂×#103×C₇₉

—

C₂×#97×C₇₉

C₂²

yes

characteristic

#391×C₇₉

—

#14579161

—

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

#14579101

—

#14579101

—

C₂×C₇₉⋊C₁₆

—

#14579095

—

#14579095

D₄

yes

—

#14579093

—

#14579093

—

C₂²×Q₈×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

—

C₂×Q₁₆×C₇₉

Q₈

yes

complemented

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

#106×C₇₉

—

#103×C₇₉

—

#97×C₇₉

—

#95×C₇₉

—

C₂×Dic₁₅₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

Q₁₆×C₇₉

—

Q₁₆×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

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

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

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#124

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

100

C₄×C₇₉

—

101

C₄×C₇₉

—

102

C₄×C₇₉

#393

yes

characteristic

103

C₄×C₇₉

#448

yes

characteristic

104

C₄×C₇₉

#476

yes

characteristic

105

C₄×C₇₉

#489

yes

characteristic

106

Dic₇₉

—

107

#13368

—

maximal, Sylow

108

C₂×C₇₉

—

109

C₂×C₇₉

—

110

C₂×C₇₉

#2023

yes

characteristic

111

C₂×C₇₉

#2191

yes

—

112

#3162

—

113

#2141

—

114

#1971

D₇₉

yes

characteristic

115

#1386

—

116

#1377

—

117

C₇₉

#13368

yes

characteristic, complemented, Sylow

118

C₂²×Q₁₆

C₂×D₇₉

yes

characteristic

119

#533

—

158

—

120

C₂×M₃₂

—

121

#501

—

158

—

122

#427

—

123

#427

—

158

—

124

C₂×#106

C₂×D₇₉

yes

characteristic

125

C₂×#103

—

126

C₂×#97

C₂×D₇₉

yes

characteristic

127

#391

—

128

#358

—

158

—

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

Dic₁₅₈

yes

complemented

134

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

135

C₂²×C₈

C₂²×D₇₉

yes

characteristic

136

C₄⋊Q₈

—

158

—

137

#125

—

158

—

138

C₂²⋊Q₈

—

316

—

139

C₄×Q₈

—

316

—

140

#117

—

158

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

143

M₃₂

—

158

—

144

M₃₂

—

158

—

145

C₂×C₁₆

—

158

—

146

#107

—

147

#107

—

158

—

148

#106

—

149

#106

—

158

—

150

#106

—

158

—

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

—

316

—

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

#11929609

yes

—

164

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

#11929612

yes

—

170

C₂×C₈

D₄×D₇₉

yes

—

171

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

172

C₂×C₈

#11929623

yes

—

173

C₂×C₈

#11929624

yes

characteristic

174

C₄⋊C₄

—

175

C₄⋊C₄

—

176

C₄⋊C₄

—

316

—

177

C₄⋊C₄

—

316

—

178

C₄⋊C₄

—

158

—

179

C₂²⋊C₄

—

316

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

316

—

183

C₁₆

—

316

—

184

C₂³

C₂×#11929609

yes

characteristic

185

Q₈

—

186

Q₈

—

187

Q₈

—

188

Q₈

—

189

Q₈

—

632

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

316

—

198

C₂×C₄

#14579129

yes

characteristic

199

C₂×C₄

#14579140

yes

characteristic

200

C₂×C₄

#14579148

yes

characteristic

201

C₂×C₄

C₂×#11929612

yes

characteristic

202

C₂×C₄

#14579186

yes

characteristic

203

C₂×C₄

#14579191

yes

characteristic

204

C₈

—

205

C₈

#14579138

yes

—

206

C₈

#14579147

yes

—

207

C₂²

—

208

C₂²

#19098696

yes

characteristic

209

C₂²

#19098843

yes

characteristic

210

C₂²

#19098909

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

632

—

215

C₄

#19098475

yes

characteristic

216

C₄

#19098701

yes

characteristic

217

C₄

#19098853

yes

characteristic

218

C₄

#19098922

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#23764536

yes

characteristic

222

C₂

#23765350

yes

—

	subgroup	G/H
center	C₂²	#19098696
commutator, cosocle	C₂²×C₈×C₇₉	C₂³
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1971	C₂
socle	C₂²×C₇₉	#447
prime radical	C₂³×C₇₉	C₂×D₈
hypercenter	#1971	D₇₉
hypocenter	C₇₉	#13368
LCS	G ▷ C₂²×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

subgroup

G/H

center

C₂²

#19098696

commutator, cosocle

C₂²×C₈×C₇₉

C₂³

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1971

C₂

socle

C₂²×C₇₉

#447

prime radical

C₂³×C₇₉

C₂×D₈

D₇₉

C₇₉

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1971

N	H ≅ G/N	type
33. C₂×Q₁₆×C₇₉	Q₈	semidirect
117. C₇₉	#13368	semidirect
133. C₂×Q₁₆	Dic₁₅₈	semidirect

H ≅ G/N

type

33. C₂×Q₁₆×C₇₉

Q₈

semidirect

117. C₇₉

#13368

semidirect

133. C₂×Q₁₆

Dic₁₅₈

semidirect

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

subgroup	index
#23773924	2
#23765120	2
#23761507	2
#23761484	2
C₇₉×#1971	2
#13368	79

subgroup

index

C₇₉×#1971

Subgroups — #34207358

Conjugacy classes of subgroups

Complements

Maximal subgroups