Group #34191359 — subgroups

Subgroups — #34191359

Order 20224.

416 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 101 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	#23764377	C₂	1	1	yes	maximal, characteristic
2	#23761963	C₂	1	2	yes	maximal
3	#23761258	C₂	1	2	yes	maximal
4	#23761233	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#2862	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098616	C₂²	1	2	yes	—
8	#19098587	C₂²	1	1	yes	characteristic
9	#19098323	—	2	1	no	—
10	#19098323	—	2	1	no	—
11	#19098299	—	2	4	no	—
12	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
13	#391×C₇₉	—	2	4	no	—
14	#382×C₇₉	—	2	1	no	—
15	#382×C₇₉	C₂²	1	2	yes	—
16	C₄³×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	1	no	—
18	#14579161	Q₈	1	2	yes	complemented
19	#14579153	—	2	1	no	—
20	#14579140	—	2	2	no	—
21	C₄×Dic₁₅₈	—	2	2	no	—
22	#14579120	—	2	1	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₇₉	—	2	2	no	—
28	C₄⋊Q₈×C₇₉	—	4	2	no	—
29	C₄⋊Q₈×C₇₉	—	4	2	no	—
30	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	—	2	2	no	—
33	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
36	#95×C₇₉	—	2	4	no	—
37	#95×C₇₉	—	2	2	no	—
38	#95×C₇₉	D₄	1	4	yes	—
39	C₂×Dic₁₅₈	—	2	2	no	—
40	#11929608	—	4	1	no	—
41	#11929608	—	4	1	no	—
42	C₂×C₇₉⋊C₈	—	4	2	no	—
43	C₂×C₇₉⋊C₈	—	4	1	no	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	C₄⋊Dic₇₉	—	2	2	no	—
46	Dic₇₉⋊C₄	—	4	2	no	—
47	C₂²⋊Dic₇₉	—	4	1	no	—
48	C₄×Dic₇₉	—	4	1	no	—
49	C₂×Q₈×C₇₉	—	4	4	no	—
50	C₂×Q₈×C₇₉	—	4	8	no	—
51	C₂²×C₄×C₇₉	—	2	1	no	—
52	C₂²×C₄×C₇₉	—	2	1	no	—
53	C₂²×C₄×C₇₉	—	2	4	no	—
54	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₇₉	—	4	4	no	—
57	C₄⋊C₄×C₇₉	—	4	2	no	—
58	C₄⋊C₄×C₇₉	—	4	2	no	—
59	C₄²×C₇₉	—	2	1	no	—
60	C₄²×C₇₉	—	2	1	no	—
61	C₄²×C₇₉	—	4	4	no	—
62	C₄²×C₇₉	SD₁₆	1	2	yes	—
63	C₄²×C₇₉	Q₁₆	1	2	yes	—
64	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₇₉	#41	1	1	yes	characteristic
66	C₄²×C₇₉	#41	1	2	yes	—
67	C₂³×C₇₉	#120	1	1	yes	characteristic
68	Q₈×C₇₉	—	8	8	no	—
69	C₂×C₄×C₇₉	—	2	4	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	—	4	8	no	—
75	C₂×C₄×C₇₉	—	4	4	no	—
76	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#124	1	2	yes	—
79	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
84	C₂×Dic₇₉	—	4	2	no	—
85	Dic₁₅₈	—	8	2	no	—
86	C₇₉⋊C₈	—	8	2	no	—
87	C₇₉⋊C₈	—	8	1	no	—
88	C₂²×C₇₉	—	2	2	no	—
89	C₂²×C₇₉	#393	1	1	yes	characteristic
90	C₂²×C₇₉	#450	1	1	yes	characteristic
91	C₂²×C₇₉	#451	1	1	yes	characteristic
92	C₄×C₇₉	—	2	4	no	—
93	C₄×C₇₉	—	2	2	no	—
94	C₄×C₇₉	—	2	2	no	—
95	C₄×C₇₉	—	4	2	no	—
96	C₄×C₇₉	—	8	4	no	—
97	C₄×C₇₉	#475	1	1	yes	characteristic
98	C₄×C₇₉	#477	1	1	yes	characteristic
99	C₄×C₇₉	#487	1	1	yes	characteristic
100	C₄×C₇₉	#489	1	1	yes	characteristic
101	Dic₇₉	—	8	2	no	—
102	#10363	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	2	1	no	—
104	C₂×C₇₉	—	2	1	no	—
105	C₂×C₇₉	#2024	1	1	yes	characteristic
106	C₂×C₇₉	#2031	1	2	yes	—
107	#2862	D₇₉	1	1	yes	characteristic
108	#1994	—	79	1	no	—
109	#1518	—	79	2	no	—
110	#1286	—	79	2	no	—
111	#1276	—	79	1	no	—
112	C₇₉	#10363	1	1	yes	characteristic, complemented, Sylow
113	#533	—	79	1	no	—
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	#431	—	79	2	no	—
116	#423	—	79	1	no	—
117	#391	—	2	4	no	—
118	#382	—	2	1	no	—
119	#382	C₂×D₇₉	1	2	yes	—
120	C₄³	C₂×D₇₉	1	1	yes	characteristic
121	#339	—	158	1	no	—
122	#339	—	158	1	no	—
123	#328	—	158	4	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×M₁₆	—	158	1	no	—
126	C₄⋊Q₈	—	4	2	no	—
127	C₄⋊Q₈	—	4	2	no	—
128	C₄⋊Q₈	—	79	2	no	—
129	#125	—	158	1	no	—
130	C₂²⋊Q₈	—	158	2	no	—
131	C₄×Q₈	—	158	2	no	—
132	#117	—	158	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
139	C₄⋊C₈	—	158	2	no	—
140	C₄⋊C₈	—	158	2	no	—
141	C₂²⋊C₈	—	158	2	no	—
142	C₄×C₈	—	158	2	no	—
143	#95	—	2	4	no	—
144	#95	—	2	2	no	—
145	#95	C₂²⋊D₇₉	1	4	yes	—
146	C₂×Q₈	—	4	4	no	—
147	C₂×Q₈	—	4	8	no	—
148	C₂×Q₈	—	158	2	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
153	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	M₁₆	—	316	1	no	—
155	M₁₆	—	316	1	no	—
156	C₂×C₈	—	316	2	no	—
157	C₂×C₈	—	316	1	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	316	2	no	—
162	C₄⋊C₄	—	158	2	no	—
163	C₄⋊C₄	—	158	2	no	—
164	C₂²⋊C₄	—	316	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	4	no	—
168	C₄²	—	316	1	no	—
169	C₄²	#11929612	1	2	yes	—
170	C₄²	#11929614	1	2	yes	—
171	C₄²	Q₈×D₇₉	1	1	yes	characteristic
172	C₄²	#11929625	1	1	yes	characteristic
173	C₄²	#11929625	1	2	yes	—
174	C₂³	#14579130	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	Q₈	—	632	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	316	2	no	—
185	C₂×C₄	#14579137	1	2	yes	—
186	C₂×C₄	#14579137	1	1	yes	characteristic
187	C₂×C₄	#14579150	1	2	yes	—
188	C₂×C₄	#14579150	1	1	yes	characteristic
189	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
190	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
191	C₂×C₄	#14579187	1	1	yes	characteristic
192	C₂×C₄	#14579192	1	1	yes	characteristic
193	C₈	—	632	2	no	—
194	C₈	—	632	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#19098474	1	1	yes	characteristic
197	C₂²	#19098714	1	1	yes	characteristic
198	C₂²	#19098719	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	8	4	no	—
204	C₄	—	632	2	no	—
205	C₄	#19098845	1	1	yes	characteristic
206	C₄	#19098855	1	1	yes	characteristic
207	C₄	#19098915	1	1	yes	characteristic
208	C₄	#19098925	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	#23764538	1	1	yes	characteristic
212	C₂	#23764569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23764377

C₂

yes

maximal, characteristic

#23761963

C₂

yes

maximal

#23761258

C₂

yes

maximal

#23761233

C₂

yes

maximal, characteristic

C₇₉×#2862

C₂

yes

maximal, characteristic

#19099120

C₂²

yes

characteristic

#19098616

C₂²

yes

—

#19098587

C₂²

yes

—

—

—

C₂×C₄⋊Q₈×C₇₉

—

#391×C₇₉

—

#382×C₇₉

—

#382×C₇₉

C₂²

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

#14579161

Q₈

yes

complemented

#14579153

—

#14579140

—

C₄×Dic₁₅₈

—

#14579120

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₄⋊Q₈×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

characteristic

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#95×C₇₉

—

#95×C₇₉

—

#95×C₇₉

D₄

yes

—

C₂×Dic₁₅₈

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

SD₁₆

yes

—

C₄²×C₇₉

Q₁₆

yes

—

C₄²×C₇₉

C₂×Q₈

yes

C₄²×C₇₉

yes

C₄²×C₇₉

yes

—

C₂³×C₇₉

#120

yes

characteristic

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

#124

yes

—

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

Dic₁₅₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

100

C₄×C₇₉

#489

yes

characteristic

101

Dic₇₉

—

102

#10363

—

maximal, Sylow

103

C₂×C₇₉

—

104

C₂×C₇₉

—

105

C₂×C₇₉

#2024

yes

characteristic

106

C₂×C₇₉

#2031

yes

—

107

#2862

D₇₉

yes

characteristic

108

#1994

—

109

#1518

—

110

#1286

—

111

#1276

—

112

C₇₉

#10363

yes

characteristic, complemented, Sylow

113

#533

—

114

C₂×C₄⋊Q₈

—

115

#431

—

116

#423

—

117

#391

—

118

#382

—

119

#382

C₂×D₇₉

yes

—

120

C₄³

C₂×D₇₉

yes

characteristic

121

#339

—

158

—

122

#339

—

158

—

123

#328

—

158

—

124

C₂²×Q₈

—

125

C₂×M₁₆

—

158

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

#125

—

158

—

130

C₂²⋊Q₈

—

158

—

131

C₄×Q₈

—

158

—

132

#117

—

158

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

137

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

138

C₂×C₄²

C₂²×D₇₉

yes

characteristic

139

C₄⋊C₈

—

158

—

140

C₄⋊C₈

—

158

—

141

C₂²⋊C₈

—

158

—

142

C₄×C₈

—

158

—

143

#95

—

144

#95

—

145

#95

C₂²⋊D₇₉

yes

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

158

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

153

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

154

M₁₆

—

316

—

155

M₁₆

—

316

—

156

C₂×C₈

—

316

—

157

C₂×C₈

—

316

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

316

—

162

C₄⋊C₄

—

158

—

163

C₄⋊C₄

—

158

—

164

C₂²⋊C₄

—

316

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

316

—

169

C₄²

#11929612

yes

—

170

C₄²

#11929614

yes

—

171

C₄²

Q₈×D₇₉

yes

characteristic

172

C₄²

#11929625

yes

characteristic

173

C₄²

#11929625

yes

—

174

C₂³

#14579130

yes

characteristic

175

Q₈

—

176

Q₈

—

632

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

316

—

185

C₂×C₄

#14579137

yes

—

186

C₂×C₄

#14579137

yes

characteristic

187

C₂×C₄

#14579150

yes

—

188

C₂×C₄

#14579150

yes

characteristic

189

C₂×C₄

C₂×#11929612

yes

characteristic

190

C₂×C₄

C₂×#11929614

yes

characteristic

191

C₂×C₄

#14579187

yes

characteristic

192

C₂×C₄

#14579192

yes

characteristic

193

C₈

—

632

—

194

C₈

—

632

—

195

C₂²

—

196

C₂²

#19098474

yes

characteristic

197

C₂²

#19098714

yes

characteristic

198

C₂²

#19098719

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

632

—

205

C₄

#19098845

yes

characteristic

206

C₄

#19098855

yes

characteristic

207

C₄

#19098915

yes

characteristic

208

C₄

#19098925

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#23764538

yes

characteristic

212

C₂

#23764569

yes

—

	subgroup	G/H
center	C₂²	#19098474
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2862	C₂
socle	C₂²×C₇₉	#393
prime radical	C₂³×C₇₉	#120
hypercenter	#2862	D₇₉
hypocenter	C₇₉	#10363
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

subgroup

G/H

center

C₂²

#19098474

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2862

C₂

C₂²×C₇₉

C₂³×C₇₉

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2862

N	H ≅ G/N	type
18. #14579161	Q₈	semidirect
112. C₇₉	#10363	semidirect

H ≅ G/N

type

18. #14579161

Q₈

semidirect

112. C₇₉

#10363

semidirect

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

subgroup	index
#23764377	2
#23761963	2
#23761258	2
#23761233	2
C₇₉×#2862	2
#10363	79

subgroup

index

C₇₉×#2862

Subgroups — #34191359

Conjugacy classes of subgroups

Complements

Maximal subgroups