Group #34193340 — subgroups

Subgroups — #34193340

Order 20224.

494 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 145 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	#23764206	C₂	1	1	yes	maximal, characteristic
2	#23764205	C₂	1	2	yes	maximal
3	#23763527	C₂	1	2	yes	maximal, complemented
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579158	C₂²	1	1	yes	characteristic
7	C₂×#14579083	—	2	2	no	—
8	C₂×#14579081	—	2	4	no	—
9	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
10	#19098433	C₂²	1	1	yes	characteristic
11	#19098325	—	2	4	no	—
12	C₂×#106×C₇₉	—	2	2	no	—
13	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
14	#384×C₇₉	C₂²	1	2	yes	—
15	C₂²×D₃₁₆	—	2	2	no	—
16	C₂²×D₃₁₆	—	2	1	no	—
17	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
18	#14579158	—	2	2	no	—
19	#14579158	—	2	1	no	—
20	#14579158	Q₈	1	2	yes	—
21	C₂×C₄⋊Dic₇₉	—	2	1	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	#14579083	—	4	4	no	—
24	#14579081	—	4	8	no	—
25	C₄×C₇₉⋊C₈	—	2	2	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₇₉	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
32	#106×C₇₉	—	4	4	no	—
33	C₄×C₈×C₇₉	—	2	2	no	—
34	C₂³×D₇₉	—	4	1	no	—
35	C₂×D₃₁₆	—	2	2	no	—
36	C₂×D₃₁₆	—	2	4	no	—
37	C₂×D₃₁₆	—	4	1	no	—
38	C₂×D₃₁₆	—	4	2	no	—
39	C₂×D₃₁₆	—	4	2	no	—
40	C₂×D₃₁₆	—	4	4	no	—
41	C₂×D₃₁₆	—	4	4	no	—
42	C₂×D₃₁₆	—	4	2	no	—
43	C₂²×Dic₇₉	—	4	1	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	—	2	4	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₄⋊Dic₇₉	—	4	2	no	—
48	C₄⋊Dic₇₉	—	4	4	no	—
49	C₂²×C₄×C₇₉	—	4	2	no	—
50	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₇₉	—	2	2	no	—
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
55	C₄⋊C₄×C₇₉	—	4	4	no	—
56	C₄⋊C₄×C₇₉	—	4	4	no	—
57	C₄⋊C₄×C₇₉	—	4	4	no	—
58	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₉	#41	1	2	yes	—
60	C₄²×C₇₉	#41	1	1	yes	characteristic
61	C₂³×C₇₉	#127	1	1	yes	characteristic
62	C₂²×D₇₉	—	4	2	no	—
63	C₂²×D₇₉	—	8	1	no	—
64	C₂²×D₇₉	—	8	2	no	—
65	C₂²×D₇₉	—	8	2	no	—
66	C₂²×D₇₉	—	8	1	no	—
67	C₂×C₄×C₇₉	—	8	2	no	—
68	C₂×C₄×C₇₉	—	8	2	no	—
69	C₂×C₄×C₇₉	—	8	2	no	—
70	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
72	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
73	C₂×C₄×C₇₉	#123	1	2	yes	—
74	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
75	C₂×C₄×C₇₉	#135	1	4	yes	—
76	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
78	C₂×Dic₇₉	—	8	1	no	—
79	C₂×Dic₇₉	—	8	2	no	—
80	D₃₁₆	—	4	4	no	—
81	D₃₁₆	—	4	8	no	—
82	D₃₁₆	—	8	2	no	—
83	D₃₁₆	—	8	4	no	—
84	C₈×C₇₉	—	2	4	no	—
85	C₇₉⋊C₈	—	4	4	no	—
86	C₂²×C₇₉	#399	1	1	yes	characteristic
87	C₂²×C₇₉	#492	1	1	yes	characteristic
88	C₂²×C₇₉	#495	1	2	yes	—
89	C₂²×C₇₉	#496	1	2	yes	—
90	C₂²×C₇₉	#496	1	1	yes	characteristic
91	C₂×D₇₉	—	8	4	no	—
92	C₂×D₇₉	—	8	2	no	—
93	C₂×D₇₉	—	16	1	no	—
94	C₂×D₇₉	—	16	1	no	—
95	C₂×D₇₉	—	16	2	no	—
96	C₄×C₇₉	—	16	2	no	—
97	C₄×C₇₉	#476	1	2	yes	—
98	C₄×C₇₉	#478	1	4	yes	—
99	C₄×C₇₉	#481	1	2	yes	—
100	C₄×C₇₉	#482	1	4	yes	—
101	Dic₇₉	—	16	1	no	—
102	#10677	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	#1708	1	1	yes	characteristic
104	C₂×C₇₉	#1715	1	1	yes	characteristic
105	C₂×C₇₉	#1717	1	2	yes	—
106	C₂×C₇₉	#2092	1	2	yes	—
107	C₂×C₇₉	#2094	1	1	yes	characteristic
108	D₇₉	—	16	2	no	—
109	#1957	—	79	1	no	—
110	#1957	—	79	2	no	—
111	#1844	D₇₉	1	1	yes	characteristic, complemented
112	#1834	—	79	2	no	—
113	#1273	—	79	1	no	—
114	C₇₉	#10677	1	1	yes	characteristic, complemented, Sylow
115	C₂×#127	—	79	1	no	—
116	C₂×#106	—	2	2	no	—
117	C₂×#102	—	158	4	no	—
118	C₂×#102	—	158	2	no	—
119	C₂×C₄×C₈	—	79	2	no	—
120	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
121	#384	—	79	1	no	—
122	#384	C₂×D₇₉	1	2	yes	—
123	#340	—	158	4	no	—
124	C₂²×D₄	—	158	2	no	—
125	C₂²×D₄	—	158	1	no	—
126	C₂²×C₈	—	79	4	no	—
127	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
128	#127	—	79	2	no	—
129	#127	—	158	2	no	—
130	#127	—	158	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	158	2	no	—
135	C₂×C₄⋊C₄	—	158	1	no	—
136	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
137	#106	—	4	4	no	—
138	#102	—	316	8	no	—
139	#102	—	316	4	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	158	2	no	—
142	C₄×C₈	—	158	2	no	—
143	C₂⁴	—	316	1	no	—
144	C₂×D₄	—	316	1	no	—
145	C₂×D₄	—	316	2	no	—
146	C₂×D₄	—	316	2	no	—
147	C₂×D₄	—	316	4	no	—
148	C₂×D₄	—	316	2	no	—
149	C₂×D₄	—	316	4	no	—
150	C₂×D₄	—	158	2	no	—
151	C₂×D₄	—	158	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	—	316	1	no	—
154	C₂²×C₄	D₄×D₇₉	1	2	yes	—
155	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	158	4	no	—
159	C₂×C₈	—	158	4	no	—
160	C₂×C₈	—	158	4	no	—
161	C₂×C₈	#11929611	1	4	yes	—
162	C₄⋊C₄	—	4	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	316	2	no	—
166	C₄⋊C₄	—	316	4	no	—
167	C₄²	Q₈×D₇₉	1	1	yes	characteristic
168	C₄²	#11929624	1	2	yes	—
169	C₄²	#11929625	1	1	yes	characteristic
170	C₂³	—	316	2	no	—
171	C₂³	—	632	1	no	—
172	C₂³	—	632	2	no	—
173	C₂³	—	632	2	no	—
174	C₂³	—	632	1	no	—
175	C₂³	#14579157	1	1	yes	characteristic
176	D₄	—	316	4	no	—
177	D₄	—	316	8	no	—
178	D₄	—	632	2	no	—
179	D₄	—	632	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	632	1	no	—
184	C₂×C₄	—	632	2	no	—
185	C₂×C₄	#14579137	1	1	yes	characteristic
186	C₂×C₄	#14579139	1	2	yes	—
187	C₂×C₄	#14579144	1	1	yes	characteristic
188	C₂×C₄	#14579146	1	2	yes	—
189	C₂×C₄	C₂×#11929611	1	2	yes	—
190	C₂×C₄	#14579182	1	4	yes	—
191	C₂×C₄	#14579187	1	2	yes	—
192	C₂×C₄	#14579188	1	4	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	316	4	no	—
195	C₂²	—	632	4	no	—
196	C₂²	—	632	2	no	—
197	C₂²	—	1264	1	no	—
198	C₂²	—	1264	1	no	—
199	C₂²	—	1264	2	no	—
200	C₂²	#19098498	1	1	yes	characteristic
201	C₂²	#19098937	1	1	yes	characteristic
202	C₂²	#19098944	1	2	yes	—
203	C₂²	#19098949	1	2	yes	—
204	C₂²	#19098950	1	1	yes	characteristic
205	C₄	—	16	2	no	—
206	C₄	—	1264	1	no	—
207	C₄	#19098850	1	2	yes	—
208	C₄	#19098862	1	4	yes	—
209	C₄	#19098881	1	2	yes	—
210	C₄	#19098889	1	4	yes	—
211	C₂	—	1264	2	no	—
212	C₂	#23763002	1	1	yes	characteristic
213	C₂	#23763029	1	1	yes	characteristic
214	C₂	#23763037	1	2	yes	—
215	C₂	#23764922	1	2	yes	—
216	C₂	#23764934	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764206

C₂

yes

maximal, characteristic

#23764205

C₂

yes

maximal

#23763527

C₂

yes

maximal, complemented

C₇₉⋊#1273

C₂

yes

maximal, characteristic, complemented

C₇₉×#1844

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

characteristic

C₂×#14579083

—

C₂×#14579081

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098433

C₂²

yes

characteristic

#19098325

—

C₂×#106×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

C₂²

yes

—

C₂²×D₃₁₆

—

C₂²×D₃₁₆

—

C₂²×C₇₉⋊C₈

D₄

yes

—

#14579158

—

#14579158

—

#14579158

Q₈

yes

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

—

#14579083

—

#14579081

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#106×C₇₉

—

C₄×C₈×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₄²×C₇₉

#41

yes

—

C₄²×C₇₉

yes

C₂³×C₇₉

yes

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₇₉

#123

yes

characteristic

C₂×C₄×C₇₉

#123

yes

—

C₂×C₄×C₇₉

C₂×SD₁₆

yes

—

C₂×C₄×C₇₉

#135

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#496

yes

—

C₂²×C₇₉

#496

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#476

yes

—

C₄×C₇₉

#478

yes

—

C₄×C₇₉

#481

yes

—

100

C₄×C₇₉

#482

yes

—

101

Dic₇₉

—

102

#10677

—

maximal, Sylow

103

C₂×C₇₉

#1708

yes

characteristic

104

C₂×C₇₉

#1715

yes

characteristic

105

C₂×C₇₉

#1717

yes

—

106

C₂×C₇₉

#2092

yes

—

107

C₂×C₇₉

#2094

yes

characteristic

108

D₇₉

—

109

#1957

—

110

#1957

—

111

#1844

D₇₉

yes

characteristic, complemented

112

#1834

—

113

#1273

—

114

C₇₉

#10677

yes

characteristic, complemented, Sylow

115

C₂×#127

—

116

C₂×#106

—

117

C₂×#102

—

158

—

118

C₂×#102

—

158

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

121

#384

—

122

#384

C₂×D₇₉

yes

—

123

#340

—

158

—

124

C₂²×D₄

—

158

—

125

C₂²×D₄

—

158

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²⋊D₇₉

yes

—

128

#127

—

129

#127

—

158

—

130

#127

—

158

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

158

—

135

C₂×C₄⋊C₄

—

158

—

136

C₂×C₄²

C₂²×D₇₉

yes

characteristic

137

#106

—

138

#102

—

316

—

139

#102

—

316

—

140

C₄×C₈

—

141

C₄×C₈

—

158

—

142

C₄×C₈

—

158

—

143

C₂⁴

—

316

—

144

C₂×D₄

—

316

—

145

C₂×D₄

—

316

—

146

C₂×D₄

—

316

—

147

C₂×D₄

—

316

—

148

C₂×D₄

—

316

—

149

C₂×D₄

—

316

—

150

C₂×D₄

—

158

—

151

C₂×D₄

—

158

—

152

C₂²×C₄

—

153

C₂²×C₄

—

316

—

154

C₂²×C₄

D₄×D₇₉

yes

—

155

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

158

—

159

C₂×C₈

—

158

—

160

C₂×C₈

—

158

—

161

C₂×C₈

#11929611

yes

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

316

—

166

C₄⋊C₄

—

316

—

167

C₄²

Q₈×D₇₉

yes

characteristic

168

C₄²

#11929624

yes

—

169

C₄²

#11929625

yes

characteristic

170

C₂³

—

316

—

171

C₂³

—

632

—

172

C₂³

—

632

—

173

C₂³

—

632

—

174

C₂³

—

632

—

175

C₂³

#14579157

yes

characteristic

176

D₄

—

316

—

177

D₄

—

316

—

178

D₄

—

632

—

179

D₄

—

632

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

632

—

184

C₂×C₄

—

632

—

185

C₂×C₄

#14579137

yes

characteristic

186

C₂×C₄

#14579139

yes

—

187

C₂×C₄

#14579144

yes

characteristic

188

C₂×C₄

#14579146

yes

—

189

C₂×C₄

C₂×#11929611

yes

—

190

C₂×C₄

#14579182

yes

—

191

C₂×C₄

#14579187

yes

—

192

C₂×C₄

#14579188

yes

—

193

C₈

—

194

C₈

—

316

—

195

C₂²

—

632

—

196

C₂²

—

632

—

197

C₂²

—

1264

—

198

C₂²

—

1264

—

199

C₂²

—

1264

—

200

C₂²

#19098498

yes

characteristic

201

C₂²

#19098937

yes

characteristic

202

C₂²

#19098944

yes

—

203

C₂²

#19098949

yes

—

204

C₂²

#19098950

yes

characteristic

205

C₄

—

206

C₄

—

1264

—

207

C₄

#19098850

yes

—

208

C₄

#19098862

yes

—

209

C₄

#19098881

yes

—

210

C₄

#19098889

yes

—

211

C₂

—

1264

—

212

C₂

#23763002

yes

characteristic

213

C₂

#23763029

yes

characteristic

214

C₂

#23763037

yes

—

215

C₂

#23764922

yes

—

216

C₂

#23764934

yes

characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1844	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#14579158	C₂²
hypercenter	#1844	D₇₉
hypocenter	C₇₉	#10677
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14579157

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1844

C₂

C₂³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
3. #23763527	C₂	semidirect
4. C₇₉⋊#1273	C₂	semidirect
5. C₇₉×#1844	C₂	semidirect
111. #1844	D₇₉	semidirect
114. C₇₉	#10677	semidirect

H ≅ G/N

type

3. #23763527

C₂

semidirect

4. C₇₉⋊#1273

C₂

semidirect

5. C₇₉×#1844

C₂

semidirect

111. #1844

D₇₉

semidirect

114. C₇₉

#10677

semidirect

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

subgroup	index
#23764206	2
#23764205	2
#23763527	2
C₇₉⋊#1273	2
C₇₉×#1844	2
#10677	79

subgroup

index

#23764206

#23764205

#23763527

C₇₉⋊#1273

C₇₉×#1844

#10677

Subgroups — #34193340

Conjugacy classes of subgroups

Complements

Maximal subgroups