Group #34193317 — subgroups

Subgroups — #34193317

Order 20224.

526 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 153 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	#23763559	C₂	1	2	yes	maximal, complemented
4	C₇₉⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579158	C₂²	1	1	yes	characteristic
7	C₂×#14579094	—	2	4	no	—
8	C₂×#14579083	—	2	2	no	—
9	C₂×#14579081	—	2	4	no	—
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
11	#19098433	C₂²	1	1	yes	characteristic
12	C₂×#107×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	#14579094	—	4	8	no	—
24	#14579083	—	4	4	no	—
25	#14579081	—	4	8	no	—
26	C₄×C₇₉⋊C₈	—	2	2	no	—
27	C₄×C₇₉⋊C₈	—	2	2	no	—
28	C₂²×C₈×C₇₉	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	#107×C₇₉	—	4	4	no	—
34	C₄×C₈×C₇₉	—	2	2	no	—
35	C₂³×D₇₉	—	4	1	no	—
36	C₂×D₃₁₆	—	2	2	no	—
37	C₂×D₃₁₆	—	2	4	no	—
38	C₂×D₃₁₆	—	4	1	no	—
39	C₂×D₃₁₆	—	4	2	no	—
40	C₂×D₃₁₆	—	4	4	no	—
41	C₂×D₃₁₆	—	4	2	no	—
42	C₂×D₃₁₆	—	4	2	no	—
43	C₂×D₃₁₆	—	4	4	no	—
44	C₂²×Dic₇₉	—	4	1	no	—
45	C₂×C₇₉⋊C₈	—	2	4	no	—
46	C₂×C₇₉⋊C₈	—	2	4	no	—
47	C₂×C₇₉⋊C₈	D₈	1	8	yes	—
48	C₄⋊Dic₇₉	—	4	2	no	—
49	C₄⋊Dic₇₉	—	4	4	no	—
50	C₂²×C₄×C₇₉	—	4	2	no	—
51	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₉	—	2	2	no	—
54	C₂×C₈×C₇₉	—	2	2	no	—
55	C₂×C₈×C₇₉	D₈	1	4	yes	—
56	C₄⋊C₄×C₇₉	—	4	4	no	—
57	C₄⋊C₄×C₇₉	—	4	4	no	—
58	C₄⋊C₄×C₇₉	—	4	4	no	—
59	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₇₉	#41	1	2	yes	—
61	C₄²×C₇₉	#41	1	1	yes	characteristic
62	C₂³×C₇₉	#127	1	1	yes	characteristic
63	C₂²×D₇₉	—	4	2	no	—
64	C₂²×D₇₉	—	8	2	no	—
65	C₂²×D₇₉	—	8	1	no	—
66	C₂²×D₇₉	—	8	1	no	—
67	C₂²×D₇₉	—	8	2	no	—
68	C₂×C₄×C₇₉	—	8	2	no	—
69	C₂×C₄×C₇₉	—	8	2	no	—
70	C₂×C₄×C₇₉	—	8	2	no	—
71	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
74	C₂×C₄×C₇₉	#123	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₇₉	#137	1	2	yes	—
78	C₂×C₄×C₇₉	#137	1	4	yes	—
79	C₂×Dic₇₉	—	8	2	no	—
80	C₂×Dic₇₉	—	8	1	no	—
81	D₃₁₆	—	4	4	no	—
82	D₃₁₆	—	4	8	no	—
83	D₃₁₆	—	8	2	no	—
84	D₃₁₆	—	8	4	no	—
85	C₈×C₇₉	—	2	4	no	—
86	C₇₉⋊C₈	—	2	8	no	—
87	C₂²×C₇₉	#399	1	1	yes	characteristic
88	C₂²×C₇₉	#493	1	2	yes	—
89	C₂²×C₇₉	#493	1	1	yes	characteristic
90	C₂²×C₇₉	#497	1	1	yes	characteristic
91	C₂²×C₇₉	#497	1	2	yes	—
92	C₂×D₇₉	—	8	4	no	—
93	C₂×D₇₉	—	8	2	no	—
94	C₂×D₇₉	—	16	1	no	—
95	C₂×D₇₉	—	16	2	no	—
96	C₂×D₇₉	—	16	1	no	—
97	C₄×C₇₉	—	16	2	no	—
98	C₄×C₇₉	#474	1	2	yes	—
99	C₄×C₇₉	#474	1	4	yes	—
100	C₄×C₇₉	#480	1	2	yes	—
101	C₄×C₇₉	#480	1	4	yes	—
102	Dic₇₉	—	16	1	no	—
103	#10673	—	79	1	no	maximal, Sylow
104	C₂×C₇₉	#1712	1	1	yes	characteristic
105	C₂×C₇₉	#1722	1	2	yes	—
106	C₂×C₇₉	#1722	1	1	yes	characteristic
107	C₂×C₇₉	#2093	1	2	yes	—
108	C₂×C₇₉	#2093	1	1	yes	characteristic
109	D₇₉	—	16	2	no	—
110	#1957	—	79	1	no	—
111	#1957	—	79	2	no	—
112	#1845	—	79	2	no	—
113	#1845	D₇₉	1	1	yes	characteristic, complemented
114	#1273	—	79	1	no	—
115	C₇₉	#10673	1	1	yes	characteristic, complemented, Sylow
116	C₂×#127	—	79	1	no	—
117	C₂×#107	—	2	2	no	—
118	C₂×#107	—	158	4	no	—
119	C₂×#102	—	158	4	no	—
120	C₂×#102	—	158	2	no	—
121	C₂×C₄×C₈	—	79	2	no	—
122	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
123	#384	—	79	1	no	—
124	#384	C₂×D₇₉	1	2	yes	—
125	C₂²×D₄	—	158	2	no	—
126	C₂²×D₄	—	158	1	no	—
127	C₂²×C₈	—	79	4	no	—
128	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
129	#127	—	79	2	no	—
130	#127	—	158	2	no	—
131	#127	—	158	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	158	2	no	—
136	C₂×C₄⋊C₄	—	158	1	no	—
137	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
138	#107	—	4	4	no	—
139	#107	—	316	8	no	—
140	#102	—	316	8	no	—
141	#102	—	316	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	158	2	no	—
144	C₄×C₈	—	158	2	no	—
145	C₂⁴	—	316	1	no	—
146	C₂×D₄	—	316	1	no	—
147	C₂×D₄	—	316	2	no	—
148	C₂×D₄	—	316	4	no	—
149	C₂×D₄	—	316	2	no	—
150	C₂×D₄	—	316	2	no	—
151	C₂×D₄	—	316	4	no	—
152	C₂×D₄	—	158	2	no	—
153	C₂×D₄	—	158	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	316	1	no	—
156	C₂²×C₄	D₄×D₇₉	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	79	8	no	—
161	C₂×C₈	—	158	4	no	—
162	C₂×C₈	—	158	4	no	—
163	C₂×C₈	#11929609	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	316	4	no	—
168	C₄⋊C₄	—	316	2	no	—
169	C₄²	Q₈×D₇₉	1	1	yes	characteristic
170	C₄²	#11929624	1	2	yes	—
171	C₄²	#11929625	1	1	yes	characteristic
172	C₂³	—	316	2	no	—
173	C₂³	—	632	2	no	—
174	C₂³	—	632	1	no	—
175	C₂³	—	632	1	no	—
176	C₂³	—	632	2	no	—
177	C₂³	#14579157	1	1	yes	characteristic
178	D₄	—	316	4	no	—
179	D₄	—	316	8	no	—
180	D₄	—	632	2	no	—
181	D₄	—	632	4	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	632	2	no	—
186	C₂×C₄	—	632	1	no	—
187	C₂×C₄	#14579137	1	1	yes	characteristic
188	C₂×C₄	#14579139	1	2	yes	—
189	C₂×C₄	#14579144	1	1	yes	characteristic
190	C₂×C₄	#14579146	1	2	yes	—
191	C₂×C₄	D₈×D₇₉	1	4	yes	—
192	C₂×C₄	C₂×#11929609	1	2	yes	—
193	C₂×C₄	#14579192	1	2	yes	—
194	C₂×C₄	#14579193	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	158	8	no	—
197	C₂²	—	632	4	no	—
198	C₂²	—	632	2	no	—
199	C₂²	—	1264	1	no	—
200	C₂²	—	1264	2	no	—
201	C₂²	—	1264	1	no	—
202	C₂²	#19098498	1	1	yes	characteristic
203	C₂²	#19098938	1	2	yes	—
204	C₂²	#19098939	1	1	yes	characteristic
205	C₂²	#19098955	1	1	yes	characteristic
206	C₂²	#19098956	1	2	yes	—
207	C₄	—	16	2	no	—
208	C₄	—	1264	1	no	—
209	C₄	#19098840	1	2	yes	—
210	C₄	#19098842	1	4	yes	—
211	C₄	#19098876	1	2	yes	—
212	C₄	#19098878	1	4	yes	—
213	C₂	—	1264	2	no	—
214	C₂	#23763014	1	1	yes	characteristic
215	C₂	#23763070	1	2	yes	—
216	C₂	#23763072	1	1	yes	characteristic
217	C₂	#23764925	1	2	yes	—
218	C₂	#23764929	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23764206

C₂

yes

maximal, characteristic

#23764205

C₂

yes

maximal

#23763559

C₂

yes

maximal, complemented

C₇₉⋊#1273

C₂

yes

maximal, characteristic, complemented

C₇₉×#1845

C₂

yes

maximal, characteristic, complemented

C₂×#14579158

C₂²

yes

—

—

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098433

C₂²

yes

characteristic

C₂×#107×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

—

#14579094

—

#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

#107×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

D₈

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

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

D₈

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×D₈

yes

—

C₂×C₄×C₇₉

C₂×D₈

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#399

yes

characteristic

C₂²×C₇₉

#493

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#474

yes

—

C₄×C₇₉

#474

yes

—

100

C₄×C₇₉

#480

yes

—

101

C₄×C₇₉

#480

yes

—

102

Dic₇₉

—

103

#10673

—

maximal, Sylow

104

C₂×C₇₉

#1712

yes

characteristic

105

C₂×C₇₉

#1722

yes

—

106

C₂×C₇₉

#1722

yes

characteristic

107

C₂×C₇₉

#2093

yes

—

108

C₂×C₇₉

#2093

yes

characteristic

109

D₇₉

—

110

#1957

—

111

#1957

—

112

#1845

—

113

#1845

D₇₉

yes

characteristic, complemented

114

#1273

—

115

C₇₉

#10673

yes

characteristic, complemented, Sylow

116

C₂×#127

—

117

C₂×#107

—

118

C₂×#107

—

158

—

119

C₂×#102

—

158

—

120

C₂×#102

—

158

—

121

C₂×C₄×C₈

—

122

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

123

#384

—

124

#384

C₂×D₇₉

yes

—

125

C₂²×D₄

—

158

—

126

C₂²×D₄

—

158

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₇₉

yes

—

129

#127

—

130

#127

—

158

—

131

#127

—

158

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

158

—

136

C₂×C₄⋊C₄

—

158

—

137

C₂×C₄²

C₂²×D₇₉

yes

characteristic

138

#107

—

139

#107

—

316

—

140

#102

—

316

—

141

#102

—

316

—

142

C₄×C₈

—

143

C₄×C₈

—

158

—

144

C₄×C₈

—

158

—

145

C₂⁴

—

316

—

146

C₂×D₄

—

316

—

147

C₂×D₄

—

316

—

148

C₂×D₄

—

316

—

149

C₂×D₄

—

316

—

150

C₂×D₄

—

316

—

151

C₂×D₄

—

316

—

152

C₂×D₄

—

158

—

153

C₂×D₄

—

158

—

154

C₂²×C₄

—

155

C₂²×C₄

—

316

—

156

C₂²×C₄

D₄×D₇₉

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

158

—

162

C₂×C₈

—

158

—

163

C₂×C₈

#11929609

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

316

—

168

C₄⋊C₄

—

316

—

169

C₄²

Q₈×D₇₉

yes

characteristic

170

C₄²

#11929624

yes

—

171

C₄²

#11929625

yes

characteristic

172

C₂³

—

316

—

173

C₂³

—

632

—

174

C₂³

—

632

—

175

C₂³

—

632

—

176

C₂³

—

632

—

177

C₂³

#14579157

yes

characteristic

178

D₄

—

316

—

179

D₄

—

316

—

180

D₄

—

632

—

181

D₄

—

632

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

632

—

186

C₂×C₄

—

632

—

187

C₂×C₄

#14579137

yes

characteristic

188

C₂×C₄

#14579139

yes

—

189

C₂×C₄

#14579144

yes

characteristic

190

C₂×C₄

#14579146

yes

—

191

C₂×C₄

D₈×D₇₉

yes

—

192

C₂×C₄

C₂×#11929609

yes

—

193

C₂×C₄

#14579192

yes

—

194

C₂×C₄

#14579193

yes

—

195

C₈

—

196

C₈

—

158

—

197

C₂²

—

632

—

198

C₂²

—

632

—

199

C₂²

—

1264

—

200

C₂²

—

1264

—

201

C₂²

—

1264

—

202

C₂²

#19098498

yes

characteristic

203

C₂²

#19098938

yes

—

204

C₂²

#19098939

yes

characteristic

205

C₂²

#19098955

yes

characteristic

206

C₂²

#19098956

yes

—

207

C₄

—

208

C₄

—

1264

—

209

C₄

#19098840

yes

—

210

C₄

#19098842

yes

—

211

C₄

#19098876

yes

—

212

C₄

#19098878

yes

—

213

C₂

—

1264

—

214

C₂

#23763014

yes

characteristic

215

C₂

#23763070

yes

—

216

C₂

#23763072

yes

characteristic

217

C₂

#23764925

yes

—

218

C₂

#23764929

yes

characteristic

	subgroup	G/H
center	C₂³	#14579157
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle	C₂³×C₇₉	#127
prime radical	C₂×#14579158	C₂²
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#10673
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14579157

commutator, cosocle

C₂×C₄²×C₇₉

C₂³

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1845

C₂

C₂³×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
3. #23763559	C₂	semidirect
4. C₇₉⋊#1273	C₂	semidirect
5. C₇₉×#1845	C₂	semidirect
113. #1845	D₇₉	semidirect
115. C₇₉	#10673	semidirect

H ≅ G/N

type

3. #23763559

C₂

semidirect

4. C₇₉⋊#1273

C₂

semidirect

5. C₇₉×#1845

C₂

semidirect

113. #1845

D₇₉

semidirect

115. C₇₉

#10673

semidirect

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

subgroup	index
#23764206	2
#23764205	2
#23763559	2
C₇₉⋊#1273	2
C₇₉×#1845	2
#10673	79

subgroup

index

#23764206

#23764205

#23763559

C₇₉⋊#1273

C₇₉×#1845

#10673

Subgroups — #34193317

Conjugacy classes of subgroups

Complements

Maximal subgroups