Group #34181259 — subgroups

Subgroups — #34181259

Order 20224.

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

normal subgroups only

This group has 388 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	#23772112	C₂	1	1	yes	maximal, characteristic
2	#23762161	C₂	1	2	yes	maximal, complemented
3	#23762035	C₂	1	1	yes	maximal, characteristic
4	#23761310	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1536	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579168	C₄	1	2	yes	—
7	#19098641	—	2	2	no	—
8	D₄×C₇₉⋊C₈	—	2	2	no	—
9	#19098620	C₂²	1	2	yes	—
10	#19098587	C₂²	1	1	yes	characteristic
11	#19098530	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#19098315	—	2	2	no	—
14	#19098313	—	2	2	no	—
15	#19098303	—	2	2	no	—
16	#532×C₇₉	C₂²	1	1	yes	characteristic
17	#432×C₇₉	C₂²	1	2	yes	—
18	#329×C₇₉	—	2	2	no	—
19	#14579168	—	4	4	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	C₂×C₄	1	2	yes	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
27	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
28	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
29	#14579072	—	2	2	no	—
30	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
31	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
32	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₇₉	Q₈	1	1	yes	characteristic
34	#127×C₇₉	Q₈	1	1	yes	characteristic
35	#124×C₇₉	D₄	1	2	yes	—
36	#121×C₇₉	—	2	2	no	—
37	C₄×D₄×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₉	—	2	2	no	—
40	C₂²⋊C₈×C₇₉	—	2	2	no	—
41	#97×C₇₉	—	2	2	no	—
42	#11929608	—	4	2	no	—
43	#11929608	—	4	4	no	—
44	C₂×C₇₉⋊C₈	—	2	2	no	—
45	C₂×C₇₉⋊C₈	—	2	2	no	—
46	C₂×C₇₉⋊C₈	—	4	1	no	—
47	C₂×C₇₉⋊C₈	—	4	4	no	—
48	C₂×C₇₉⋊C₈	—	4	2	no	—
49	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
50	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
52	#41×C₇₉	—	4	2	no	—
53	#41×C₇₉	—	4	2	no	—
54	C₂×Q₈×C₇₉	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₇₉	—	2	2	no	—
56	C₂×D₄×C₇₉	—	2	2	no	—
57	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₉	—	2	2	no	—
59	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₉	#41	1	2	yes	—
61	C₂×C₈×C₇₉	—	4	2	no	—
62	C₄⋊C₄×C₇₉	—	2	2	no	—
63	C₂²⋊C₄×C₇₉	—	2	4	no	—
64	C₄²×C₇₉	C₂×D₄	1	2	yes	—
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
67	C₂³×C₇₉	—	2	2	no	—
68	C₂³×C₇₉	#117	1	1	yes	characteristic
69	Q₈×C₇₉	—	4	2	no	—
70	D₄×C₇₉	—	4	2	no	—
71	D₄×C₇₉	—	4	4	no	—
72	D₄×C₇₉	—	4	4	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	2	1	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	4	2	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	—	4	2	no	—
79	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#121	1	2	yes	—
82	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₇₉	—	8	2	no	—
84	C₇₉⋊C₈	—	2	1	no	—
85	C₇₉⋊C₈	—	2	2	no	—
86	C₇₉⋊C₈	—	4	2	no	—
87	C₇₉⋊C₈	—	8	2	no	—
88	C₇₉⋊C₈	#100	1	2	yes	—
89	C₂²×C₇₉	—	2	1	no	—
90	C₂²×C₇₉	—	2	2	no	—
91	C₂²×C₇₉	—	4	2	no	—
92	C₂²×C₇₉	—	4	4	no	—
93	C₂²×C₇₉	#380	1	1	yes	characteristic
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	2	no	—
96	C₄×C₇₉	—	2	2	no	—
97	C₄×C₇₉	—	4	2	no	—
98	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
99	C₄×C₇₉	#413	1	1	yes	characteristic
100	#8630	—	79	1	no	maximal, Sylow
101	C₂×C₇₉	—	4	1	no	—
102	C₂×C₇₉	—	8	2	no	—
103	C₂×C₇₉	#1800	1	1	yes	characteristic
104	C₂×C₇₉	#1814	1	2	yes	—
105	#2931	—	79	1	no	—
106	#1568	—	79	2	no	—
107	#1536	—	79	1	no	—
108	#1536	D₇₉	1	1	yes	characteristic
109	#1307	—	79	2	no	—
110	C₇₉	#8630	1	1	yes	characteristic, complemented, Sylow
111	C₂×#131	—	79	2	no	—
112	#532	C₂×D₇₉	1	1	yes	characteristic
113	#436	—	158	2	no	—
114	C₈×D₄	—	158	2	no	—
115	#432	—	79	2	no	—
116	#432	C₂×D₇₉	1	2	yes	—
117	#423	—	79	1	no	—
118	#408	—	79	2	no	—
119	C₂×C₄×C₈	—	79	1	no	—
120	#335	—	158	2	no	—
121	#334	—	158	2	no	—
122	#329	—	2	2	no	—
123	#329	—	158	2	no	—
124	C₂×#41	C₄×D₇₉	1	2	yes	—
125	#131	—	316	4	no	—
126	C₂×M₁₆	—	79	2	no	—
127	C₂×M₁₆	—	158	2	no	—
128	C₂²×C₈	—	79	1	no	—
129	C₂²×C₈	—	79	1	no	—
130	C₂²×C₈	—	158	2	no	—
131	C₄⋊Q₈	Dic₁₅₈	1	1	yes	characteristic
132	#127	Dic₁₅₈	1	1	yes	characteristic
133	#124	C₂²⋊D₇₉	1	2	yes	—
134	#121	—	2	2	no	—
135	C₄×D₄	—	2	2	no	—
136	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	158	2	no	—
139	C₄⋊C₈	—	158	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	158	2	no	—
142	C₂²⋊C₈	—	158	4	no	—
143	#97	—	2	2	no	—
144	#97	—	158	2	no	—
145	C₄×C₈	—	79	2	no	—
146	C₄×C₈	—	79	2	no	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	Dic₇₉⋊C₄	1	2	yes	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	Dic₇₉⋊C₄	1	2	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
155	C₂²×C₄	#11929623	1	2	yes	—
156	M₁₆	—	316	4	no	—
157	M₁₆	—	316	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	316	4	no	—
160	C₂×C₈	—	316	2	no	—
161	C₂×C₈	—	316	1	no	—
162	C₂×C₈	—	79	1	no	—
163	C₂×C₈	—	79	2	no	—
164	C₂×C₈	—	79	1	no	—
165	C₂×C₈	—	158	2	no	—
166	C₂×C₈	—	158	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₄²	D₄×D₇₉	1	2	yes	—
170	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
171	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
172	C₂³	—	2	2	no	—
173	C₂³	#14579117	1	1	yes	characteristic
174	Q₈	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
185	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
186	C₂×C₄	#14579133	1	2	yes	—
187	C₂×C₄	#14579138	1	2	yes	—
188	C₈	—	8	2	no	—
189	C₈	—	316	2	no	—
190	C₈	—	79	2	no	—
191	C₈	—	632	2	no	—
192	C₈	—	158	1	no	—
193	C₈	—	158	2	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	4	no	—
198	C₂²	#19098421	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	#100×D₇₉	1	1	yes	characteristic
204	C₄	#19098544	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	2	no	—
207	C₂	#23763390	1	1	yes	characteristic
208	C₂	#23763452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772112

C₂

yes

maximal, characteristic

#23762161

C₂

yes

maximal, complemented

#23762035

C₂

yes

maximal, characteristic

#23761310

C₂

yes

maximal, complemented

C₇₉×#1536

C₂

yes

maximal, characteristic

C₂×#14579168

C₄

yes

—

#19098641

—

D₄×C₇₉⋊C₈

—

#19098620

C₂²

yes

—

#19098587

C₂²

yes

characteristic

#19098530

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

—

#532×C₇₉

C₂²

yes

characteristic

#432×C₇₉

C₂²

yes

—

#329×C₇₉

—

#14579168

—

C₂×#11929608

—

C₂×#11929608

C₂×C₄

yes

—

C₂²×C₇₉⋊C₈

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×#41×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

Q₈

yes

characteristic

#127×C₇₉

Q₈

yes

characteristic

#124×C₇₉

D₄

yes

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

#97×C₇₉

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₄⋊C₄

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

#117

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#100

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#380

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

100

—

101

C₂×C₇₉

—

102

C₂×C₇₉

—

103

C₂×C₇₉

#1800

yes

characteristic

104

C₂×C₇₉

#1814

yes

—

105

#2931

—

106

#1568

—

107

#1536

—

108

#1536

D₇₉

yes

characteristic

109

#1307

—

110

C₇₉

#8630

yes

characteristic, complemented, Sylow

111

C₂×#131

—

112

#532

C₂×D₇₉

yes

characteristic

113

#436

—

158

—

114

C₈×D₄

—

158

—

115

#432

—

116

#432

C₂×D₇₉

yes

—

117

#423

—

118

#408

—

119

C₂×C₄×C₈

—

120

#335

—

158

—

121

#334

—

158

—

122

#329

—

123

#329

—

158

—

124

C₂×#41

C₄×D₇₉

yes

—

125

#131

—

316

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

158

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

—

158

—

131

C₄⋊Q₈

Dic₁₅₈

yes

characteristic

132

#127

Dic₁₅₈

yes

characteristic

133

#124

C₂²⋊D₇₉

yes

—

134

#121

—

135

C₄×D₄

—

136

C₂×C₄²

C₂²×D₇₉

yes

characteristic

137

C₄⋊C₈

—

138

C₄⋊C₈

—

158

—

139

C₄⋊C₈

—

158

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

158

—

142

C₂²⋊C₈

—

158

—

143

#97

—

144

#97

—

158

—

145

C₄×C₈

—

146

C₄×C₈

—

147

#41

—

148

#41

—

149

C₂×Q₈

Dic₇₉⋊C₄

yes

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

Dic₇₉⋊C₄

yes

—

153

C₂²×C₄

—

154

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

155

C₂²×C₄

#11929623

yes

—

156

M₁₆

—

316

—

157

M₁₆

—

316

—

158

C₂×C₈

—

159

C₂×C₈

—

316

—

160

C₂×C₈

—

316

—

161

C₂×C₈

—

316

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

158

—

166

C₂×C₈

—

158

—

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

D₄×D₇₉

yes

—

170

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

171

C₄²

C₂×Dic₁₅₈

yes

characteristic

172

C₂³

—

173

C₂³

#14579117

yes

characteristic

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

C₂²⋊C₄×D₇₉

yes

characteristic

185

C₂×C₄

C₂×Dic₇₉⋊C₄

yes

characteristic

186

C₂×C₄

#14579133

yes

—

187

C₂×C₄

#14579138

yes

—

188

C₈

—

189

C₈

—

316

—

190

C₈

—

191

C₈

—

632

—

192

C₈

—

158

—

193

C₈

—

158

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#19098421

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

#100×D₇₉

yes

characteristic

204

C₄

#19098544

yes

characteristic

205

C₂

—

206

C₂

—

207

C₂

#23763390

yes

characteristic

208

C₂

#23763452

yes

—

	subgroup	G/H
center	C₂²	#19098421
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1536	C₂
socle	C₂²×C₇₉	#380
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1536	D₇₉
hypocenter	C₇₉	#8630
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

subgroup

G/H

center

C₂²

#19098421

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1536

C₂

socle

C₂²×C₇₉

#380

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1536

N	H ≅ G/N	type
2. #23762161	C₂	semidirect
4. #23761310	C₂	semidirect
110. C₇₉	#8630	semidirect

H ≅ G/N

type

2. #23762161

C₂

semidirect

4. #23761310

C₂

semidirect

110. C₇₉

#8630

semidirect

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

subgroup	index
#23772112	2
#23762161	2
#23762035	2
#23761310	2
C₇₉×#1536	2
#8630	79

subgroup

index

C₇₉×#1536

Subgroups — #34181259

Conjugacy classes of subgroups

Complements

Maximal subgroups