Group #34183586 — subgroups

Subgroups — #34183586

Order 20224.

404 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 111 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	#23772502	C₂	1	1	yes	maximal, characteristic
2	#23761995	C₂	1	1	yes	maximal, characteristic
3	#23761821	C₂	1	2	yes	maximal
4	#23761237	C₂	1	2	yes	maximal
5	C₇₉×#3160	C₂	1	1	yes	maximal, characteristic
6	#19098694	—	2	2	no	—
7	Q₈×C₇₉⋊C₈	—	2	2	no	—
8	#19098624	—	2	2	no	—
9	#19098616	C₂²	1	2	yes	—
10	C₄×#11929608	C₂²	1	2	yes	—
11	#19098305	—	2	2	no	—
12	Q₈⋊(C₇₉⋊C₈)	—	2	2	no	—
13	Q₈⋊(C₇₉⋊C₈)	—	2	2	no	—
14	#533×C₇₉	C₂²	1	1	yes	characteristic
15	#533×C₇₉	C₂²	1	1	yes	characteristic
16	#500×C₇₉	C₄	1	2	yes	—
17	#499×C₇₉	—	2	2	no	—
18	#498×C₇₉	C₄	1	2	yes	—
19	#427×C₇₉	—	2	2	no	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#11929608	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
23	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
24	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
25	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
26	#14579072	—	2	2	no	—
27	#14579072	—	2	2	no	—
28	C₄×C₇₉⋊C₈	—	2	2	no	—
29	C₄×C₇₉⋊C₈	—	2	2	no	—
30	C₄×C₇₉⋊C₈	—	2	2	no	—
31	C₂²×C₈×C₇₉	—	2	1	no	—
32	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
34	#125×C₇₉	—	2	1	no	—
35	#125×C₇₉	D₄	1	2	yes	—
36	C₂²⋊Q₈×C₇₉	—	2	2	no	—
37	C₂²⋊Q₈×C₇₉	—	2	2	no	—
38	C₄×Q₈×C₇₉	—	2	2	no	—
39	C₄×Q₈×C₇₉	—	2	2	no	—
40	#117×C₇₉	—	2	1	no	—
41	#117×C₇₉	D₄	1	2	yes	—
42	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
43	#107×C₇₉	—	2	4	no	—
44	#106×C₇₉	—	2	4	no	—
45	C₄×C₈×C₇₉	—	2	1	no	—
46	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
48	#11929608	—	2	4	no	—
49	#11929608	—	4	2	no	—
50	C₂×C₇₉⋊C₈	—	2	2	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₂×Q₈×C₇₉	—	2	2	no	—
54	C₂×Q₈×C₇₉	—	2	2	no	—
55	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	#41	1	2	yes	—
57	C₂×C₈×C₇₉	—	2	4	no	—
58	C₂×C₈×C₇₉	—	2	1	no	—
59	C₂×C₈×C₇₉	—	2	1	no	—
60	C₄⋊C₄×C₇₉	—	2	2	no	—
61	C₄⋊C₄×C₇₉	—	2	4	no	—
62	C₄⋊C₄×C₇₉	—	2	2	no	—
63	C₄⋊C₄×C₇₉	—	2	2	no	—
64	C₄⋊C₄×C₇₉	—	4	2	no	—
65	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	4	yes	—
66	C₂²⋊C₄×C₇₉	—	2	2	no	—
67	C₂²⋊C₄×C₇₉	—	4	1	no	—
68	C₄²×C₇₉	—	2	2	no	—
69	C₄²×C₇₉	—	4	1	no	—
70	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₇₉	C₂×D₄	1	2	yes	—
73	C₂³×C₇₉	#123	1	1	yes	characteristic
74	Q₈×C₇₉	—	4	2	no	—
75	Q₈×C₇₉	—	4	2	no	—
76	Q₈×C₇₉	—	4	2	no	—
77	Q₈×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	—	2	2	no	—
79	C₂×C₄×C₇₉	—	2	2	no	—
80	C₂×C₄×C₇₉	—	2	2	no	—
81	C₂×C₄×C₇₉	—	2	4	no	—
82	C₂×C₄×C₇₉	—	4	2	no	—
83	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
86	C₂×C₄×C₇₉	#121	1	2	yes	—
87	C₈×C₇₉	—	2	2	no	—
88	C₈×C₇₉	—	2	2	no	—
89	C₇₉⋊C₈	—	4	4	no	—
90	C₇₉⋊C₈	—	4	2	no	—
91	C₇₉⋊C₈	—	4	2	no	—
92	C₂²×C₇₉	—	2	2	no	—
93	C₂²×C₇₉	—	2	1	no	—
94	C₂²×C₇₉	#386	1	1	yes	characteristic
95	C₄×C₇₉	—	2	2	no	—
96	C₄×C₇₉	—	2	1	no	—
97	C₄×C₇₉	—	4	4	no	—
98	C₄×C₇₉	—	8	2	no	—
99	C₄×C₇₉	#443	1	2	yes	—
100	C₄×C₇₉	#444	1	2	yes	—
101	C₄×C₇₉	#456	1	2	yes	—
102	#9068	—	79	1	no	maximal, Sylow
103	C₂×C₇₉	—	4	1	no	—
104	C₂×C₇₉	#1876	1	2	yes	—
105	C₂×C₇₉	#1886	1	1	yes	characteristic
106	#3160	D₇₉	1	1	yes	characteristic
107	#2986	—	79	1	no	—
108	#1526	—	79	1	no	—
109	#1485	—	79	2	no	—
110	#1278	—	79	2	no	—
111	C₇₉	#9068	1	1	yes	characteristic, complemented, Sylow
112	#533	C₂×D₇₉	1	1	yes	characteristic
113	#533	C₂×D₇₉	1	1	yes	characteristic
114	#500	Dic₇₉	1	2	yes	—
115	#499	—	2	2	no	—
116	#498	Dic₇₉	1	2	yes	—
117	#446	—	158	2	no	—
118	C₈×Q₈	—	158	2	no	—
119	#433	—	158	2	no	—
120	#431	—	79	2	no	—
121	#427	—	2	2	no	—
122	C₄×M₁₆	—	79	2	no	—
123	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
124	#330	—	158	2	no	—
125	Q₈⋊C₈	—	158	2	no	—
126	Q₈⋊C₈	—	158	2	no	—
127	C₂×M₁₆	—	158	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
130	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
131	#125	—	2	1	no	—
132	#125	C₂²⋊D₇₉	1	2	yes	—
133	C₂²⋊Q₈	—	2	2	no	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₄×Q₈	—	2	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	#117	—	2	1	no	—
138	#117	C₂²⋊D₇₉	1	2	yes	—
139	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
140	#107	—	2	4	no	—
141	#106	—	2	4	no	—
142	C₄⋊C₈	—	79	2	no	—
143	C₄⋊C₈	—	79	2	no	—
144	C₄⋊C₈	—	158	4	no	—
145	C₂²⋊C₈	—	158	2	no	—
146	#97	—	158	2	no	—
147	#97	—	158	2	no	—
148	C₄×C₈	—	2	1	no	—
149	C₄×C₈	—	158	2	no	—
150	C₄×C₈	—	158	2	no	—
151	C₄×C₈	—	158	2	no	—
152	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
153	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
157	C₂²×C₄	#11929623	1	2	yes	—
158	M₁₆	—	316	2	no	—
159	M₁₆	—	158	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	316	2	no	—
164	C₂×C₈	—	158	2	no	—
165	C₂×C₈	—	158	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	C₂²⋊Dic₇₉	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	4	1	no	—
176	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
177	C₄²	D₄×D₇₉	1	2	yes	—
178	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
179	C₂³	#14579142	1	1	yes	characteristic
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
190	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
191	C₂×C₄	#14579129	1	1	yes	characteristic
192	C₂×C₄	#14579133	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	316	4	no	—
196	C₈	—	316	2	no	—
197	C₈	—	316	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19098443	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	2	1	no	—
203	C₄	—	4	4	no	—
204	C₄	—	8	2	no	—
205	C₄	#19098680	1	2	yes	—
206	C₄	#19098685	1	2	yes	—
207	C₄	#19098745	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	#23763729	1	2	yes	—
210	C₂	#23763794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23772502

C₂

yes

maximal, characteristic

#23761995

C₂

yes

maximal, characteristic

#23761821

C₂

yes

maximal

#23761237

C₂

yes

maximal

C₇₉×#3160

C₂

yes

maximal, characteristic

#19098694

—

Q₈×C₇₉⋊C₈

—

#19098624

—

#19098616

C₂²

yes

—

C₄×#11929608

C₂²

yes

—

#19098305

—

Q₈⋊(C₇₉⋊C₈)

—

Q₈⋊(C₇₉⋊C₈)

—

#533×C₇₉

C₂²

yes

characteristic

#533×C₇₉

C₂²

yes

characteristic

#500×C₇₉

C₄

yes

—

#499×C₇₉

—

#498×C₇₉

C₄

yes

—

#427×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₄⋊(C₇₉⋊C₈)

D₄

yes

—

C₂²⋊(C₇₉⋊C₈)

—

#14579072

—

#14579072

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

—

C₂²×C₈×C₇₉

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

#125×C₇₉

—

#125×C₇₉

D₄

yes

—

C₂²⋊Q₈×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

—

#117×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#107×C₇₉

—

#106×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄×C₈×C₇₉

C₂×C₄

yes

characteristic

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

#123

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#120

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#386

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#443

yes

—

100

C₄×C₇₉

#444

yes

—

101

C₄×C₇₉

#456

yes

—

102

#9068

—

maximal, Sylow

103

C₂×C₇₉

—

104

C₂×C₇₉

#1876

yes

—

105

C₂×C₇₉

#1886

yes

characteristic

106

#3160

D₇₉

yes

characteristic

107

#2986

—

108

#1526

—

109

#1485

—

110

#1278

—

111

C₇₉

#9068

yes

characteristic, complemented, Sylow

112

#533

C₂×D₇₉

yes

characteristic

113

#533

C₂×D₇₉

yes

characteristic

114

#500

Dic₇₉

yes

—

115

#499

—

116

#498

Dic₇₉

yes

—

117

#446

—

158

—

118

C₈×Q₈

—

158

—

119

#433

—

158

—

120

#431

—

121

#427

—

122

C₄×M₁₆

—

123

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

124

#330

—

158

—

125

Q₈⋊C₈

—

158

—

126

Q₈⋊C₈

—

158

—

127

C₂×M₁₆

—

158

—

128

C₂²×C₈

—

129

C₄⋊Q₈

C₂×Dic₇₉

yes

—

130

C₄⋊Q₈

C₂×Dic₇₉

yes

—

131

#125

—

132

#125

C₂²⋊D₇₉

yes

—

133

C₂²⋊Q₈

—

134

C₂²⋊Q₈

—

135

C₄×Q₈

—

136

C₄×Q₈

—

137

#117

—

138

#117

C₂²⋊D₇₉

yes

—

139

C₂×C₄²

C₂²×D₇₉

yes

characteristic

140

#107

—

141

#106

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

158

—

145

C₂²⋊C₈

—

158

—

146

#97

—

158

—

147

#97

—

158

—

148

C₄×C₈

—

149

C₄×C₈

—

158

—

150

C₄×C₈

—

158

—

151

C₄×C₈

—

158

—

152

C₄×C₈

C₂×Dic₇₉

yes

characteristic

153

C₄×C₈

C₂×Dic₇₉

yes

characteristic

154

C₂×Q₈

—

155

C₂×Q₈

—

156

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

157

C₂²×C₄

#11929623

yes

—

158

M₁₆

—

316

—

159

M₁₆

—

158

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

316

—

164

C₂×C₈

—

158

—

165

C₂×C₈

—

158

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂²⋊Dic₇₉

yes

—

172

C₂²⋊C₄

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

—

176

C₄²

C₂²×Dic₇₉

yes

characteristic

177

C₄²

D₄×D₇₉

yes

—

178

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

179

C₂³

#14579142

yes

characteristic

180

Q₈

—

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

190

C₂×C₄

D₄×Dic₇₉

yes

—

191

C₂×C₄

#14579129

yes

characteristic

192

C₂×C₄

#14579133

yes

—

193

C₈

—

194

C₈

—

195

C₈

—

316

—

196

C₈

—

316

—

197

C₈

—

316

—

198

C₂²

—

199

C₂²

—

200

C₂²

#19098443

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

#19098680

yes

—

206

C₄

#19098685

yes

—

207

C₄

#19098745

yes

—

208

C₂

—

209

C₂

#23763729

yes

—

210

C₂

#23763794

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3160	C₂
socle	C₂²×C₇₉	#386
prime radical	C₂³×C₇₉	#123
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3160	D₇₉
hypocenter	C₇₉	#9068
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#14579129

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3160

C₂

C₂²×C₇₉

C₂³×C₇₉

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
111. C₇₉	#9068	semidirect

H ≅ G/N

type

111. C₇₉

#9068

semidirect

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

subgroup	index
#23772502	2
#23761995	2
#23761821	2
#23761237	2
C₇₉×#3160	2
#9068	79

subgroup

index

C₇₉×#3160

Subgroups — #34183586

Conjugacy classes of subgroups

Complements

Maximal subgroups