Group #23764302 — subgroups

Subgroups — #23764302

Order 10112.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#11929611	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579096	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098567	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579080	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14579077	C₂	1	1	yes	maximal, characteristic
6	#19098519	C₂	1	1	yes	maximal, characteristic, complemented
7	#419×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₁₆	C₂²	1	1	yes	characteristic
9	C₂×#11929611	—	2	1	no	—
10	C₂×#11929611	—	2	1	no	—
11	C₂×#11929611	—	2	2	no	—
12	C₂×#11929611	C₄	1	4	yes	complemented
13	C₂×#11929608	C₂²	1	1	yes	characteristic
14	C₂×#11929608	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	#14579096	—	2	2	no	—
17	#14579081	—	2	2	no	—
18	#14579080	—	2	2	no	—
19	#14579077	—	2	2	no	—
20	#14579072	—	2	1	no	—
21	C₄×C₇₉⋊C₈	—	2	1	no	—
22	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	#117×C₇₉	C₂²	1	1	yes	characteristic
25	#103×C₇₉	—	2	2	no	—
26	C₂³×D₇₉	—	2	1	no	—
27	C₂×D₃₁₆	—	2	1	no	—
28	C₂×D₃₁₆	—	2	1	no	—
29	C₂×D₃₁₆	—	2	1	no	—
30	C₂×D₃₁₆	—	2	1	no	—
31	C₂×D₃₁₆	—	2	1	no	—
32	C₂×D₃₁₆	C₂×C₄	1	2	yes	—
33	#11929611	—	2	4	no	—
34	#11929611	—	2	4	no	—
35	#11929608	—	2	1	no	—
36	#11929608	—	2	1	no	—
37	#11929608	—	2	1	no	—
38	#11929608	D₄	1	2	yes	—
39	C₂×C₇₉⋊C₈	—	2	1	no	—
40	C₂×C₇₉⋊C₈	—	2	1	no	—
41	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₇₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
45	C₂×Q₈×C₇₉	—	2	1	no	—
46	C₂×Q₈×C₇₉	—	2	1	no	—
47	C₂×Q₈×C₇₉	—	2	1	no	—
48	C₂×Q₈×C₇₉	—	2	1	no	—
49	C₂×Q₈×C₇₉	—	2	1	no	—
50	C₂×Q₈×C₇₉	C₂×C₄	1	2	yes	—
51	C₂²×C₄×C₇₉	—	2	1	no	—
52	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
53	M₁₆×C₇₉	—	2	1	no	—
54	M₁₆×C₇₉	—	2	1	no	—
55	C₂×C₈×C₇₉	—	2	1	no	—
56	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
57	C₂²⋊C₄×C₇₉	—	2	1	no	—
58	C₄²×C₇₉	—	2	1	no	—
59	C₂³×C₇₉	#41	1	1	yes	characteristic
60	C₂²×D₇₉	—	2	2	no	—
61	C₂²×D₇₉	—	4	1	no	—
62	C₂²×D₇₉	—	4	1	no	—
63	C₂²×D₇₉	—	8	2	no	—
64	Q₈×C₇₉	—	2	2	no	—
65	Q₈×C₇₉	—	2	2	no	—
66	Q₈×C₇₉	—	4	1	no	—
67	Q₈×C₇₉	—	4	1	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	2	no	—
70	C₂×C₄×C₇₉	—	4	1	no	—
71	C₂×C₄×C₇₉	—	4	1	no	—
72	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
78	D₃₁₆	—	2	2	no	—
79	D₃₁₆	—	2	2	no	—
80	D₃₁₆	—	4	1	no	—
81	D₃₁₆	—	4	1	no	—
82	C₈×C₇₉	—	4	1	no	—
83	C₇₉⋊C₈	—	2	2	no	—
84	C₇₉⋊C₈	—	4	1	no	—
85	C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
86	C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₇₉	—	2	2	no	—
88	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₇₉	#121	1	1	yes	characteristic
90	C₂²×C₇₉	#124	1	1	yes	characteristic
91	C₂×D₇₉	—	4	2	no	—
92	C₂×D₇₉	—	8	2	no	—
93	C₂×D₇₉	—	8	2	no	—
94	C₂×D₇₉	—	8	2	no	—
95	C₄×C₇₉	—	4	2	no	—
96	C₄×C₇₉	—	4	2	no	—
97	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
99	C₄×C₇₉	#121	1	1	yes	characteristic
100	C₄×C₇₉	#127	1	1	yes	characteristic
101	C₂×C₇₉	—	2	1	no	—
102	C₂×C₇₉	—	2	1	no	—
103	C₂×C₇₉	#390	1	1	yes	characteristic
104	C₂×C₇₉	#471	1	2	yes	—
105	D₇₉	—	8	2	no	—
106	#1977	—	79	1	no	maximal, Sylow
107	C₇₉	#1977	1	1	yes	characteristic, complemented, Sylow
108	C₂²×SD₁₆	—	79	1	no	—
109	C₂×#108	—	79	1	no	—
110	#419	D₇₉	1	1	yes	characteristic, complemented
111	#418	—	79	1	no	—
112	C₂×#101	—	79	1	no	—
113	C₂×#100	—	79	1	no	—
114	#405	—	79	1	no	—
115	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
116	C₂²×D₄	—	79	1	no	—
117	C₂×SD₁₆	—	79	4	no	—
118	C₂×SD₁₆	—	158	1	no	—
119	C₂×SD₁₆	—	158	1	no	—
120	C₂×SD₁₆	—	158	2	no	—
121	C₂×M₁₆	—	79	1	no	—
122	C₂×M₁₆	—	79	1	no	—
123	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
124	C₂²×C₈	—	79	1	no	—
125	#117	C₂×D₇₉	1	1	yes	characteristic
126	#108	—	158	2	no	—
127	#103	—	2	2	no	—
128	#102	—	158	2	no	—
129	#101	—	158	2	no	—
130	#100	—	158	2	no	—
131	#97	—	158	1	no	—
132	C₄×C₈	—	158	1	no	—
133	C₂⁴	—	158	1	no	—
134	C₂×Q₈	—	2	1	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₄×D₇₉	1	2	yes	—
140	C₂×D₄	—	79	2	no	—
141	C₂×D₄	—	158	1	no	—
142	C₂×D₄	—	158	1	no	—
143	C₂×D₄	—	158	1	no	—
144	C₂×D₄	—	158	1	no	—
145	C₂×D₄	—	158	1	no	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
148	SD₁₆	—	158	4	no	—
149	SD₁₆	—	158	4	no	—
150	M₁₆	—	2	1	no	—
151	M₁₆	—	2	1	no	—
152	M₁₆	—	79	2	no	—
153	M₁₆	—	158	1	no	—
154	M₁₆	—	158	1	no	—
155	M₁₆	—	158	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	79	2	no	—
158	C₂×C₈	—	79	1	no	—
159	C₂×C₈	—	79	1	no	—
160	C₂×C₈	—	79	2	no	—
161	C₂×C₈	—	158	1	no	—
162	C₂×C₈	—	158	1	no	—
163	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₂³	—	316	1	no	—
167	C₂³	—	316	1	no	—
168	C₂³	—	632	2	no	—
169	C₂³	—	158	2	no	—
170	C₂³	#11929623	1	1	yes	characteristic
171	Q₈	—	2	2	no	—
172	Q₈	—	2	2	no	—
173	Q₈	—	4	1	no	—
174	Q₈	—	4	1	no	—
175	D₄	—	316	1	no	—
176	D₄	—	316	1	no	—
177	D₄	—	158	2	no	—
178	D₄	—	158	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
184	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
185	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
186	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂×C₄	#11929624	1	1	yes	characteristic
189	C₈	—	4	1	no	—
190	C₈	—	316	1	no	—
191	C₈	—	79	2	no	—
192	C₈	—	79	2	no	—
193	C₈	—	158	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	316	2	no	—
196	C₂²	—	632	2	no	—
197	C₂²	—	632	2	no	—
198	C₂²	—	632	2	no	—
199	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
200	C₂²	#14579133	1	1	yes	characteristic
201	C₂²	#14579147	1	1	yes	characteristic
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
205	C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
206	C₄	#14579134	1	1	yes	characteristic
207	C₄	#14579157	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	—	632	2	no	—
211	C₂	#19098463	1	1	yes	characteristic
212	C₂	#19098828	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929611

C₂

yes

maximal, characteristic

C₂×#14579096

C₂

yes

maximal, characteristic, complemented

#19098567

C₂

yes

maximal, characteristic

C₂×#14579080

C₂

yes

maximal, characteristic, complemented

C₂×#14579077

C₂

yes

maximal, characteristic

#19098519

C₂

yes

maximal, characteristic, complemented

#419×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₁₆

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#11929608

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₇₉⋊C₈

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂³×D₇₉

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂×C₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

characteristic

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

#41

yes

characteristic

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

#41

yes

characteristic

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

C₄×D₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

#121

yes

characteristic

100

C₄×C₇₉

#127

yes

characteristic

101

C₂×C₇₉

—

102

C₂×C₇₉

—

103

C₂×C₇₉

#390

yes

characteristic

104

C₂×C₇₉

#471

yes

—

105

D₇₉

—

106

#1977

—

maximal, Sylow

107

C₇₉

#1977

yes

characteristic, complemented, Sylow

108

C₂²×SD₁₆

—

109

C₂×#108

—

110

#419

D₇₉

yes

characteristic, complemented

111

#418

—

112

C₂×#101

—

113

C₂×#100

—

114

#405

—

115

C₂²×Q₈

C₂×D₇₉

yes

characteristic

116

C₂²×D₄

—

117

C₂×SD₁₆

—

118

C₂×SD₁₆

—

158

—

119

C₂×SD₁₆

—

158

—

120

C₂×SD₁₆

—

158

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₂×D₇₉

yes

characteristic

124

C₂²×C₈

—

125

#117

C₂×D₇₉

yes

characteristic

126

#108

—

158

—

127

#103

—

128

#102

—

158

—

129

#101

—

158

—

130

#100

—

158

—

131

#97

—

158

—

132

C₄×C₈

—

158

—

133

C₂⁴

—

158

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

C₄×D₇₉

yes

—

140

C₂×D₄

—

141

C₂×D₄

—

158

—

142

C₂×D₄

—

158

—

143

C₂×D₄

—

158

—

144

C₂×D₄

—

158

—

145

C₂×D₄

—

158

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂²×D₇₉

yes

characteristic

148

SD₁₆

—

158

—

149

SD₁₆

—

158

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

158

—

154

M₁₆

—

158

—

155

M₁₆

—

158

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

158

—

162

C₂×C₈

—

158

—

163

C₄⋊C₄

C₂²⋊D₇₉

yes

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₂³

—

316

—

167

C₂³

—

316

—

168

C₂³

—

632

—

169

C₂³

—

158

—

170

C₂³

#11929623

yes

characteristic

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

Q₈

—

175

D₄

—

316

—

176

D₄

—

316

—

177

D₄

—

158

—

178

D₄

—

158

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

184

C₂×C₄

D₄×D₇₉

yes

characteristic

185

C₂×C₄

D₄×D₇₉

yes

characteristic

186

C₂×C₄

D₄×D₇₉

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

188

C₂×C₄

#11929624

yes

characteristic

189

C₈

—

190

C₈

—

316

—

191

C₈

—

192

C₈

—

193

C₈

—

158

—

194

C₂²

—

195

C₂²

—

316

—

196

C₂²

—

632

—

197

C₂²

—

632

—

198

C₂²

—

632

—

199

C₂²

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

200

C₂²

#14579133

yes

characteristic

201

C₂²

#14579147

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

C₂²⋊C₄×D₇₉

yes

characteristic

205

C₄

C₄×C₂²⋊D₇₉

yes

characteristic

206

C₄

#14579134

yes

characteristic

207

C₄

#14579157

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

—

632

—

211

C₂

#19098463

yes

characteristic

212

C₂

#19098828

yes

—

	subgroup	G/H
center	C₂²	#14579133
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#419×C₇₉	C₂
socle	C₂²×C₇₉	#121
prime radical	C₂²×D₃₁₆	C₂²
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#419	D₇₉
hypocenter	C₇₉	#1977
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14579133

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#419×C₇₉

C₂

socle

C₂²×C₇₉

#121

prime radical

C₂²×D₃₁₆

C₂²

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

N	H ≅ G/N	type
2. C₂×#14579096	C₂	semidirect
4. C₂×#14579080	C₂	semidirect
6. #19098519	C₂	semidirect
7. #419×C₇₉	C₂	semidirect
12. C₂×#11929611	C₄	semidirect
107. C₇₉	#1977	semidirect
110. #419	D₇₉	semidirect

H ≅ G/N

type

2. C₂×#14579096

C₂

semidirect

4. C₂×#14579080

C₂

semidirect

6. #19098519

C₂

semidirect

7. #419×C₇₉

C₂

semidirect

12. C₂×#11929611

C₄

semidirect

107. C₇₉

#1977

semidirect

110. #419

D₇₉

semidirect

Subgroups — #23764302

Conjugacy classes of subgroups

Complements