Group #23774533 — subgroups

Subgroups — #23774533

Order 10112.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19099118	C₂	1	1	yes	maximal, characteristic, complemented
2	#19099111	C₂	1	1	yes	maximal, characteristic
3	#19098966	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098961	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098909	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098886	C₂	1	4	yes	maximal, complemented
7	#19098865	C₂	1	4	yes	maximal, complemented
8	#19098614	C₂	1	1	yes	maximal, characteristic, complemented
9	#486×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929623	—	2	1	no	—
11	#14579161	C₂²	1	1	yes	characteristic
12	#14579161	C₂²	1	1	yes	characteristic
13	#14579152	C₂²	1	2	yes	—
14	#14579148	—	2	1	no	—
15	#14579141	—	2	1	no	—
16	#14579137	—	2	1	no	—
17	#14579132	—	2	1	no	—
18	#14579132	C₂²	1	2	yes	—
19	Q₈×Dic₇₉	—	2	1	no	—
20	#14579126	—	2	1	no	—
21	D₄×Dic₇₉	—	2	1	no	—
22	D₄×Dic₇₉	C₂²	1	2	yes	—
23	#14579119	C₂²	1	2	yes	—
24	C₄²×D₇₉	C₂²	1	1	yes	characteristic
25	#14579094	C₂²	1	4	yes	—
26	#14579093	C₂²	1	4	yes	—
27	Dic₇₉⋊C₈	C₂²	1	2	yes	—
28	#14579082	C₂²	1	4	yes	—
29	(C₂×D₇₉)⋊C₈	C₂²	1	2	yes	—
30	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
31	C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
32	#127×C₇₉	C₂²	1	1	yes	characteristic
33	#102×C₇₉	C₂²	1	4	yes	—
34	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	#11929623	—	4	4	no	—
36	C₂×Dic₁₅₈	—	2	1	no	—
37	C₂×Dic₁₅₈	—	2	1	no	—
38	C₂×C₂²⋊D₇₉	—	2	2	no	—
39	C₂²×Dic₇₉	—	2	2	no	—
40	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄×D₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
43	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
44	C₄⋊Dic₇₉	—	2	2	no	—
45	C₄⋊Dic₇₉	—	2	1	no	—
46	C₄⋊Dic₇₉	—	2	1	no	—
47	C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	C₄⋊Dic₇₉	C₂³	1	2	yes	—
49	Dic₇₉⋊C₄	—	2	2	no	—
50	C₂²⋊Dic₇₉	—	2	2	no	—
51	C₂²⋊Dic₇₉	—	2	2	no	—
52	(C₂×D₇₉)⋊C₄	—	2	2	no	—
53	C₄×Dic₇₉	—	2	2	no	—
54	C₄×Dic₇₉	D₄	1	1	yes	characteristic
55	C₄×Dic₇₉	D₄	1	1	yes	characteristic
56	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
57	C₂×Q₈×C₇₉	—	2	1	no	—
58	C₂×D₄×C₇₉	—	2	1	no	—
59	C₂×D₄×C₇₉	—	2	1	no	—
60	C₂×D₄×C₇₉	C₂³	1	2	yes	—
61	C₂×C₈×C₇₉	C₂³	1	2	yes	—
62	C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
64	C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	C₂³×C₇₉	—	2	2	no	—
66	C₂²×D₇₉	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₇₉	—	4	2	no	—
68	D₄×C₇₉	—	2	4	no	—
69	D₄×C₇₉	—	4	2	no	—
70	D₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
75	C₂²⋊D₇₉	—	4	4	no	—
76	C₂×Dic₇₉	—	2	2	no	—
77	C₂×Dic₇₉	—	2	2	no	—
78	C₂×Dic₇₉	—	4	4	no	—
79	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄×D₇₉	—	2	2	no	—
83	C₄×D₇₉	—	2	2	no	—
84	C₄×D₇₉	—	2	2	no	—
85	Dic₁₅₈	—	4	2	no	—
86	Dic₁₅₈	—	4	2	no	—
87	C₈×C₇₉	—	2	2	no	—
88	C₇₉⋊C₈	—	2	2	no	—
89	C₂²×C₇₉	—	4	2	no	—
90	C₂²×C₇₉	—	4	2	no	—
91	C₂²×C₇₉	—	4	2	no	—
92	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
93	C₂×D₇₉	—	2	1	no	—
94	C₂×D₇₉	—	2	1	no	—
95	C₂×D₇₉	—	2	1	no	—
96	C₄×C₇₉	—	4	2	no	—
97	C₄×C₇₉	#135	1	2	yes	—
98	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
99	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
100	Dic₇₉	—	2	1	no	—
101	Dic₇₉	—	2	1	no	—
102	Dic₇₉	—	2	1	no	—
103	Dic₇₉	—	4	2	no	—
104	Dic₇₉	—	4	2	no	—
105	C₂×C₇₉	—	8	2	no	—
106	C₂×C₇₉	#536	1	1	yes	characteristic
107	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
108	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
109	D₇₉	—	4	1	no	—
110	#3228	—	79	1	no	maximal, Sylow
111	C₇₉	#3228	1	1	yes	characteristic, complemented, Sylow
112	#533	—	79	1	no	—
113	#532	—	79	1	no	—
114	#500	—	79	1	no	—
115	#498	—	79	1	no	—
116	#486	—	79	1	no	—
117	#486	D₇₉	1	1	yes	characteristic, complemented
118	#482	—	79	4	no	—
119	#478	—	79	4	no	—
120	#431	—	79	1	no	—
121	C₂×#41	—	158	1	no	—
122	C₄⋊Q₈	—	79	1	no	—
123	C₄⋊Q₈	—	79	1	no	—
124	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
125	#127	C₂×D₇₉	1	1	yes	characteristic
126	#125	—	79	2	no	—
127	#124	—	158	1	no	—
128	C₂²⋊Q₈	—	158	1	no	—
129	C₂²⋊Q₈	—	158	1	no	—
130	#121	—	79	2	no	—
131	#121	—	158	1	no	—
132	C₄×Q₈	—	158	1	no	—
133	C₄×Q₈	—	158	1	no	—
134	C₄×D₄	—	79	2	no	—
135	C₄×D₄	—	158	1	no	—
136	#117	—	79	2	no	—
137	C₂×C₄²	—	79	1	no	—
138	#107	—	79	4	no	—
139	#106	—	79	4	no	—
140	C₄⋊C₈	—	79	2	no	—
141	#102	—	79	4	no	—
142	#102	C₂×D₇₉	1	4	yes	—
143	C₂²⋊C₈	—	79	2	no	—
144	C₄×C₈	—	79	1	no	—
145	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	#41	—	316	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	158	1	no	—
149	C₂×Q₈	—	158	1	no	—
150	C₂×D₄	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	158	2	no	—
153	C₂×D₄	C₂²×D₇₉	1	2	yes	—
154	C₂²×C₄	—	79	1	no	—
155	C₂²×C₄	—	79	1	no	—
156	C₂²×C₄	—	79	1	no	—
157	C₂²×C₄	—	158	2	no	—
158	C₂×C₈	—	79	2	no	—
159	C₂×C₈	C₂²×D₇₉	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	79	2	no	—
162	C₄⋊C₄	—	79	2	no	—
163	C₄⋊C₄	—	158	2	no	—
164	C₄⋊C₄	—	158	2	no	—
165	C₄⋊C₄	—	158	1	no	—
166	C₄⋊C₄	—	158	1	no	—
167	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
168	C₂²⋊C₄	—	158	2	no	—
169	C₂²⋊C₄	—	158	2	no	—
170	C₂²⋊C₄	—	158	2	no	—
171	C₄²	—	79	1	no	—
172	C₄²	—	79	1	no	—
173	C₄²	—	79	1	no	—
174	C₄²	—	158	2	no	—
175	C₄²	C₂²×D₇₉	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	—	79	1	no	—
178	Q₈	—	4	2	no	—
179	Q₈	—	316	2	no	—
180	Q₈	—	316	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	2	no	—
183	D₄	—	4	2	no	—
184	D₄	—	316	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	316	4	no	—
187	C₂×C₄	—	79	1	no	—
188	C₂×C₄	—	79	1	no	—
189	C₂×C₄	—	79	1	no	—
190	C₂×C₄	—	158	2	no	—
191	C₂×C₄	—	158	2	no	—
192	C₂×C₄	—	158	2	no	—
193	C₂×C₄	—	158	2	no	—
194	C₂×C₄	—	158	2	no	—
195	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
196	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	158	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	158	1	no	—
204	C₂²	—	158	1	no	—
205	C₂²	—	158	1	no	—
206	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
207	C₄	—	4	2	no	—
208	C₄	—	316	2	no	—
209	C₄	—	316	2	no	—
210	C₄	—	158	1	no	—
211	C₄	—	158	1	no	—
212	C₄	—	158	1	no	—
213	C₄	#14579182	1	2	yes	—
214	C₄	#14579185	1	2	yes	—
215	C₄	#14579208	1	2	yes	—
216	C₂	—	8	2	no	—
217	C₂	—	316	1	no	—
218	C₂	#19099133	1	1	yes	characteristic
219	C₂	C₂×#14579182	1	1	yes	characteristic
220	C₂	C₂×#14579185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099118

C₂

yes

maximal, characteristic, complemented

#19099111

C₂

yes

maximal, characteristic

#19098966

C₂

yes

maximal, characteristic, complemented

#19098961

C₂

yes

maximal, characteristic, complemented

#19098909

C₂

yes

maximal, characteristic, complemented

#19098886

C₂

yes

maximal, complemented

#19098865

C₂

yes

maximal, complemented

#19098614

C₂

yes

maximal, characteristic, complemented

#486×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

characteristic

#14579152

C₂²

yes

—

—

—

—

—

C₂²

yes

—

Q₈×Dic₇₉

—

#14579126

—

D₄×Dic₇₉

—

D₄×Dic₇₉

C₂²

yes

—

#14579119

C₂²

yes

—

C₄²×D₇₉

C₂²

yes

characteristic

#14579094

C₂²

yes

—

#14579093

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

#14579082

C₂²

yes

—

(C₂×D₇₉)⋊C₈

C₂²

yes

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₉

C₂²

yes

characteristic

#127×C₇₉

C₂²

yes

characteristic

#102×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

#11929623

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

C₂³

yes

characteristic

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

characteristic

C₄×Dic₇₉

D₄

yes

characteristic

C₄×Dic₇₉

C₂³

yes

characteristic

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₂×Dic₇₉

C₂×D₄

yes

characteristic

C₄×D₇₉

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#135

yes

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

2₋¹⁺⁴

yes

—

100

Dic₇₉

—

101

Dic₇₉

—

102

Dic₇₉

—

103

Dic₇₉

—

104

Dic₇₉

—

105

C₂×C₇₉

—

106

C₂×C₇₉

#536

yes

characteristic

107

C₂×C₇₉

C₂×#135

yes

characteristic

108

C₂×C₇₉

C₂×AGL(1,ℤ/8)

yes

characteristic

109

D₇₉

—

110

#3228

—

maximal, Sylow

111

C₇₉

#3228

yes

characteristic, complemented, Sylow

112

#533

—

113

#532

—

114

#500

—

115

#498

—

116

#486

—

117

#486

D₇₉

yes

characteristic, complemented

118

#482

—

119

#478

—

120

#431

—

121

C₂×#41

—

158

—

122

C₄⋊Q₈

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

125

#127

C₂×D₇₉

yes

characteristic

126

#125

—

127

#124

—

158

—

128

C₂²⋊Q₈

—

158

—

129

C₂²⋊Q₈

—

158

—

130

#121

—

131

#121

—

158

—

132

C₄×Q₈

—

158

—

133

C₄×Q₈

—

158

—

134

C₄×D₄

—

135

C₄×D₄

—

158

—

136

#117

—

137

C₂×C₄²

—

138

#107

—

139

#106

—

140

C₄⋊C₈

—

141

#102

—

142

#102

C₂×D₇₉

yes

—

143

C₂²⋊C₈

—

144

C₄×C₈

—

145

C₄×C₈

C₂×D₇₉

yes

characteristic

146

#41

—

316

—

147

C₂×Q₈

—

148

C₂×Q₈

—

158

—

149

C₂×Q₈

—

158

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

158

—

153

C₂×D₄

C₂²×D₇₉

yes

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

—

158

C₂×C₈

—

159

C₂×C₈

C₂²×D₇₉

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

158

—

164

C₄⋊C₄

—

158

—

165

C₄⋊C₄

—

158

—

166

C₄⋊C₄

—

158

—

167

C₄⋊C₄

C₂²×D₇₉

yes

—

168

C₂²⋊C₄

—

158

—

169

C₂²⋊C₄

—

158

—

170

C₂²⋊C₄

—

158

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

158

—

175

C₄²

C₂²×D₇₉

yes

characteristic

176

C₂³

—

177

C₂³

—

178

Q₈

—

179

Q₈

—

316

—

180

Q₈

—

316

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

316

—

185

C₂×C₄

—

186

C₂×C₄

—

316

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

158

—

191

C₂×C₄

—

158

—

192

C₂×C₄

—

158

—

193

C₂×C₄

—

158

—

194

C₂×C₄

—

158

—

195

C₂×C₄

D₄×D₇₉

yes

characteristic

196

C₂×C₄

D₄×D₇₉

yes

characteristic

197

C₂×C₄

C₂³×D₇₉

yes

characteristic

198

C₈

—

199

C₈

—

158

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

158

—

204

C₂²

—

158

—

205

C₂²

—

158

—

206

C₂²

C₂×D₄×D₇₉

yes

characteristic

207

C₄

—

208

C₄

—

316

—

209

C₄

—

316

—

210

C₄

—

158

—

211

C₄

—

158

—

212

C₄

—

158

—

213

C₄

#14579182

yes

—

214

C₄

#14579185

yes

—

215

C₄

#14579208

yes

—

216

C₂

—

217

C₂

—

316

—

218

C₂

#19099133

yes

characteristic

219

C₂

C₂×#14579182

yes

characteristic

220

C₂

C₂×#14579185

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#486×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099111	C₂
hypercenter	#486	D₇₉
hypocenter	C₇₉	#3228
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#486×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #19099118	C₂	semidirect
3. #19098966	C₂	semidirect
4. #19098961	C₂	semidirect
5. #19098909	C₂	semidirect
6. #19098886	C₂	semidirect
7. #19098865	C₂	semidirect
8. #19098614	C₂	semidirect
9. #486×C₇₉	C₂	semidirect
111. C₇₉	#3228	semidirect
117. #486	D₇₉	semidirect

H ≅ G/N

type

1. #19099118

C₂

semidirect

3. #19098966

C₂

semidirect

4. #19098961

C₂

semidirect

5. #19098909

C₂

semidirect

6. #19098886

C₂

semidirect

7. #19098865

C₂

semidirect

8. #19098614

C₂

semidirect

9. #486×C₇₉

C₂

semidirect

111. C₇₉

#3228

semidirect

117. #486

D₇₉

semidirect

Subgroups — #23774533

Conjugacy classes of subgroups

Complements