Group #23774117 — subgroups

Subgroups — #23774117

Order 10112.

402 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 121 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	#19098935	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098914	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098808	C₂	1	4	yes	maximal, complemented
7	#19098660	C₂	1	4	yes	maximal, complemented
8	C₄×#11929607	C₂	1	1	yes	maximal, characteristic, complemented
9	#486×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929623	—	2	1	no	—
11	C₂×#11929612	C₂²	1	4	yes	—
12	C₂×#11929607	C₂²	1	2	yes	—
13	#14579161	C₂²	1	1	yes	characteristic
14	#14579161	C₂²	1	1	yes	characteristic
15	#14579152	—	2	1	no	—
16	#14579148	—	2	1	no	—
17	#14579141	C₂²	1	2	yes	—
18	#14579137	—	2	1	no	—
19	#14579132	—	2	1	no	—
20	#14579132	C₂²	1	2	yes	—
21	Q₈×Dic₇₉	—	2	1	no	—
22	#14579126	C₂²	1	2	yes	—
23	D₄×Dic₇₉	—	2	1	no	—
24	D₄×Dic₇₉	C₂²	1	2	yes	—
25	#14579119	—	2	1	no	—
26	C₄²×D₇₉	C₂²	1	1	yes	characteristic
27	#14579094	C₂²	1	4	yes	—
28	#14579086	C₂²	1	4	yes	—
29	#14579071	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	C₂³	1	2	yes	—
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	#11929612	—	2	8	no	—
44	#11929607	—	2	4	no	—
45	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
46	C₄⋊Dic₇₉	—	2	2	no	—
47	C₄⋊Dic₇₉	—	2	1	no	—
48	C₄⋊Dic₇₉	—	2	1	no	—
49	C₄⋊Dic₇₉	—	2	1	no	—
50	C₄⋊Dic₇₉	C₂³	1	2	yes	—
51	Dic₇₉⋊C₄	—	2	2	no	—
52	C₂²⋊Dic₇₉	—	2	2	no	—
53	C₂²⋊Dic₇₉	—	2	2	no	—
54	(C₂×D₇₉)⋊C₄	—	2	2	no	—
55	C₄×Dic₇₉	—	2	2	no	—
56	C₄×Dic₇₉	D₄	1	1	yes	characteristic
57	C₄×Dic₇₉	D₄	1	1	yes	characteristic
58	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂×Q₈×C₇₉	—	2	1	no	—
60	C₂×D₄×C₇₉	—	2	1	no	—
61	C₂×D₄×C₇₉	—	2	1	no	—
62	C₂×D₄×C₇₉	C₂³	1	2	yes	—
63	C₂×C₈×C₇₉	C₂³	1	2	yes	—
64	C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
66	C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	C₂³×C₇₉	—	2	2	no	—
68	C₂²×D₇₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₇₉	—	4	2	no	—
70	D₄×C₇₉	—	2	4	no	—
71	D₄×C₇₉	—	4	2	no	—
72	D₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
77	C₂²⋊D₇₉	—	4	4	no	—
78	C₂×Dic₇₉	—	2	2	no	—
79	C₂×Dic₇₉	—	2	2	no	—
80	C₂×Dic₇₉	—	4	4	no	—
81	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
84	C₄×D₇₉	—	2	2	no	—
85	C₄×D₇₉	—	2	2	no	—
86	C₄×D₇₉	—	2	2	no	—
87	Dic₁₅₈	—	2	4	no	—
88	Dic₁₅₈	—	4	2	no	—
89	C₈×C₇₉	—	2	2	no	—
90	C₇₉⋊C₈	#41	1	4	yes	complemented
91	C₂²×C₇₉	—	4	2	no	—
92	C₂²×C₇₉	—	4	2	no	—
93	C₂²×C₇₉	—	4	2	no	—
94	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×D₇₉	—	2	1	no	—
96	C₂×D₇₉	—	2	1	no	—
97	C₂×D₇₉	—	2	1	no	—
98	C₄×C₇₉	—	4	2	no	—
99	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
100	C₄×C₇₉	#137	1	2	yes	—
101	C₄×C₇₉	C₂×#41	1	2	yes	—
102	Dic₇₉	—	2	1	no	—
103	Dic₇₉	—	2	1	no	—
104	Dic₇₉	—	2	1	no	—
105	Dic₇₉	—	4	2	no	—
106	Dic₇₉	—	4	2	no	—
107	C₂×C₇₉	—	8	2	no	—
108	C₂×C₇₉	#532	1	1	yes	characteristic
109	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₇₉	C₂×#137	1	1	yes	characteristic
111	D₇₉	—	4	1	no	—
112	#3182	—	79	1	no	maximal, Sylow
113	C₇₉	#3182	1	1	yes	characteristic, complemented, Sylow
114	#533	—	79	1	no	—
115	#532	—	79	1	no	—
116	#500	—	79	1	no	—
117	#492	—	79	1	no	—
118	#487	—	79	1	no	—
119	#486	D₇₉	1	1	yes	characteristic, complemented
120	#468	—	79	4	no	—
121	#440	—	79	4	no	—
122	C₄×M₁₆	—	79	1	no	—
123	C₂×#41	—	158	1	no	—
124	C₂×SD₁₆	—	79	4	no	—
125	C₂×M₁₆	—	79	2	no	—
126	C₄⋊Q₈	—	79	1	no	—
127	C₄⋊Q₈	—	79	1	no	—
128	C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
129	#127	C₂×D₇₉	1	1	yes	characteristic
130	#125	—	158	1	no	—
131	#124	—	158	1	no	—
132	C₂²⋊Q₈	—	79	2	no	—
133	C₂²⋊Q₈	—	158	1	no	—
134	#121	—	79	2	no	—
135	#121	—	158	1	no	—
136	C₄×Q₈	—	79	2	no	—
137	C₄×Q₈	—	158	1	no	—
138	C₄×D₄	—	79	2	no	—
139	C₄×D₄	—	158	1	no	—
140	#117	—	158	1	no	—
141	C₂×C₄²	—	79	1	no	—
142	#107	—	79	4	no	—
143	#103	—	79	4	no	—
144	#102	C₂×D₇₉	1	4	yes	—
145	#97	—	79	2	no	—
146	C₄×C₈	—	79	1	no	—
147	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
148	#41	—	316	4	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×Q₈	—	79	2	no	—
151	C₂×Q₈	—	158	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	158	2	no	—
155	C₂×D₄	C₂²×D₇₉	1	2	yes	—
156	C₂²×C₄	—	79	1	no	—
157	C₂²×C₄	—	79	1	no	—
158	C₂²×C₄	—	79	1	no	—
159	C₂²×C₄	—	158	2	no	—
160	SD₁₆	—	158	8	no	—
161	M₁₆	—	158	4	no	—
162	C₂×C₈	—	79	2	no	—
163	C₂×C₈	C₂²×D₇₉	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	79	2	no	—
166	C₄⋊C₄	—	158	2	no	—
167	C₄⋊C₄	—	158	2	no	—
168	C₄⋊C₄	—	158	1	no	—
169	C₄⋊C₄	—	158	1	no	—
170	C₄⋊C₄	—	158	1	no	—
171	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
172	C₂²⋊C₄	—	158	2	no	—
173	C₂²⋊C₄	—	158	2	no	—
174	C₂²⋊C₄	—	158	2	no	—
175	C₄²	—	79	1	no	—
176	C₄²	—	79	1	no	—
177	C₄²	—	79	1	no	—
178	C₄²	—	158	2	no	—
179	C₄²	C₂²×D₇₉	1	1	yes	characteristic
180	C₂³	—	2	2	no	—
181	C₂³	—	79	1	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	316	2	no	—
184	Q₈	—	158	4	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	2	no	—
188	D₄	—	316	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	316	4	no	—
191	C₂×C₄	—	79	1	no	—
192	C₂×C₄	—	79	1	no	—
193	C₂×C₄	—	79	1	no	—
194	C₂×C₄	—	158	2	no	—
195	C₂×C₄	—	158	2	no	—
196	C₂×C₄	—	158	2	no	—
197	C₂×C₄	—	158	2	no	—
198	C₂×C₄	—	158	2	no	—
199	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
200	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	79	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	158	1	no	—
208	C₂²	—	158	1	no	—
209	C₂²	—	158	1	no	—
210	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	316	2	no	—
213	C₄	—	316	2	no	—
214	C₄	—	158	1	no	—
215	C₄	—	158	1	no	—
216	C₄	—	158	1	no	—
217	C₄	#14579185	1	2	yes	—
218	C₄	#14579190	1	2	yes	—
219	C₄	#41×D₇₉	1	2	yes	—
220	C₂	—	8	2	no	—
221	C₂	—	316	1	no	—
222	C₂	#19099112	1	1	yes	characteristic
223	C₂	C₂×#14579185	1	1	yes	characteristic
224	C₂	C₂×#14579190	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

#19098935

C₂

yes

maximal, characteristic, complemented

#19098914

C₂

yes

maximal, characteristic, complemented

#19098808

C₂

yes

maximal, complemented

#19098660

C₂

yes

maximal, complemented

C₄×#11929607

C₂

yes

maximal, characteristic, complemented

#486×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929623

—

C₂×#11929612

C₂²

yes

—

C₂×#11929607

C₂²

yes

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

—

—

C₂²

yes

—

#14579137

—

#14579132

—

#14579132

C₂²

yes

—

Q₈×Dic₇₉

—

#14579126

C₂²

yes

—

D₄×Dic₇₉

—

D₄×Dic₇₉

C₂²

yes

—

#14579119

—

C₄²×D₇₉

C₂²

yes

characteristic

#14579094

C₂²

yes

—

#14579086

C₂²

yes

—

#14579071

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

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

D₄

yes

characteristic

C₂×C₄×D₇₉

C₂³

yes

characteristic

#11929612

—

#11929607

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

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

#41

yes

complemented

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

100

C₄×C₇₉

#137

yes

—

101

C₄×C₇₉

C₂×#41

yes

—

102

Dic₇₉

—

103

Dic₇₉

—

104

Dic₇₉

—

105

Dic₇₉

—

106

Dic₇₉

—

107

C₂×C₇₉

—

108

C₂×C₇₉

#532

yes

characteristic

109

C₂×C₇₉

C₂×AGL(1,ℤ/8)

yes

characteristic

110

C₂×C₇₉

C₂×#137

yes

characteristic

111

D₇₉

—

112

#3182

—

maximal, Sylow

113

C₇₉

#3182

yes

characteristic, complemented, Sylow

114

#533

—

115

#532

—

116

#500

—

117

#492

—

118

#487

—

119

#486

D₇₉

yes

characteristic, complemented

120

#468

—

121

#440

—

122

C₄×M₁₆

—

123

C₂×#41

—

158

—

124

C₂×SD₁₆

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

C₂×D₇₉

yes

characteristic

129

#127

C₂×D₇₉

yes

characteristic

130

#125

—

158

—

131

#124

—

158

—

132

C₂²⋊Q₈

—

133

C₂²⋊Q₈

—

158

—

134

#121

—

135

#121

—

158

—

136

C₄×Q₈

—

137

C₄×Q₈

—

158

—

138

C₄×D₄

—

139

C₄×D₄

—

158

—

140

#117

—

158

—

141

C₂×C₄²

—

142

#107

—

143

#103

—

144

#102

C₂×D₇₉

yes

—

145

#97

—

146

C₄×C₈

—

147

C₄×C₈

C₂×D₇₉

yes

characteristic

148

#41

—

316

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

158

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

158

—

155

C₂×D₄

C₂²×D₇₉

yes

—

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

158

—

160

SD₁₆

—

158

—

161

M₁₆

—

158

—

162

C₂×C₈

—

163

C₂×C₈

C₂²×D₇₉

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

158

—

167

C₄⋊C₄

—

158

—

168

C₄⋊C₄

—

158

—

169

C₄⋊C₄

—

158

—

170

C₄⋊C₄

—

158

—

171

C₄⋊C₄

C₂²×D₇₉

yes

—

172

C₂²⋊C₄

—

158

—

173

C₂²⋊C₄

—

158

—

174

C₂²⋊C₄

—

158

—

175

C₄²

—

176

C₄²

—

177

C₄²

—

178

C₄²

—

158

—

179

C₄²

C₂²×D₇₉

yes

characteristic

180

C₂³

—

181

C₂³

—

182

Q₈

—

183

Q₈

—

316

—

184

Q₈

—

158

—

185

D₄

—

186

D₄

—

187

D₄

—

188

D₄

—

316

—

189

C₂×C₄

—

190

C₂×C₄

—

316

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

158

—

195

C₂×C₄

—

158

—

196

C₂×C₄

—

158

—

197

C₂×C₄

—

158

—

198

C₂×C₄

—

158

—

199

C₂×C₄

D₄×D₇₉

yes

characteristic

200

C₂×C₄

D₄×D₇₉

yes

characteristic

201

C₂×C₄

C₂³×D₇₉

yes

characteristic

202

C₈

—

203

C₈

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

158

—

208

C₂²

—

158

—

209

C₂²

—

158

—

210

C₂²

C₂×D₄×D₇₉

yes

characteristic

211

C₄

—

212

C₄

—

316

—

213

C₄

—

316

—

214

C₄

—

158

—

215

C₄

—

158

—

216

C₄

—

158

—

217

C₄

#14579185

yes

—

218

C₄

#14579190

yes

—

219

C₄

#41×D₇₉

yes

—

220

C₂

—

221

C₂

—

316

—

222

C₂

#19099112

yes

characteristic

223

C₂

C₂×#14579185

yes

characteristic

224

C₂

C₂×#14579190

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	#3182
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. #19098935	C₂	semidirect
5. #19098914	C₂	semidirect
6. #19098808	C₂	semidirect
7. #19098660	C₂	semidirect
8. C₄×#11929607	C₂	semidirect
9. #486×C₇₉	C₂	semidirect
90. C₇₉⋊C₈	#41	semidirect
113. C₇₉	#3182	semidirect
119. #486	D₇₉	semidirect

H ≅ G/N

type

1. #19099118

C₂

semidirect

3. #19098966

C₂

semidirect

4. #19098935

C₂

semidirect

5. #19098914

C₂

semidirect

6. #19098808

C₂

semidirect

7. #19098660

C₂

semidirect

8. C₄×#11929607

C₂

semidirect

9. #486×C₇₉

C₂

semidirect

90. C₇₉⋊C₈

#41

semidirect

113. C₇₉

#3182

semidirect

119. #486

D₇₉

semidirect

Subgroups — #23774117

Conjugacy classes of subgroups

Complements