Group #23762464 — subgroups

Subgroups — #23762464

Order 10112.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19099234	C₂	1	1	yes	maximal, characteristic, complemented
2	#19098939	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098760	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098306	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098293	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊(C₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11929624	—	2	1	no	—
9	C₂×#11929612	—	2	1	no	—
10	C₂×#11929609	—	2	2	no	—
11	#14579158	C₂²	1	1	yes	characteristic
12	#14579157	—	2	1	no	—
13	#14579135	—	2	1	no	—
14	#14579134	—	2	2	no	—
15	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
16	#14579123	—	2	1	no	—
17	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
18	#14579082	—	2	1	no	—
19	C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×D₈×C₇₉	—	2	1	no	—
21	#127×C₇₉	C₂²	1	1	yes	characteristic
22	C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
24	#102×C₇₉	—	2	1	no	—
25	#11929624	—	4	2	no	—
26	#11929624	—	4	2	no	—
27	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
28	C₂×D₃₁₆	—	2	1	no	—
29	C₂×D₃₁₆	—	2	1	no	—
30	C₂×D₃₁₆	D₄	1	2	yes	—
31	C₂×C₂²⋊D₇₉	—	2	2	no	—
32	C₂×C₂²⋊D₇₉	—	2	2	no	—
33	C₂×C₄×D₇₉	—	2	2	no	—
34	#11929612	—	4	2	no	—
35	#11929609	—	4	4	no	—
36	C₂×C₇₉⋊C₈	—	2	1	no	—
37	C₂×C₇₉⋊C₈	—	2	1	no	—
38	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	Dic₇₉⋊C₄	—	2	1	no	—
40	(C₂×D₇₉)⋊C₄	—	2	2	no	—
41	C₄×Dic₇₉	—	2	1	no	—
42	C₂×D₄×C₇₉	—	2	1	no	—
43	C₂×D₄×C₇₉	—	2	1	no	—
44	C₂×D₄×C₇₉	—	2	1	no	—
45	C₂×D₄×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₉	—	2	1	no	—
47	D₈×C₇₉	—	4	2	no	—
48	C₂×C₈×C₇₉	—	2	1	no	—
49	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
50	C₂²⋊C₄×C₇₉	—	2	1	no	—
51	C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₂³×C₇₉	—	2	1	no	—
53	C₂³×C₇₉	—	4	1	no	—
54	C₂²×D₇₉	—	2	2	no	—
55	D₄×C₇₉	—	2	1	no	—
56	D₄×C₇₉	—	2	1	no	—
57	D₄×C₇₉	—	4	2	no	—
58	D₄×C₇₉	—	4	1	no	—
59	D₄×C₇₉	—	4	1	no	—
60	D₄×C₇₉	—	4	1	no	—
61	D₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	2	1	no	—
63	C₂×C₄×C₇₉	—	4	1	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂²⋊D₇₉	—	4	4	no	—
69	C₂²⋊D₇₉	—	8	2	no	—
70	C₂×Dic₇₉	—	2	1	no	—
71	C₂×Dic₇₉	—	2	1	no	—
72	D₃₁₆	—	4	2	no	—
73	D₃₁₆	—	4	1	no	—
74	D₃₁₆	—	4	1	no	—
75	D₃₁₆	—	4	1	no	—
76	D₃₁₆	—	4	1	no	—
77	C₄×D₇₉	—	4	4	no	—
78	Dic₁₅₈	—	2	1	no	—
79	Dic₁₅₈	D₈	1	2	yes	complemented
80	C₈×C₇₉	—	4	1	no	—
81	C₇₉⋊C₈	—	2	2	no	—
82	C₇₉⋊C₈	—	4	1	no	—
83	C₂²×C₇₉	—	4	1	no	—
84	C₂²×C₇₉	—	4	1	no	—
85	C₂²×C₇₉	—	4	1	no	—
86	C₂²×C₇₉	—	8	1	no	—
87	C₂²×C₇₉	—	8	1	no	—
88	C₂²×C₇₉	—	8	1	no	—
89	C₂²×C₇₉	#120	1	1	yes	characteristic
90	C₂×D₇₉	—	4	2	no	—
91	C₂×D₇₉	—	4	2	no	—
92	C₂×D₇₉	—	4	2	no	—
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	4	1	no	—
95	C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
96	C₄×C₇₉	#135	1	1	yes	characteristic
97	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
99	Dic₇₉	—	2	2	no	—
100	Dic₇₉	—	4	1	no	—
101	C₂×C₇₉	—	8	1	no	—
102	C₂×C₇₉	—	16	1	no	—
103	C₂×C₇₉	#447	1	1	yes	characteristic
104	C₂×C₇₉	#449	1	1	yes	characteristic
105	C₂×C₇₉	#453	1	1	yes	characteristic
106	D₇₉	—	8	2	no	—
107	#1621	—	79	1	no	maximal, Sylow
108	C₇₉	#1621	1	1	yes	characteristic, complemented, Sylow
109	#550	—	79	1	no	—
110	#493	—	79	1	no	—
111	#460	—	79	1	no	—
112	#459	D₇₉	1	1	yes	characteristic, complemented
113	#331	—	79	1	no	—
114	Q₈⋊C₈	—	79	1	no	—
115	D₄⋊C₈	—	79	1	no	—
116	C₂×#41	—	158	1	no	—
117	C₂×SD₁₆	—	158	1	no	—
118	C₂×D₈	—	2	1	no	—
119	C₂×D₈	—	158	2	no	—
120	#127	—	79	1	no	—
121	#127	—	158	1	no	—
122	#127	C₂×D₇₉	1	1	yes	characteristic
123	#121	—	158	2	no	—
124	#121	—	158	1	no	—
125	C₄×Q₈	—	79	1	no	—
126	C₄×D₄	—	158	1	no	—
127	C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
128	C₄⋊C₈	—	79	1	no	—
129	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
130	#102	—	2	1	no	—
131	#102	—	158	1	no	—
132	C₄×C₈	—	79	1	no	—
133	#41	—	316	2	no	—
134	#41	—	316	2	no	—
135	C₂×Q₈	—	79	1	no	—
136	C₂×D₄	—	2	1	no	—
137	C₂×D₄	—	2	1	no	—
138	C₂×D₄	—	2	1	no	—
139	C₂×D₄	—	79	2	no	—
140	C₂×D₄	—	158	2	no	—
141	C₂×D₄	—	158	2	no	—
142	C₂×D₄	—	158	1	no	—
143	C₂×D₄	—	158	1	no	—
144	C₂×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
145	C₂²×C₄	—	2	1	no	—
146	C₂²×C₄	—	158	2	no	—
147	SD₁₆	—	316	2	no	—
148	D₈	—	4	2	no	—
149	D₈	—	316	4	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	158	1	no	—
152	C₂×C₈	—	158	1	no	—
153	C₄⋊C₄	—	79	1	no	—
154	C₄⋊C₄	—	158	1	no	—
155	C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
156	C₂²⋊C₄	—	2	1	no	—
157	C₂²⋊C₄	—	158	2	no	—
158	C₄²	—	158	1	no	—
159	C₄²	C₂²×D₇₉	1	1	yes	characteristic
160	C₂³	—	2	1	no	—
161	C₂³	—	4	1	no	—
162	C₂³	—	158	2	no	—
163	Q₈	—	79	2	no	—
164	Q₈	—	158	1	no	—
165	D₄	—	2	1	no	—
166	D₄	—	2	1	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	1	no	—
169	D₄	—	4	1	no	—
170	D₄	—	4	1	no	—
171	D₄	—	4	1	no	—
172	D₄	—	316	2	no	—
173	D₄	—	316	4	no	—
174	D₄	—	316	1	no	—
175	D₄	—	316	1	no	—
176	D₄	—	316	1	no	—
177	D₄	—	316	1	no	—
178	D₄	—	632	2	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	316	4	no	—
183	C₂×C₄	—	158	1	no	—
184	C₂×C₄	—	158	1	no	—
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	C₈	—	4	1	no	—
189	C₈	—	316	1	no	—
190	C₈	—	158	2	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	1	no	—
193	C₂²	—	4	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	—	8	1	no	—
196	C₂²	—	8	1	no	—
197	C₂²	—	316	2	no	—
198	C₂²	—	316	2	no	—
199	C₂²	—	316	2	no	—
200	C₂²	#14579129	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	4	1	no	—
203	C₄	—	316	1	no	—
204	C₄	—	158	2	no	—
205	C₄	D₈×D₇₉	1	1	yes	characteristic
206	C₄	#14579181	1	1	yes	characteristic
207	C₄	#14579185	1	1	yes	characteristic
208	C₄	#14579187	1	1	yes	characteristic
209	C₂	—	8	1	no	—
210	C₂	—	16	1	no	—
211	C₂	—	632	2	no	—
212	C₂	#19098695	1	1	yes	characteristic
213	C₂	#19098706	1	1	yes	characteristic
214	C₂	#19098729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099234

C₂

yes

maximal, characteristic, complemented

#19098939

C₂

yes

maximal, characteristic, complemented

#19098760

C₂

yes

maximal, characteristic, complemented

#19098306

C₂

yes

maximal, characteristic, complemented

#19098293

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

#459×C₇₉

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

—

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

#14579123

—

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

#14579082

—

C₄×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₇₉

—

#127×C₇₉

C₂²

yes

characteristic

C₄×D₄×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

#102×C₇₉

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

D₄

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929612

—

#11929609

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₄

yes

characteristic

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂²×D₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

D₈

yes

complemented

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#120

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×D₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

100

Dic₇₉

—

101

C₂×C₇₉

—

102

C₂×C₇₉

—

103

C₂×C₇₉

#447

yes

characteristic

104

C₂×C₇₉

#449

yes

characteristic

105

C₂×C₇₉

#453

yes

characteristic

106

D₇₉

—

107

#1621

—

maximal, Sylow

108

C₇₉

#1621

yes

characteristic, complemented, Sylow

109

#550

—

110

#493

—

111

#460

—

112

#459

D₇₉

yes

characteristic, complemented

113

#331

—

114

Q₈⋊C₈

—

115

D₄⋊C₈

—

116

C₂×#41

—

158

—

117

C₂×SD₁₆

—

158

—

118

C₂×D₈

—

119

C₂×D₈

—

158

—

120

#127

—

121

#127

—

158

—

122

#127

C₂×D₇₉

yes

characteristic

123

#121

—

158

—

124

#121

—

158

—

125

C₄×Q₈

—

126

C₄×D₄

—

158

—

127

C₄×D₄

C₂×D₇₉

yes

characteristic

128

C₄⋊C₈

—

129

C₄⋊C₈

C₂×D₇₉

yes

characteristic

130

#102

—

131

#102

—

158

—

132

C₄×C₈

—

133

#41

—

316

—

134

#41

—

316

—

135

C₂×Q₈

—

136

C₂×D₄

—

137

C₂×D₄

—

138

C₂×D₄

—

139

C₂×D₄

—

140

C₂×D₄

—

158

—

141

C₂×D₄

—

158

—

142

C₂×D₄

—

158

—

143

C₂×D₄

—

158

—

144

C₂×D₄

C₂²⋊D₇₉

yes

characteristic

145

C₂²×C₄

—

146

C₂²×C₄

—

158

—

147

SD₁₆

—

316

—

148

D₈

—

149

D₈

—

316

—

150

C₂×C₈

—

151

C₂×C₈

—

158

—

152

C₂×C₈

—

158

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

158

—

155

C₄⋊C₄

C₂²⋊D₇₉

yes

characteristic

156

C₂²⋊C₄

—

157

C₂²⋊C₄

—

158

—

158

C₄²

—

158

—

159

C₄²

C₂²×D₇₉

yes

characteristic

160

C₂³

—

161

C₂³

—

162

C₂³

—

158

—

163

Q₈

—

164

Q₈

—

158

—

165

D₄

—

166

D₄

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

D₄

—

172

D₄

—

316

—

173

D₄

—

316

—

174

D₄

—

316

—

175

D₄

—

316

—

176

D₄

—

316

—

177

D₄

—

316

—

178

D₄

—

632

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

316

—

183

C₂×C₄

—

158

—

184

C₂×C₄

—

158

—

185

C₂×C₄

D₄×D₇₉

yes

characteristic

186

C₂×C₄

D₄×D₇₉

yes

characteristic

187

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

188

C₈

—

189

C₈

—

316

—

190

C₈

—

158

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

316

—

198

C₂²

—

316

—

199

C₂²

—

316

—

200

C₂²

#14579129

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

316

—

204

C₄

—

158

—

205

C₄

D₈×D₇₉

yes

characteristic

206

C₄

#14579181

yes

characteristic

207

C₄

#14579185

yes

characteristic

208

C₄

#14579187

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

—

632

—

212

C₂

#19098695

yes

characteristic

213

C₂

#19098706

yes

characteristic

214

C₂

#19098729

yes

characteristic

	subgroup	G/H
center	C₂²	#14579129
commutator, cosocle	C₄²×C₇₉	C₂³
Frattini	C₄²	C₂²×D₇₉
Fitting	#459×C₇₉	C₂
socle	C₂²×C₇₉	#120
hypercenter	#459	D₇₉
hypocenter	C₇₉	#1621
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

subgroup

G/H

center

C₂²

#14579129

commutator, cosocle

C₄²×C₇₉

C₂³

Frattini

C₄²

C₂²×D₇₉

Fitting

#459×C₇₉

C₂

C₂²×C₇₉

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #459

N	H ≅ G/N	type
1. #19099234	C₂	semidirect
2. #19098939	C₂	semidirect
3. #19098760	C₂	semidirect
4. #19098306	C₂	semidirect
5. #19098293	C₂	semidirect
6. D₄⋊(C₇₉⋊C₈)	C₂	semidirect
7. #459×C₇₉	C₂	semidirect
79. Dic₁₅₈	D₈	semidirect
108. C₇₉	#1621	semidirect
112. #459	D₇₉	semidirect

H ≅ G/N

type

1. #19099234

C₂

semidirect

2. #19098939

C₂

semidirect

3. #19098760

C₂

semidirect

4. #19098306

C₂

semidirect

5. #19098293

C₂

semidirect

6. D₄⋊(C₇₉⋊C₈)

C₂

semidirect

7. #459×C₇₉

C₂

semidirect

79. Dic₁₅₈

D₈

semidirect

108. C₇₉

#1621

semidirect

112. #459

D₇₉

semidirect

Subgroups — #23762464

Conjugacy classes of subgroups

Complements