Group #23774861 — subgroups

Subgroups — #23774861

Order 10112.

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

normal subgroups only

This group has 346 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	#19099184	C₂	1	2	yes	maximal, complemented
2	#19098859	C₂	1	2	yes	maximal, complemented
3	#19098844	C₂	1	2	yes	maximal, complemented
4	#19098809	C₂	1	2	yes	maximal, complemented
5	#19098804	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098799	C₂	1	1	yes	maximal, characteristic
7	#19098786	C₂	1	2	yes	maximal, complemented
8	#19098779	C₂	1	2	yes	maximal, complemented
9	#424×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×#11929613	C₂²	1	2	yes	—
12	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
13	#14579161	C₂²	1	2	yes	—
14	#14579156	—	2	2	no	—
15	#14579149	C₂²	1	2	yes	—
16	#14579147	—	2	2	no	—
17	#14579143	—	2	2	no	—
18	#14579140	C₂²	1	2	yes	—
19	#14579139	—	2	2	no	—
20	#14579135	C₂²	1	2	yes	—
21	C₄×Dic₁₅₈	C₂²	1	2	yes	—
22	C₄×D₃₁₆	C₂²	1	2	yes	—
23	#14579095	C₂²	1	1	yes	characteristic
24	#14579095	C₂²	1	1	yes	characteristic
25	#14579095	C₂²	1	2	yes	—
26	#14579086	C₂²	1	2	yes	—
27	#14579086	C₂²	1	2	yes	—
28	#14579083	C₂²	1	2	yes	—
29	#14579083	C₂²	1	2	yes	—
30	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	#117×C₇₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
34	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
35	C₂×Dic₁₅₈	—	2	2	no	—
36	C₂×Dic₁₅₈	C₂³	1	2	yes	—
37	C₂×D₃₁₆	C₂³	1	2	yes	—
38	C₂×C₂²⋊D₇₉	—	2	2	no	—
39	C₂×C₄×D₇₉	—	2	2	no	—
40	Dic₃₁₆	—	4	1	no	—
41	#11929613	—	4	2	no	—
42	D₆₃₂	—	4	1	no	—
43	C₄⋊Dic₇₉	—	2	2	no	—
44	C₄⋊Dic₇₉	C₂³	1	2	yes	—
45	C₄⋊Dic₇₉	C₂³	1	2	yes	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	Dic₇₉⋊C₄	—	2	2	no	—
48	C₂²⋊Dic₇₉	—	2	2	no	—
49	(C₂×D₇₉)⋊C₄	—	2	2	no	—
50	(C₂×D₇₉)⋊C₄	—	2	2	no	—
51	(C₂×D₇₉)⋊C₄	—	2	2	no	—
52	(C₂×D₇₉)⋊C₄	—	2	2	no	—
53	C₄×Dic₇₉	—	2	2	no	—
54	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
55	M₁₆×C₇₉	—	4	1	no	—
56	C₂×C₈×C₇₉	—	4	1	no	—
57	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₂³	1	2	yes	—
60	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
63	C₄²×C₇₉	C₂³	1	2	yes	—
64	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	1	no	—
68	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₇₉	—	8	2	no	—
73	C₂×Dic₇₉	—	2	2	no	—
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	D₃₁₆	—	2	2	no	—
77	D₃₁₆	—	2	2	no	—
78	C₄×D₇₉	—	8	2	no	—
79	Dic₁₅₈	—	2	2	no	—
80	Dic₁₅₈	—	2	2	no	—
81	Dic₁₅₈	—	8	2	no	—
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	—	2	2	no	—
84	C₈×C₇₉	—	2	1	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	—	2	1	no	—
88	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₇₉	—	4	2	no	—
90	C₂×D₇₉	—	4	2	no	—
91	C₂×D₇₉	—	4	2	no	—
92	C₄×C₇₉	—	2	1	no	—
93	C₄×C₇₉	—	2	2	no	—
94	C₄×C₇₉	—	2	2	no	—
95	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₇₉	—	4	2	no	—
98	Dic₇₉	—	4	2	no	—
99	Dic₇₉	—	4	2	no	—
100	C₂×C₇₉	—	4	1	no	—
101	C₂×C₇₉	#537	1	1	yes	characteristic
102	C₂×C₇₉	#576	1	1	yes	characteristic
103	C₂×C₇₉	#578	1	1	yes	characteristic
104	D₇₉	—	8	2	no	—
105	#3262	—	79	1	no	maximal, Sylow
106	C₇₉	#3262	1	1	yes	characteristic, complemented, Sylow
107	#542	—	79	2	no	—
108	#477	—	79	2	no	—
109	#474	—	79	2	no	—
110	#468	—	79	2	no	—
111	#467	—	79	1	no	—
112	#466	—	79	1	no	—
113	#464	—	79	2	no	—
114	#463	—	79	2	no	—
115	#424	D₇₉	1	1	yes	characteristic, complemented
116	C₂×Q₁₆	—	79	1	no	—
117	C₂×SD₁₆	—	79	2	no	—
118	C₂×D₈	—	79	1	no	—
119	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
120	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
121	C₄⋊Q₈	—	79	2	no	—
122	#126	—	158	2	no	—
123	#124	—	79	2	no	—
124	#124	—	158	2	no	—
125	#123	—	158	2	no	—
126	C₂²⋊Q₈	—	79	2	no	—
127	C₂²⋊Q₈	—	158	2	no	—
128	#121	—	79	2	no	—
129	C₄×Q₈	—	79	2	no	—
130	C₄×D₄	—	79	2	no	—
131	#117	C₂×D₇₉	1	1	yes	characteristic
132	#107	—	79	1	no	—
133	#107	—	79	1	no	—
134	#107	—	79	2	no	—
135	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
136	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
137	#103	—	79	2	no	—
138	#103	—	79	2	no	—
139	#102	—	79	2	no	—
140	#102	—	79	2	no	—
141	C₂×Q₈	—	79	2	no	—
142	C₂×Q₈	—	158	2	no	—
143	C₂×D₄	—	79	2	no	—
144	C₂×D₄	—	158	2	no	—
145	C₂²×C₄	—	158	2	no	—
146	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
147	Q₁₆	—	316	1	no	—
148	SD₁₆	—	316	2	no	—
149	D₈	—	316	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
153	C₂×C₈	C₂²×D₇₉	1	2	yes	—
154	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
155	C₄⋊C₄	—	79	2	no	—
156	C₄⋊C₄	—	79	2	no	—
157	C₄⋊C₄	—	158	2	no	—
158	C₄⋊C₄	—	158	2	no	—
159	C₄⋊C₄	—	158	2	no	—
160	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
161	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
162	C₂²⋊C₄	—	158	2	no	—
163	C₂²⋊C₄	—	158	2	no	—
164	C₂²⋊C₄	—	158	2	no	—
165	C₂²⋊C₄	—	158	2	no	—
166	C₂²⋊C₄	—	158	2	no	—
167	C₂²⋊C₄	D₃₁₆	1	2	yes	—
168	C₄²	—	158	2	no	—
169	C₄²	C₂²×D₇₉	1	2	yes	—
170	C₂³	—	158	2	no	—
171	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
172	Q₈	—	632	2	no	—
173	Q₈	—	158	2	no	—
174	Q₈	—	158	2	no	—
175	D₄	—	632	2	no	—
176	D₄	—	158	2	no	—
177	D₄	—	158	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	632	2	no	—
181	C₂×C₄	—	158	2	no	—
182	C₂×C₄	—	158	2	no	—
183	C₂×C₄	—	158	2	no	—
184	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
185	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
186	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
187	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	316	2	no	—
195	C₂²	—	316	2	no	—
196	C₂²	—	316	2	no	—
197	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	316	2	no	—
202	C₄	—	316	2	no	—
203	C₄	—	316	2	no	—
204	C₄	#14579206	1	1	yes	characteristic
205	C₄	#14579209	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	632	2	no	—
208	C₂	#19099144	1	1	yes	characteristic
209	C₂	#19099368	1	1	yes	characteristic
210	C₂	#19099380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099184

C₂

yes

maximal, complemented

#19098859

C₂

yes

maximal, complemented

#19098844

C₂

yes

maximal, complemented

#19098809

C₂

yes

maximal, complemented

#19098804

C₂

yes

maximal, characteristic, complemented

#19098799

C₂

yes

maximal, characteristic

#19098786

C₂

yes

maximal, complemented

#19098779

C₂

yes

maximal, complemented

#424×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929613

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

#14579161

C₂²

yes

—

#14579156

—

#14579149

C₂²

yes

—

#14579147

—

#14579143

—

#14579140

C₂²

yes

—

#14579139

—

#14579135

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

#14579095

C₂²

yes

characteristic

#14579095

C₂²

yes

characteristic

#14579095

C₂²

yes

—

#14579086

C₂²

yes

—

#14579086

C₂²

yes

—

#14579083

C₂²

yes

—

#14579083

C₂²

yes

—

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

#11929613

—

D₆₃₂

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₂²⋊C₄×C₇₉

D₄

yes

—

C₄²×C₇₉

C₂³

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

100

C₂×C₇₉

—

101

C₂×C₇₉

#537

yes

characteristic

102

C₂×C₇₉

#576

yes

characteristic

103

C₂×C₇₉

#578

yes

characteristic

104

D₇₉

—

105

#3262

—

maximal, Sylow

106

C₇₉

#3262

yes

characteristic, complemented, Sylow

107

#542

—

108

#477

—

109

#474

—

110

#468

—

111

#467

—

112

#466

—

113

#464

—

114

#463

—

115

#424

D₇₉

yes

characteristic, complemented

116

C₂×Q₁₆

—

117

C₂×SD₁₆

—

118

C₂×D₈

—

119

C₂×M₁₆

C₂×D₇₉

yes

characteristic

120

C₂²×C₈

C₂×D₇₉

yes

characteristic

121

C₄⋊Q₈

—

122

#126

—

158

—

123

#124

—

124

#124

—

158

—

125

#123

—

158

—

126

C₂²⋊Q₈

—

127

C₂²⋊Q₈

—

158

—

128

#121

—

129

C₄×Q₈

—

130

C₄×D₄

—

131

#117

C₂×D₇₉

yes

characteristic

132

#107

—

133

#107

—

134

#107

—

135

C₄⋊C₈

C₂×D₇₉

yes

—

136

C₄⋊C₈

C₂×D₇₉

yes

—

137

#103

—

138

#103

—

139

#102

—

140

#102

—

141

C₂×Q₈

—

142

C₂×Q₈

—

158

—

143

C₂×D₄

—

144

C₂×D₄

—

158

—

145

C₂²×C₄

—

158

—

146

C₂²×C₄

C₂²×D₇₉

yes

characteristic

147

Q₁₆

—

316

—

148

SD₁₆

—

316

—

149

D₈

—

316

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

C₂²×D₇₉

yes

characteristic

153

C₂×C₈

C₂²×D₇₉

yes

—

154

C₂×C₈

C₂²×D₇₉

yes

characteristic

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

—

158

C₄⋊C₄

—

158

—

159

C₄⋊C₄

—

158

—

160

C₄⋊C₄

D₃₁₆

yes

characteristic

161

C₄⋊C₄

D₃₁₆

yes

characteristic

162

C₂²⋊C₄

—

158

—

163

C₂²⋊C₄

—

158

—

164

C₂²⋊C₄

—

158

—

165

C₂²⋊C₄

—

158

—

166

C₂²⋊C₄

—

158

—

167

C₂²⋊C₄

D₃₁₆

yes

—

168

C₄²

—

158

—

169

C₄²

C₂²×D₇₉

yes

—

170

C₂³

—

158

—

171

C₂³

C₂×D₃₁₆

yes

characteristic

172

Q₈

—

632

—

173

Q₈

—

158

—

174

Q₈

—

158

—

175

D₄

—

632

—

176

D₄

—

158

—

177

D₄

—

158

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

632

—

181

C₂×C₄

—

158

—

182

C₂×C₄

—

158

—

183

C₂×C₄

—

158

—

184

C₂×C₄

C₂×D₃₁₆

yes

—

185

C₂×C₄

C₂×D₃₁₆

yes

—

186

C₂×C₄

C₂×D₃₁₆

yes

characteristic

187

C₂×C₄

C₂³×D₇₉

yes

characteristic

188

C₈

—

189

C₈

—

190

C₈

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

316

—

195

C₂²

—

316

—

196

C₂²

—

316

—

197

C₂²

C₂²×D₃₁₆

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

316

—

202

C₄

—

316

—

203

C₄

—

316

—

204

C₄

#14579206

yes

characteristic

205

C₄

#14579209

yes

characteristic

206

C₂

—

207

C₂

—

632

—

208

C₂

#19099144

yes

characteristic

209

C₂

#19099368

yes

characteristic

210

C₂

#19099380

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#424×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19098799	C₂
hypercenter	#424	D₇₉
hypocenter	C₇₉	#3262
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#424×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #19099184	C₂	semidirect
2. #19098859	C₂	semidirect
3. #19098844	C₂	semidirect
4. #19098809	C₂	semidirect
5. #19098804	C₂	semidirect
7. #19098786	C₂	semidirect
8. #19098779	C₂	semidirect
9. #424×C₇₉	C₂	semidirect
106. C₇₉	#3262	semidirect
115. #424	D₇₉	semidirect

H ≅ G/N

type

1. #19099184

C₂

semidirect

2. #19098859

C₂

semidirect

3. #19098844

C₂

semidirect

4. #19098809

C₂

semidirect

5. #19098804

C₂

semidirect

7. #19098786

C₂

semidirect

8. #19098779

C₂

semidirect

9. #424×C₇₉

C₂

semidirect

106. C₇₉

#3262

semidirect

115. #424

D₇₉

semidirect

Subgroups — #23774861

Conjugacy classes of subgroups

Complements