Group #23763790 — subgroups

Subgroups — #23763790

Order 10112.

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

normal subgroups only

This group has 324 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	C₂×#14579159	C₂	1	1	yes	maximal, characteristic
2	#19099040	C₂	1	1	yes	maximal, characteristic
3	#19098572	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579079	C₂	1	1	yes	maximal, characteristic
5	#19098324	C₂	1	1	yes	maximal, characteristic
6	#19098322	C₂	1	1	yes	maximal, characteristic
7	#419×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×#11929608	C₂²	1	1	yes	characteristic
10	#14579159	—	2	2	no	—
11	#14579159	C₄	1	4	yes	complemented
12	C₄×Dic₁₅₈	—	2	2	no	—
13	#14579126	—	2	2	no	—
14	#14579120	C₂²	1	1	yes	characteristic
15	#14579117	—	2	2	no	—
16	C₂×Dic₇₉⋊C₄	—	2	2	no	—
17	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
18	#14579085	—	2	2	no	—
19	#14579079	—	2	2	no	—
20	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
22	#117×C₇₉	C₂²	1	1	yes	characteristic
23	#103×C₇₉	—	2	2	no	—
24	C₂×Dic₁₅₈	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	1	no	—
28	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₅₈	D₄	1	2	yes	—
30	C₂²×Dic₇₉	—	2	1	no	—
31	C₂²×Dic₇₉	D₄	1	2	yes	—
32	#11929608	—	2	1	no	—
33	#11929608	—	2	1	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₄⋊Dic₇₉	D₄	1	2	yes	—
36	Dic₇₉⋊C₄	—	2	4	no	—
37	Dic₇₉⋊C₄	—	2	8	no	—
38	C₂²⋊Dic₇₉	—	2	1	no	—
39	C₄×Dic₇₉	—	2	4	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₄×Dic₇₉	—	2	1	no	—
42	C₄×Dic₇₉	C₂×C₄	1	2	yes	—
43	C₂×Q₈×C₇₉	—	2	1	no	—
44	C₂×Q₈×C₇₉	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₇₉	—	2	1	no	—
50	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
51	M₁₆×C₇₉	—	2	1	no	—
52	M₁₆×C₇₉	—	2	1	no	—
53	C₂×C₈×C₇₉	—	2	1	no	—
54	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
55	C₂²⋊C₄×C₇₉	—	2	1	no	—
56	C₄²×C₇₉	—	2	1	no	—
57	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
58	Q₈×C₇₉	—	2	2	no	—
59	Q₈×C₇₉	—	2	2	no	—
60	Q₈×C₇₉	—	4	1	no	—
61	Q₈×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	4	1	no	—
65	C₂×C₄×C₇₉	—	4	1	no	—
66	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
71	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
72	C₂×Dic₇₉	—	2	4	no	—
73	C₂×Dic₇₉	—	2	2	no	—
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	—	2	2	no	—
77	C₂×Dic₇₉	C₂²⋊C₄	1	4	yes	—
78	Dic₁₅₈	—	2	2	no	—
79	Dic₁₅₈	—	2	2	no	—
80	Dic₁₅₈	—	2	2	no	—
81	Dic₁₅₈	—	2	2	no	—
82	C₈×C₇₉	—	4	1	no	—
83	C₇₉⋊C₈	—	4	1	no	—
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₇₉	#120	1	1	yes	characteristic
87	C₂²×C₇₉	#123	1	1	yes	characteristic
88	C₄×C₇₉	—	4	2	no	—
89	C₄×C₇₉	—	4	2	no	—
90	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
91	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₇₉	#121	1	1	yes	characteristic
93	C₄×C₇₉	#121	1	1	yes	characteristic
94	Dic₇₉	—	2	4	no	—
95	Dic₇₉	—	2	4	no	—
96	Dic₇₉	—	4	2	no	—
97	C₂×C₇₉	—	2	1	no	—
98	C₂×C₇₉	—	2	1	no	—
99	C₂×C₇₉	#386	1	1	yes	characteristic
100	C₂×C₇₉	#456	1	2	yes	—
101	#1886	—	79	1	no	maximal, Sylow
102	C₇₉	#1886	1	1	yes	characteristic, complemented, Sylow
103	C₂×C₄⋊Q₈	—	79	1	no	—
104	#519	—	79	1	no	—
105	#419	—	79	1	no	—
106	#419	D₇₉	1	1	yes	characteristic
107	C₂×#101	—	79	1	no	—
108	#339	—	79	1	no	—
109	#339	—	79	1	no	—
110	C₂²×Q₈	—	79	1	no	—
111	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
112	C₂×M₁₆	—	79	1	no	—
113	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
114	C₄⋊Q₈	—	79	4	no	—
115	C₄⋊Q₈	—	158	2	no	—
116	C₄×Q₈	—	158	2	no	—
117	C₄×Q₈	—	158	2	no	—
118	#117	—	79	1	no	—
119	#117	—	158	2	no	—
120	#117	C₂×D₇₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	—	158	2	no	—
122	C₂×C₄²	—	79	1	no	—
123	#103	—	2	2	no	—
124	#103	—	158	2	no	—
125	#101	—	158	2	no	—
126	C₂×Q₈	—	2	1	no	—
127	C₂×Q₈	—	2	1	no	—
128	C₂×Q₈	—	2	1	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	79	2	no	—
132	C₂×Q₈	—	79	2	no	—
133	C₂×Q₈	—	158	1	no	—
134	C₂×Q₈	—	158	1	no	—
135	C₂×Q₈	—	158	1	no	—
136	C₂×Q₈	—	158	1	no	—
137	C₂×Q₈	C₄×D₇₉	1	2	yes	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	79	2	no	—
140	C₂²×C₄	—	158	1	no	—
141	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
142	M₁₆	—	2	1	no	—
143	M₁₆	—	2	1	no	—
144	M₁₆	—	158	1	no	—
145	M₁₆	—	158	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	158	1	no	—
148	C₄⋊C₄	—	79	2	no	—
149	C₄⋊C₄	—	158	8	no	—
150	C₄⋊C₄	—	158	4	no	—
151	C₄⋊C₄	C₂²⋊D₇₉	1	2	yes	—
152	C₂²⋊C₄	—	2	1	no	—
153	C₂²⋊C₄	—	158	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	79	2	no	—
156	C₄²	—	158	4	no	—
157	C₄²	—	158	1	no	—
158	C₄²	—	158	1	no	—
159	C₂³	D₄×D₇₉	1	1	yes	characteristic
160	Q₈	—	2	2	no	—
161	Q₈	—	2	2	no	—
162	Q₈	—	4	1	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	158	2	no	—
165	Q₈	—	158	2	no	—
166	Q₈	—	158	2	no	—
167	Q₈	—	158	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	79	4	no	—
173	C₂×C₄	—	158	2	no	—
174	C₂×C₄	—	158	4	no	—
175	C₂×C₄	—	158	2	no	—
176	C₂×C₄	—	158	2	no	—
177	C₂×C₄	—	158	2	no	—
178	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
179	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
182	C₂×C₄	#11929623	1	1	yes	characteristic
183	C₂×C₄	#11929624	1	1	yes	characteristic
184	C₈	—	4	1	no	—
185	C₈	—	316	1	no	—
186	C₂²	—	2	2	no	—
187	C₂²	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
188	C₂²	#14579129	1	1	yes	characteristic
189	C₂²	#14579143	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	4	2	no	—
192	C₄	—	316	2	no	—
193	C₄	—	158	4	no	—
194	C₄	—	158	4	no	—
195	C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
196	C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
197	C₄	#14579132	1	1	yes	characteristic
198	C₄	#14579134	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#19098439	1	1	yes	characteristic
202	C₂	#19098744	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579159

C₂

yes

maximal, characteristic

#19099040

C₂

yes

maximal, characteristic

#19098572

C₂

yes

maximal, characteristic

C₂×#14579079

C₂

yes

maximal, characteristic

#19098324

C₂

yes

maximal, characteristic

#19098322

C₂

yes

maximal, characteristic

#419×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂×#11929608

C₂²

yes

characteristic

#14579159

—

#14579159

C₄

yes

complemented

C₄×Dic₁₅₈

—

#14579126

—

#14579120

C₂²

yes

characteristic

#14579117

—

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579085

—

#14579079

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

#117×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂×Dic₁₅₈

D₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

—

#11929608

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂×C₄

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

C₂×D₄

yes

characteristic

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

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂²⋊C₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₂²⋊C₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₄×D₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#386

yes

characteristic

100

C₂×C₇₉

#456

yes

—

101

#1886

—

maximal, Sylow

102

C₇₉

#1886

yes

characteristic, complemented, Sylow

103

C₂×C₄⋊Q₈

—

104

#519

—

105

#419

—

106

#419

D₇₉

yes

characteristic

107

C₂×#101

—

108

#339

—

109

#339

—

110

C₂²×Q₈

—

111

C₂²×Q₈

C₂×D₇₉

yes

characteristic

112

C₂×M₁₆

—

113

C₂×M₁₆

C₂×D₇₉

yes

characteristic

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

158

—

116

C₄×Q₈

—

158

—

117

C₄×Q₈

—

158

—

118

#117

—

119

#117

—

158

—

120

#117

C₂×D₇₉

yes

characteristic

121

C₂×C₄⋊C₄

—

158

—

122

C₂×C₄²

—

123

#103

—

124

#103

—

158

—

125

#101

—

158

—

126

C₂×Q₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

158

—

134

C₂×Q₈

—

158

—

135

C₂×Q₈

—

158

—

136

C₂×Q₈

—

158

—

137

C₂×Q₈

C₄×D₇₉

yes

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

158

—

141

C₂²×C₄

C₂²×D₇₉

yes

characteristic

142

M₁₆

—

143

M₁₆

—

144

M₁₆

—

158

—

145

M₁₆

—

158

—

146

C₂×C₈

—

147

C₂×C₈

—

158

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

158

—

150

C₄⋊C₄

—

158

—

151

C₄⋊C₄

C₂²⋊D₇₉

yes

—

152

C₂²⋊C₄

—

153

C₂²⋊C₄

—

158

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

158

—

157

C₄²

—

158

—

158

C₄²

—

158

—

159

C₂³

D₄×D₇₉

yes

characteristic

160

Q₈

—

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

158

—

165

Q₈

—

158

—

166

Q₈

—

158

—

167

Q₈

—

158

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

158

—

174

C₂×C₄

—

158

—

175

C₂×C₄

—

158

—

176

C₂×C₄

—

158

—

177

C₂×C₄

—

158

—

178

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

179

C₂×C₄

D₄×D₇₉

yes

characteristic

180

C₂×C₄

D₄×D₇₉

yes

characteristic

181

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

182

C₂×C₄

#11929623

yes

characteristic

183

C₂×C₄

#11929624

yes

characteristic

184

C₈

—

185

C₈

—

316

—

186

C₂²

—

187

C₂²

C₂²⋊C₄×D₇₉

yes

characteristic

188

C₂²

#14579129

yes

characteristic

189

C₂²

#14579143

yes

characteristic

190

C₄

—

191

C₄

—

192

C₄

—

316

—

193

C₄

—

158

—

194

C₄

—

158

—

195

C₄

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

196

C₄

C₄×C₂²⋊D₇₉

yes

characteristic

197

C₄

#14579132

yes

characteristic

198

C₄

#14579134

yes

characteristic

199

C₂

—

200

C₂

—

201

C₂

#19098439

yes

characteristic

202

C₂

#19098744

yes

—

	subgroup	G/H
center	C₂²	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#419×C₇₉	C₂
socle	C₂²×C₇₉	#120
prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	#419	D₇₉
hypocenter	C₇₉	#1886
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

subgroup

G/H

center

C₂²

#14579129

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#419×C₇₉

C₂

socle

C₂²×C₇₉

#120

prime radical

C₂³×C₇₉

C₂×D₄

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
11. #14579159	C₄	semidirect
102. C₇₉	#1886	semidirect

H ≅ G/N

type

11. #14579159

C₄

semidirect

102. C₇₉

#1886

semidirect

Subgroups — #23763790

Conjugacy classes of subgroups

Complements