Group #23764921 — subgroups

Subgroups — #23764921

Order 10112.

274 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 85 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×#14579094	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14579092	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14579082	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579082	C₂	1	1	yes	maximal, characteristic
5	#19098429	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098318	C₂	1	1	yes	maximal, characteristic, complemented
7	#390×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
12	#14579094	—	2	2	no	—
13	#14579092	—	2	2	no	—
14	#14579082	—	2	2	no	—
15	#14579082	—	2	2	no	—
16	C₇₉⋊#95	—	2	2	no	—
17	C₂²×D₄×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
20	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×Dic₇₉	—	2	1	no	—
22	C₂²×Dic₇₉	—	2	1	no	—
23	C₂×C₇₉⋊C₈	—	2	1	no	—
24	C₂×C₇₉⋊C₈	—	2	1	no	—
25	C₂×C₇₉⋊C₈	—	2	1	no	—
26	C₂×C₇₉⋊C₈	—	2	1	no	—
27	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
28	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₇₉	—	2	1	no	—
30	C₄⋊Dic₇₉	—	2	1	no	—
31	C₄⋊Dic₇₉	—	2	1	no	—
32	C₄⋊Dic₇₉	—	2	1	no	—
33	C₂⁴×C₇₉	—	2	1	no	—
34	C₂×D₄×C₇₉	—	2	1	no	—
35	C₂×D₄×C₇₉	—	2	1	no	—
36	C₂×D₄×C₇₉	—	2	1	no	—
37	C₂×D₄×C₇₉	—	2	1	no	—
38	C₂×D₄×C₇₉	—	2	1	no	—
39	C₂×D₄×C₇₉	Q₈	1	2	yes	—
40	C₂²×C₄×C₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	D₄	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₇₉	—	2	1	no	—
44	C₄⋊C₄×C₇₉	—	2	1	no	—
45	C₂²⋊C₄×C₇₉	—	4	4	no	—
46	C₄²×C₇₉	—	2	1	no	—
47	C₄²×C₇₉	—	2	1	no	—
48	C₂³×C₇₉	—	2	2	no	—
49	C₂³×C₇₉	—	4	1	no	—
50	C₂³×C₇₉	—	4	1	no	—
51	C₂³×C₇₉	—	4	2	no	—
52	C₂³×C₇₉	—	4	2	no	—
53	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
54	D₄×C₇₉	—	2	2	no	—
55	D₄×C₇₉	—	2	2	no	—
56	D₄×C₇₉	—	4	1	no	—
57	D₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	—	2	2	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	4	2	no	—
63	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
67	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
68	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
69	C₂×Dic₇₉	—	4	1	no	—
70	C₂×Dic₇₉	—	4	1	no	—
71	C₂×Dic₇₉	—	4	2	no	—
72	C₂×Dic₇₉	—	4	2	no	—
73	C₇₉⋊C₈	—	2	2	no	—
74	C₇₉⋊C₈	—	2	2	no	—
75	C₂²×C₇₉	—	4	2	no	—
76	C₂²×C₇₉	—	4	4	no	—
77	C₂²×C₇₉	—	8	2	no	—
78	C₂²×C₇₉	—	8	1	no	—
79	C₂²×C₇₉	—	8	1	no	—
80	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂²×C₇₉	#123	1	1	yes	characteristic
82	C₂²×C₇₉	#127	1	1	yes	characteristic
83	C₂²×C₇₉	#135	1	2	yes	—
84	C₂²×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
85	C₄×C₇₉	—	4	2	no	—
86	C₄×C₇₉	—	8	1	no	—
87	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
88	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄×C₇₉	#123	1	1	yes	characteristic
90	C₄×C₇₉	#123	1	1	yes	characteristic
91	Dic₇₉	—	8	1	no	—
92	Dic₇₉	—	8	1	no	—
93	C₂×C₇₉	—	8	2	no	—
94	C₂×C₇₉	#399	1	1	yes	characteristic
95	C₂×C₇₉	#478	1	2	yes	—
96	C₂×C₇₉	#482	1	2	yes	—
97	C₂×C₇₉	#495	1	1	yes	characteristic
98	C₂×C₇₉	#496	1	1	yes	characteristic
99	#2092	—	79	1	no	maximal, Sylow
100	C₇₉	#2092	1	1	yes	characteristic, complemented, Sylow
101	C₂×#107	—	79	1	no	—
102	C₂×#106	—	79	1	no	—
103	C₂×#102	—	79	1	no	—
104	C₂×#102	—	79	1	no	—
105	#390	D₇₉	1	1	yes	characteristic
106	#382	—	79	1	no	—
107	#336	—	79	1	no	—
108	C₂²×D₄	C₂×D₇₉	1	1	yes	characteristic
109	C₂²×C₈	—	79	1	no	—
110	C₂²×C₈	—	79	1	no	—
111	C₂×C₄⋊C₄	—	79	1	no	—
112	C₂×C₄⋊C₄	—	79	1	no	—
113	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	—	2	2	no	—
115	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
116	#107	—	158	2	no	—
117	#106	—	158	2	no	—
118	#102	—	158	2	no	—
119	#102	—	158	2	no	—
120	#95	—	158	2	no	—
121	C₂⁴	—	2	1	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	2	1	no	—
125	C₂×D₄	—	2	1	no	—
126	C₂×D₄	—	2	1	no	—
127	C₂×D₄	Dic₁₅₈	1	2	yes	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	158	1	no	—
130	C₂²×C₄	—	158	1	no	—
131	C₂²×C₄	C₂²⋊D₇₉	1	2	yes	—
132	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
133	C₂×C₈	—	79	2	no	—
134	C₂×C₈	—	79	2	no	—
135	C₂×C₈	—	158	1	no	—
136	C₂×C₈	—	158	1	no	—
137	C₂×C₈	—	158	1	no	—
138	C₂×C₈	—	158	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	1	no	—
141	C₄⋊C₄	—	158	1	no	—
142	C₄⋊C₄	—	158	1	no	—
143	C₄⋊C₄	—	158	1	no	—
144	C₄⋊C₄	—	158	1	no	—
145	C₂²⋊C₄	—	4	4	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	2	1	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	4	1	no	—
150	C₂³	—	4	1	no	—
151	C₂³	—	4	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
154	D₄	—	2	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	1	no	—
157	D₄	—	4	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	316	1	no	—
164	C₂×C₄	—	316	1	no	—
165	C₂×C₄	—	316	2	no	—
166	C₂×C₄	—	316	2	no	—
167	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
168	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
170	C₂×C₄	#11929623	1	1	yes	characteristic
171	C₂×C₄	#11929623	1	1	yes	characteristic
172	C₂×C₄	#11929624	1	1	yes	characteristic
173	C₈	—	158	2	no	—
174	C₈	—	158	2	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	4	no	—
177	C₂²	—	8	2	no	—
178	C₂²	—	8	1	no	—
179	C₂²	—	8	1	no	—
180	C₂²	#14579140	1	1	yes	characteristic
181	C₂²	#14579142	1	1	yes	characteristic
182	C₂²	#14579157	1	1	yes	characteristic
183	C₂²	#14579181	1	2	yes	—
184	C₂²	#14579187	1	2	yes	—
185	C₄	—	4	2	no	—
186	C₄	—	8	1	no	—
187	C₄	—	632	1	no	—
188	C₄	—	632	1	no	—
189	C₄	#14579138	1	1	yes	characteristic
190	C₄	#14579138	1	1	yes	characteristic
191	C₄	#14579143	1	1	yes	characteristic
192	C₄	#14579143	1	1	yes	characteristic
193	C₂	—	8	2	no	—
194	C₂	#19098499	1	1	yes	characteristic
195	C₂	#19098863	1	2	yes	—
196	C₂	#19098884	1	2	yes	—
197	C₂	#19098945	1	1	yes	characteristic
198	C₂	#19098950	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579094

C₂

yes

maximal, characteristic, complemented

C₂×#14579092

C₂

yes

maximal, characteristic, complemented

C₂×#14579082

C₂

yes

maximal, characteristic

C₂×#14579082

C₂

yes

maximal, characteristic

#19098429

C₂

yes

maximal, characteristic, complemented

#19098318

C₂

yes

maximal, characteristic, complemented

#390×C₇₉

C₂

yes

maximal, characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

—

—

—

—

C₇₉⋊#95

—

C₂²×D₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₂⁴×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

Q₈

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₄

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×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×Q₈

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂²⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#399

yes

characteristic

C₂×C₇₉

#478

yes

—

C₂×C₇₉

#482

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

100

C₇₉

#2092

yes

characteristic, complemented, Sylow

101

C₂×#107

—

102

C₂×#106

—

103

C₂×#102

—

104

C₂×#102

—

105

#390

D₇₉

yes

characteristic

106

#382

—

107

#336

—

108

C₂²×D₄

C₂×D₇₉

yes

characteristic

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

114

C₂×C₂²⋊C₄

—

115

C₂×C₄²

C₂×D₇₉

yes

characteristic

116

#107

—

158

—

117

#106

—

158

—

118

#102

—

158

—

119

#102

—

158

—

120

#95

—

158

—

121

C₂⁴

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

Dic₁₅₈

yes

—

128

C₂²×C₄

—

129

C₂²×C₄

—

158

—

130

C₂²×C₄

—

158

—

131

C₂²×C₄

C₂²⋊D₇₉

yes

—

132

C₂²×C₄

C₂²×D₇₉

yes

characteristic

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

158

—

136

C₂×C₈

—

158

—

137

C₂×C₈

—

158

—

138

C₂×C₈

—

158

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

158

—

142

C₄⋊C₄

—

158

—

143

C₄⋊C₄

—

158

—

144

C₄⋊C₄

—

158

—

145

C₂²⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

C₂×C₂²⋊D₇₉

yes

characteristic

154

D₄

—

155

D₄

—

156

D₄

—

157

D₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

316

—

164

C₂×C₄

—

316

—

165

C₂×C₄

—

316

—

166

C₂×C₄

—

316

—

167

C₂×C₄

D₄×D₇₉

yes

characteristic

168

C₂×C₄

D₄×D₇₉

yes

characteristic

169

C₂×C₄

C₂×Dic₁₅₈

yes

characteristic

170

C₂×C₄

#11929623

yes

characteristic

171

C₂×C₄

#11929623

yes

characteristic

172

C₂×C₄

#11929624

yes

characteristic

173

C₈

—

158

—

174

C₈

—

158

—

175

C₂²

—

176

C₂²

—

177

C₂²

—

178

C₂²

—

179

C₂²

—

180

C₂²

#14579140

yes

characteristic

181

C₂²

#14579142

yes

characteristic

182

C₂²

#14579157

yes

characteristic

183

C₂²

#14579181

yes

—

184

C₂²

#14579187

yes

—

185

C₄

—

186

C₄

—

187

C₄

—

632

—

188

C₄

—

632

—

189

C₄

#14579138

yes

characteristic

190

C₄

#14579138

yes

characteristic

191

C₄

#14579143

yes

characteristic

192

C₄

#14579143

yes

characteristic

193

C₂

—

194

C₂

#19098499

yes

characteristic

195

C₂

#19098863

yes

—

196

C₂

#19098884

yes

—

197

C₂

#19098945

yes

characteristic

198

C₂

#19098950

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#390×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₄
prime radical	C₂²×D₄×C₇₉	C₂²
hypercenter	#390	D₇₉
hypocenter	C₇₉	#2092
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#390×C₇₉

C₂

socle

C₂³×C₇₉

C₂×D₄

prime radical

C₂²×D₄×C₇₉

C₂²

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. C₂×#14579094	C₂	semidirect
2. C₂×#14579092	C₂	semidirect
5. #19098429	C₂	semidirect
6. #19098318	C₂	semidirect
100. C₇₉	#2092	semidirect

H ≅ G/N

type

1. C₂×#14579094

C₂

semidirect

2. C₂×#14579092

C₂

semidirect

5. #19098429

C₂

semidirect

6. #19098318

C₂

semidirect

100. C₇₉

#2092

semidirect

Subgroups — #23764921

Conjugacy classes of subgroups

Complements