Group #23774866 — subgroups

Subgroups — #23774866

Order 10112.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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	#19099186	C₂	1	2	yes	maximal, complemented
2	#19098852	C₂	1	2	yes	maximal, complemented
3	#19098846	C₂	1	2	yes	maximal, complemented
4	#19098816	C₂	1	1	yes	maximal, characteristic
5	#19098794	C₂	1	2	yes	maximal, complemented
6	#19098787	C₂	1	2	yes	maximal, complemented
7	#19098781	C₂	1	2	yes	maximal, complemented
8	#19098591	C₂	1	1	yes	maximal, characteristic, complemented
9	#470×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929611	C₂²	1	2	yes	—
11	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#14579160	C₂²	1	2	yes	—
14	#14579156	—	2	2	no	—
15	#14579150	C₂²	1	2	yes	—
16	#14579147	—	2	2	no	—
17	#14579143	—	2	2	no	—
18	#14579141	—	2	2	no	—
19	#14579135	C₂²	1	2	yes	—
20	Q₈×Dic₇₉	C₂²	1	2	yes	—
21	#14579123	C₂²	1	2	yes	—
22	#14579120	C₂²	1	1	yes	characteristic
23	#14579093	C₂²	1	1	yes	characteristic
24	#14579092	C₂²	1	2	yes	—
25	Dic₇₉⋊C₈	C₂²	1	2	yes	—
26	Dic₇₉⋊C₈	C₂²	1	2	yes	—
27	#14579085	C₂²	1	2	yes	—
28	#14579083	C₂²	1	2	yes	—
29	#14579081	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
32	C₂²⋊Q₈×C₇₉	C₂²	1	2	yes	—
33	#106×C₇₉	C₂²	1	1	yes	characteristic
34	#103×C₇₉	C₂²	1	2	yes	—
35	C₂×Dic₁₅₈	—	2	2	no	—
36	C₂×D₃₁₆	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₇₉	—	2	2	no	—
38	C₂×C₄×D₇₉	—	2	2	no	—
39	#11929611	—	4	2	no	—
40	D₆₃₂	—	4	1	no	—
41	C₂×C₇₉⋊C₈	—	4	1	no	—
42	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
43	C₄⋊Dic₇₉	—	2	2	no	—
44	C₄⋊Dic₇₉	D₄	1	2	yes	—
45	C₄⋊Dic₇₉	C₂³	1	2	yes	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	C₂²⋊Dic₇₉	D₄	1	1	yes	characteristic
48	C₂²⋊Dic₇₉	D₄	1	1	yes	characteristic
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	C₄×Dic₇₉	C₂³	1	2	yes	—
55	C₂×Q₈×C₇₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
57	Q₁₆×C₇₉	—	4	1	no	—
58	M₁₆×C₇₉	—	4	1	no	—
59	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	—	2	2	no	—
62	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₇₉	—	2	2	no	—
64	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×D₇₉	—	2	2	no	—
66	Q₈×C₇₉	—	2	2	no	—
67	Q₈×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
74	C₂²⋊D₇₉	—	8	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
77	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
78	D₃₁₆	—	2	2	no	—
79	D₃₁₆	—	2	2	no	—
80	C₄×D₇₉	—	8	2	no	—
81	Dic₁₅₈	—	8	2	no	—
82	C₈×C₇₉	—	2	1	no	—
83	C₈×C₇₉	—	2	1	no	—
84	C₇₉⋊C₈	—	2	2	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	2	no	—
87	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
88	C₂×D₇₉	—	4	2	no	—
89	C₂×D₇₉	—	4	2	no	—
90	C₂×D₇₉	—	4	2	no	—
91	C₄×C₇₉	—	2	1	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	—	4	2	no	—
94	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
96	Dic₇₉	—	2	2	no	—
97	Dic₇₉	—	2	2	no	—
98	Dic₇₉	—	4	2	no	—
99	C₂×C₇₉	—	4	1	no	—
100	C₂×C₇₉	#537	1	1	yes	characteristic
101	C₂×C₇₉	#577	1	2	yes	—
102	D₇₉	—	8	2	no	—
103	#3263	—	79	1	no	maximal, Sylow
104	C₇₉	#3263	1	1	yes	characteristic, complemented, Sylow
105	#542	—	79	2	no	—
106	#476	—	79	2	no	—
107	#475	—	79	2	no	—
108	#470	D₇₉	1	1	yes	characteristic, complemented
109	#469	—	79	1	no	—
110	#465	—	79	2	no	—
111	#464	—	79	2	no	—
112	#463	—	79	2	no	—
113	#424	—	79	1	no	—
114	C₂×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
115	C₂×SD₁₆	—	79	2	no	—
116	C₂×D₈	—	79	1	no	—
117	C₂×M₁₆	C₂×D₇₉	1	1	yes	characteristic
118	C₂²×C₈	—	79	1	no	—
119	C₄⋊Q₈	—	79	2	no	—
120	#126	—	158	2	no	—
121	#124	—	79	2	no	—
122	#124	—	158	2	no	—
123	#123	—	158	2	no	—
124	C₂²⋊Q₈	—	158	2	no	—
125	C₂²⋊Q₈	C₂×D₇₉	1	2	yes	—
126	#121	—	79	2	no	—
127	C₄×Q₈	—	79	2	no	—
128	C₄×D₄	—	79	2	no	—
129	#117	—	79	1	no	—
130	#106	—	79	1	no	—
131	#106	—	79	2	no	—
132	#106	C₂×D₇₉	1	1	yes	characteristic
133	C₄⋊C₈	—	79	2	no	—
134	C₄⋊C₈	—	79	2	no	—
135	#103	—	79	2	no	—
136	#103	C₂×D₇₉	1	2	yes	—
137	#102	—	79	2	no	—
138	#102	—	79	2	no	—
139	C₂×Q₈	—	158	2	no	—
140	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
141	C₂×D₄	—	79	2	no	—
142	C₂×D₄	—	158	2	no	—
143	C₂²×C₄	—	158	2	no	—
144	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	316	2	no	—
147	D₈	—	316	1	no	—
148	M₁₆	—	4	1	no	—
149	C₂×C₈	—	316	1	no	—
150	C₂×C₈	—	79	2	no	—
151	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	79	2	no	—
155	C₄⋊C₄	—	79	2	no	—
156	C₄⋊C₄	—	158	2	no	—
157	C₄⋊C₄	—	158	2	no	—
158	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	79	1	no	—
161	C₂²⋊C₄	—	79	1	no	—
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	C₄²	—	79	2	no	—
167	C₄²	—	158	2	no	—
168	C₂³	—	158	2	no	—
169	C₂³	D₄×D₇₉	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	632	2	no	—
173	D₄	—	632	2	no	—
174	D₄	—	158	2	no	—
175	D₄	—	158	2	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	79	2	no	—
181	C₂×C₄	—	79	2	no	—
182	C₂×C₄	—	632	2	no	—
183	C₂×C₄	—	158	2	no	—
184	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	158	2	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	316	2	no	—
192	C₂²	—	316	2	no	—
193	C₂²	—	316	2	no	—
194	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	316	2	no	—
199	C₄	—	158	2	no	—
200	C₄	—	158	2	no	—
201	C₄	#14579206	1	1	yes	characteristic
202	C₄	#14579208	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	632	2	no	—
205	C₂	#19099140	1	1	yes	characteristic
206	C₂	#19099372	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19099186

C₂

yes

maximal, complemented

#19098852

C₂

yes

maximal, complemented

#19098846

C₂

yes

maximal, complemented

#19098816

C₂

yes

maximal, characteristic

#19098794

C₂

yes

maximal, complemented

#19098787

C₂

yes

maximal, complemented

#19098781

C₂

yes

maximal, complemented

#19098591

C₂

yes

maximal, characteristic, complemented

#470×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579160

C₂²

yes

—

#14579156

—

#14579150

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₇₉

C₂²

yes

—

#14579123

C₂²

yes

—

#14579120

C₂²

yes

characteristic

#14579093

C₂²

yes

characteristic

#14579092

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

#14579085

C₂²

yes

—

#14579083

C₂²

yes

—

#14579081

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂×M₁₆×C₇₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₉

C₂²

yes

—

#106×C₇₉

C₂²

yes

characteristic

#103×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929611

—

D₆₃₂

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

D₄

yes

characteristic

C₂²⋊Dic₇₉

D₄

yes

characteristic

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

Q₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

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

—

C₂×C₇₉

—

100

C₂×C₇₉

#537

yes

characteristic

101

C₂×C₇₉

#577

yes

—

102

D₇₉

—

103

#3263

—

maximal, Sylow

104

C₇₉

#3263

yes

characteristic, complemented, Sylow

105

#542

—

106

#476

—

107

#475

—

108

#470

D₇₉

yes

characteristic, complemented

109

#469

—

110

#465

—

111

#464

—

112

#463

—

113

#424

—

114

C₂×Q₁₆

C₂×D₇₉

yes

characteristic

115

C₂×SD₁₆

—

116

C₂×D₈

—

117

C₂×M₁₆

C₂×D₇₉

yes

characteristic

118

C₂²×C₈

—

119

C₄⋊Q₈

—

120

#126

—

158

—

121

#124

—

122

#124

—

158

—

123

#123

—

158

—

124

C₂²⋊Q₈

—

158

—

125

C₂²⋊Q₈

C₂×D₇₉

yes

—

126

#121

—

127

C₄×Q₈

—

128

C₄×D₄

—

129

#117

—

130

#106

—

131

#106

—

132

#106

C₂×D₇₉

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

C₂×D₇₉

yes

—

137

#102

—

138

#102

—

139

C₂×Q₈

—

158

—

140

C₂×Q₈

C₂²×D₇₉

yes

—

141

C₂×D₄

—

142

C₂×D₄

—

158

—

143

C₂²×C₄

—

158

—

144

C₂²×C₄

C₂²×D₇₉

yes

characteristic

145

Q₁₆

—

146

SD₁₆

—

316

—

147

D₈

—

316

—

148

M₁₆

—

149

C₂×C₈

—

316

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×D₇₉

yes

characteristic

152

C₂×C₈

C₂²×D₇₉

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

—

158

—

158

C₄⋊C₄

C₂²×D₇₉

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

158

—

163

C₂²⋊C₄

—

158

—

164

C₂²⋊C₄

—

158

—

165

C₂²⋊C₄

—

158

—

166

C₄²

—

167

C₄²

—

158

—

168

C₂³

—

158

—

169

C₂³

D₄×D₇₉

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

632

—

173

D₄

—

632

—

174

D₄

—

158

—

175

D₄

—

158

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

632

—

183

C₂×C₄

—

158

—

184

C₂×C₄

D₄×D₇₉

yes

characteristic

185

C₂×C₄

C₂³×D₇₉

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

158

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

316

—

192

C₂²

—

316

—

193

C₂²

—

316

—

194

C₂²

C₂×D₄×D₇₉

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

316

—

199

C₄

—

158

—

200

C₄

—

158

—

201

C₄

#14579206

yes

characteristic

202

C₄

#14579208

yes

characteristic

203

C₂

—

204

C₂

—

632

—

205

C₂

#19099140

yes

characteristic

206

C₂

#19099372

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#470×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19098816	C₂
hypercenter	#470	D₇₉
hypocenter	C₇₉	#3263
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#470×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

N	H ≅ G/N	type
1. #19099186	C₂	semidirect
2. #19098852	C₂	semidirect
3. #19098846	C₂	semidirect
5. #19098794	C₂	semidirect
6. #19098787	C₂	semidirect
7. #19098781	C₂	semidirect
8. #19098591	C₂	semidirect
9. #470×C₇₉	C₂	semidirect
104. C₇₉	#3263	semidirect
108. #470	D₇₉	semidirect

H ≅ G/N

type

1. #19099186

C₂

semidirect

2. #19098852

C₂

semidirect

3. #19098846

C₂

semidirect

5. #19098794

C₂

semidirect

6. #19098787

C₂

semidirect

7. #19098781

C₂

semidirect

8. #19098591

C₂

semidirect

9. #470×C₇₉

C₂

semidirect

104. C₇₉

#3263

semidirect

108. #470

D₇₉

semidirect

Subgroups — #23774866

Conjugacy classes of subgroups

Complements