Group #23774600 — subgroups

Subgroups — #23774600

Order 10112.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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	#19099120	C₂	1	1	yes	maximal, characteristic, complemented
2	#19099113	C₂	1	1	yes	maximal, characteristic
3	#19098972	C₂	1	2	yes	maximal, complemented
4	#19098928	C₂	1	2	yes	maximal, complemented
5	#19098888	C₂	1	4	yes	maximal, complemented
6	#19098854	C₂	1	2	yes	maximal, complemented
7	#19098844	C₂	1	2	yes	maximal, complemented
8	#432×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11929624	—	2	1	no	—
10	#14579161	C₂²	1	1	yes	characteristic
11	#14579161	C₂²	1	1	yes	characteristic
12	#14579161	C₂²	1	1	yes	characteristic
13	#14579158	C₂²	1	1	yes	characteristic
14	#14579153	C₂²	1	2	yes	—
15	#14579149	—	2	1	no	—
16	#14579140	—	2	1	no	—
17	#14579140	—	2	1	no	—
18	#14579135	—	2	1	no	—
19	#14579135	C₂²	1	2	yes	—
20	C₄×Dic₁₅₈	—	2	1	no	—
21	C₄×Dic₁₅₈	—	2	1	no	—
22	C₄×D₃₁₆	—	2	1	no	—
23	C₄×D₃₁₆	C₂²	1	2	yes	—
24	#14579120	C₂²	1	2	yes	—
25	#14579095	C₂²	1	4	yes	—
26	#14579093	C₂²	1	4	yes	—
27	#14579083	C₂²	1	4	yes	—
28	#14579083	C₂²	1	4	yes	—
29	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
31	C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
32	#97×C₇₉	C₂²	1	2	yes	—
33	#11929624	—	4	4	no	—
34	C₂×Dic₁₅₈	—	2	1	no	—
35	C₂×Dic₁₅₈	—	2	1	no	—
36	C₂×Dic₁₅₈	—	2	1	no	—
37	C₂×D₃₁₆	—	2	1	no	—
38	C₂×D₃₁₆	—	2	1	no	—
39	C₂×D₃₁₆	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₇₉	—	2	2	no	—
41	C₂×C₄×D₇₉	—	2	2	no	—
42	C₄⋊Dic₇₉	—	2	1	no	—
43	C₄⋊Dic₇₉	—	2	1	no	—
44	C₄⋊Dic₇₉	—	2	1	no	—
45	C₄⋊Dic₇₉	C₂³	1	2	yes	—
46	C₄⋊Dic₇₉	C₂³	1	2	yes	—
47	C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	Dic₇₉⋊C₄	—	2	2	no	—
49	Dic₇₉⋊C₄	—	2	2	no	—
50	C₂²⋊Dic₇₉	—	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	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₉	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂³	1	4	yes	—
58	C₄²×C₇₉	D₄	1	1	yes	characteristic
59	C₄²×C₇₉	D₄	1	1	yes	characteristic
60	C₄²×C₇₉	C₂³	1	1	yes	characteristic
61	C₄²×C₇₉	C₂³	1	1	yes	characteristic
62	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×D₇₉	—	2	2	no	—
64	C₂×C₄×C₇₉	—	2	2	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₇₉	—	4	4	no	—
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	—	2	2	no	—
77	D₃₁₆	—	2	4	no	—
78	D₃₁₆	—	4	2	no	—
79	D₃₁₆	—	4	2	no	—
80	C₄×D₇₉	—	4	4	no	—
81	Dic₁₅₈	—	4	2	no	—
82	Dic₁₅₈	—	4	2	no	—
83	Dic₁₅₈	—	4	2	no	—
84	C₈×C₇₉	—	2	4	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	1	no	—
94	C₄×C₇₉	—	2	1	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
97	C₄×C₇₉	2₋¹⁺⁴	1	2	yes	—
98	Dic₇₉	—	4	2	no	—
99	Dic₇₉	—	4	2	no	—
100	Dic₇₉	—	4	2	no	—
101	C₂×C₇₉	—	4	1	no	—
102	C₂×C₇₉	#536	1	1	yes	characteristic
103	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₇₉	#575	1	1	yes	characteristic
105	D₇₉	—	8	2	no	—
106	#3235	—	79	1	no	maximal, Sylow
107	C₇₉	#3235	1	1	yes	characteristic, complemented, Sylow
108	#533	—	79	1	no	—
109	#532	—	79	1	no	—
110	#501	—	79	2	no	—
111	#490	—	79	2	no	—
112	#482	—	79	4	no	—
113	#476	—	79	2	no	—
114	#474	—	79	2	no	—
115	#432	D₇₉	1	1	yes	characteristic, complemented
116	C₂×#41	—	158	1	no	—
117	C₄⋊Q₈	—	79	1	no	—
118	C₄⋊Q₈	—	79	1	no	—
119	C₄⋊Q₈	—	79	1	no	—
120	#127	—	79	1	no	—
121	#125	—	79	2	no	—
122	#124	—	158	1	no	—
123	C₂²⋊Q₈	—	158	1	no	—
124	C₂²⋊Q₈	—	158	1	no	—
125	#121	—	79	2	no	—
126	#121	—	158	1	no	—
127	C₄×Q₈	—	158	1	no	—
128	C₄×Q₈	—	158	1	no	—
129	C₄×D₄	—	79	2	no	—
130	C₄×D₄	—	158	1	no	—
131	#117	—	79	2	no	—
132	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
133	#107	—	79	4	no	—
134	#106	—	79	4	no	—
135	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
136	#102	—	79	4	no	—
137	#102	—	79	4	no	—
138	C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
139	#97	C₂×D₇₉	1	2	yes	—
140	#41	—	316	4	no	—
141	C₂×Q₈	—	158	1	no	—
142	C₂×Q₈	—	158	1	no	—
143	C₂×Q₈	—	158	1	no	—
144	C₂×D₄	—	79	2	no	—
145	C₂×D₄	—	158	2	no	—
146	C₂×D₄	—	158	1	no	—
147	C₂×D₄	—	158	1	no	—
148	C₂²×C₄	—	158	2	no	—
149	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
150	C₂²×C₄	D₃₁₆	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₇₉	1	4	yes	—
153	C₄⋊C₄	—	79	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	—	158	1	no	—
159	C₄⋊C₄	—	158	1	no	—
160	C₄⋊C₄	—	158	1	no	—
161	C₂²⋊C₄	—	158	2	no	—
162	C₂²⋊C₄	—	158	2	no	—
163	C₂²⋊C₄	—	158	2	no	—
164	C₄²	—	158	2	no	—
165	C₄²	D₃₁₆	1	1	yes	characteristic
166	C₄²	D₃₁₆	1	1	yes	characteristic
167	C₄²	C₂²×D₇₉	1	1	yes	characteristic
168	C₄²	C₂²×D₇₉	1	1	yes	characteristic
169	C₂³	—	158	2	no	—
170	C₂³	C₂×D₃₁₆	1	1	yes	characteristic
171	Q₈	—	316	2	no	—
172	Q₈	—	316	2	no	—
173	Q₈	—	316	2	no	—
174	D₄	—	316	2	no	—
175	D₄	—	316	2	no	—
176	D₄	—	316	4	no	—
177	D₄	—	158	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	316	4	no	—
182	C₂×C₄	—	158	2	no	—
183	C₂×C₄	—	158	2	no	—
184	C₂×C₄	—	158	2	no	—
185	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
186	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
187	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
189	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
190	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
191	C₈	—	2	4	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	316	2	no	—
196	C₂²	—	316	2	no	—
197	C₂²	—	316	2	no	—
198	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	316	2	no	—
204	C₄	—	316	2	no	—
205	C₄	—	316	2	no	—
206	C₄	#14579189	1	2	yes	—
207	C₄	#14579208	1	2	yes	—
208	C₂	—	4	1	no	—
209	C₂	—	632	2	no	—
210	C₂	#19099139	1	1	yes	characteristic
211	C₂	C₂×#14579189	1	1	yes	characteristic
212	C₂	#19099363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099120

C₂

yes

maximal, characteristic, complemented

#19099113

C₂

yes

maximal, characteristic

#19098972

C₂

yes

maximal, complemented

#19098928

C₂

yes

maximal, complemented

#19098888

C₂

yes

maximal, complemented

#19098854

C₂

yes

maximal, complemented

#19098844

C₂

yes

maximal, complemented

#432×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929624

—

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

characteristic

#14579161

C₂²

yes

characteristic

#14579158

C₂²

yes

characteristic

#14579153

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₄×Dic₁₅₈

—

C₄×Dic₁₅₈

—

C₄×D₃₁₆

—

C₄×D₃₁₆

C₂²

yes

—

#14579120

C₂²

yes

—

#14579095

C₂²

yes

—

#14579093

C₂²

yes

—

#14579083

C₂²

yes

—

#14579083

C₂²

yes

—

C₂×C₄²×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₂²⋊C₈×C₇₉

C₂²

yes

—

#97×C₇₉

C₂²

yes

—

#11929624

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

D₄

yes

characteristic

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂²×D₇₉

—

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

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

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

—

C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₄×C₇₉

2₋¹⁺⁴

yes

—

Dic₇₉

—

Dic₇₉

—

100

Dic₇₉

—

101

C₂×C₇₉

—

102

C₂×C₇₉

#536

yes

characteristic

103

C₂×C₇₉

C₂×AGL(1,ℤ/8)

yes

characteristic

104

C₂×C₇₉

#575

yes

characteristic

105

D₇₉

—

106

#3235

—

maximal, Sylow

107

C₇₉

#3235

yes

characteristic, complemented, Sylow

108

#533

—

109

#532

—

110

#501

—

111

#490

—

112

#482

—

113

#476

—

114

#474

—

115

#432

D₇₉

yes

characteristic, complemented

116

C₂×#41

—

158

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₄⋊Q₈

—

120

#127

—

121

#125

—

122

#124

—

158

—

123

C₂²⋊Q₈

—

158

—

124

C₂²⋊Q₈

—

158

—

125

#121

—

126

#121

—

158

—

127

C₄×Q₈

—

158

—

128

C₄×Q₈

—

158

—

129

C₄×D₄

—

130

C₄×D₄

—

158

—

131

#117

—

132

C₂×C₄²

C₂×D₇₉

yes

characteristic

133

#107

—

134

#106

—

135

C₄⋊C₈

C₂×D₇₉

yes

—

136

#102

—

137

#102

—

138

C₂²⋊C₈

C₂×D₇₉

yes

—

139

#97

C₂×D₇₉

yes

—

140

#41

—

316

—

141

C₂×Q₈

—

158

—

142

C₂×Q₈

—

158

—

143

C₂×Q₈

—

158

—

144

C₂×D₄

—

145

C₂×D₄

—

158

—

146

C₂×D₄

—

158

—

147

C₂×D₄

—

158

—

148

C₂²×C₄

—

158

—

149

C₂²×C₄

D₃₁₆

yes

characteristic

150

C₂²×C₄

D₃₁₆

yes

characteristic

151

C₂²×C₄

C₂²×D₇₉

yes

characteristic

152

C₂×C₈

C₂²×D₇₉

yes

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

—

158

—

158

C₄⋊C₄

—

158

—

159

C₄⋊C₄

—

158

—

160

C₄⋊C₄

—

158

—

161

C₂²⋊C₄

—

158

—

162

C₂²⋊C₄

—

158

—

163

C₂²⋊C₄

—

158

—

164

C₄²

—

158

—

165

C₄²

D₃₁₆

yes

characteristic

166

C₄²

D₃₁₆

yes

characteristic

167

C₄²

C₂²×D₇₉

yes

characteristic

168

C₄²

C₂²×D₇₉

yes

characteristic

169

C₂³

—

158

—

170

C₂³

C₂×D₃₁₆

yes

characteristic

171

Q₈

—

316

—

172

Q₈

—

316

—

173

Q₈

—

316

—

174

D₄

—

316

—

175

D₄

—

316

—

176

D₄

—

316

—

177

D₄

—

158

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

316

—

182

C₂×C₄

—

158

—

183

C₂×C₄

—

158

—

184

C₂×C₄

—

158

—

185

C₂×C₄

C₂×D₃₁₆

yes

characteristic

186

C₂×C₄

C₂×D₃₁₆

yes

characteristic

187

C₂×C₄

C₂×D₃₁₆

yes

characteristic

188

C₂×C₄

C₂×D₃₁₆

yes

characteristic

189

C₂×C₄

C₂×D₃₁₆

yes

characteristic

190

C₂×C₄

C₂³×D₇₉

yes

characteristic

191

C₈

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

316

—

196

C₂²

—

316

—

197

C₂²

—

316

—

198

C₂²

C₂²×D₃₁₆

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

316

—

204

C₄

—

316

—

205

C₄

—

316

—

206

C₄

#14579189

yes

—

207

C₄

#14579208

yes

—

208

C₂

—

209

C₂

—

632

—

210

C₂

#19099139

yes

characteristic

211

C₂

C₂×#14579189

yes

characteristic

212

C₂

#19099363

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#432×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
prime radical	#19099113	C₂
hypercenter	#432	D₇₉
hypocenter	C₇₉	#3235
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#432×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

C₂

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
1. #19099120	C₂	semidirect
3. #19098972	C₂	semidirect
4. #19098928	C₂	semidirect
5. #19098888	C₂	semidirect
6. #19098854	C₂	semidirect
7. #19098844	C₂	semidirect
8. #432×C₇₉	C₂	semidirect
107. C₇₉	#3235	semidirect
115. #432	D₇₉	semidirect

H ≅ G/N

type

1. #19099120

C₂

semidirect

3. #19098972

C₂

semidirect

4. #19098928

C₂

semidirect

5. #19098888

C₂

semidirect

6. #19098854

C₂

semidirect

7. #19098844

C₂

semidirect

8. #432×C₇₉

C₂

semidirect

107. C₇₉

#3235

semidirect

115. #432

D₇₉

semidirect

Subgroups — #23774600

Conjugacy classes of subgroups

Complements