Group #23776490 — subgroups

Subgroups — #23776490

Order 10112.

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

normal subgroups only

This group has 380 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	#19099298	C₂	1	2	yes	maximal, complemented
2	#19098952	C₂	1	2	yes	maximal, complemented
3	#19098947	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098786	C₂	1	2	yes	maximal, complemented
5	#19098779	C₂	1	2	yes	maximal, complemented
6	#19098759	C₂	1	2	yes	maximal, complemented
7	#19098665	C₂	1	2	yes	maximal, complemented
8	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#446×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×#11929613	C₂²	1	2	yes	—
13	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
14	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
15	C₂×D₆₃₂	C₂²	1	2	yes	—
16	#14579158	C₂²	1	2	yes	—
17	#14579150	—	2	2	no	—
18	#14579149	C₂²	1	2	yes	—
19	#14579149	C₂²	1	2	yes	—
20	#14579146	—	2	2	no	—
21	#14579146	—	2	2	no	—
22	C₄×Dic₁₅₈	C₂²	1	2	yes	—
23	C₄×D₃₁₆	C₂²	1	2	yes	—
24	C₄×D₃₁₆	C₂²	1	2	yes	—
25	#14579123	—	2	2	no	—
26	#14579095	C₂²	1	1	yes	characteristic
27	#14579086	C₂²	1	2	yes	—
28	#14579083	C₂²	1	2	yes	—
29	#14579083	C₂²	1	2	yes	—
30	C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
33	#97×C₇₉	C₂²	1	2	yes	—
34	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₅₈	C₂³	1	2	yes	—
36	C₂×D₃₁₆	—	2	2	no	—
37	C₂×D₃₁₆	—	2	2	no	—
38	C₂×D₃₁₆	C₂³	1	2	yes	—
39	C₂×D₃₁₆	C₂³	1	2	yes	—
40	C₂×C₄×D₇₉	—	2	2	no	—
41	C₂×C₄×D₇₉	—	2	2	no	—
42	Dic₃₁₆	—	4	1	no	—
43	#11929613	—	2	4	no	—
44	#11929613	—	4	2	no	—
45	D₆₃₂	—	2	2	no	—
46	D₆₃₂	—	4	2	no	—
47	D₆₃₂	—	4	1	no	—
48	C₄⋊Dic₇₉	C₂³	1	2	yes	—
49	Dic₇₉⋊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	(C₂×D₇₉)⋊C₄	—	2	2	no	—
54	(C₂×D₇₉)⋊C₄	—	2	2	no	—
55	(C₂×D₇₉)⋊C₄	—	2	2	no	—
56	C₄×Dic₇₉	—	2	2	no	—
57	C₂×Q₈×C₇₉	D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₇₉	C₂³	1	2	yes	—
60	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
62	C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
63	C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	C₄²×C₇₉	C₂³	1	2	yes	—
65	C₂²×D₇₉	—	2	2	no	—
66	C₂²×D₇₉	—	2	2	no	—
67	Q₈×C₇₉	—	4	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	1	yes	characteristic
71	C₂×C₄×C₇₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₇₉	—	2	2	no	—
74	C₂×Dic₇₉	—	2	2	no	—
75	D₃₁₆	—	2	2	no	—
76	D₃₁₆	—	2	2	no	—
77	D₃₁₆	—	2	2	no	—
78	D₃₁₆	—	2	2	no	—
79	D₃₁₆	—	4	4	no	—
80	D₃₁₆	—	8	2	no	—
81	C₄×D₇₉	—	8	2	no	—
82	C₄×D₇₉	—	8	2	no	—
83	Dic₁₅₈	—	2	2	no	—
84	Dic₁₅₈	—	2	2	no	—
85	C₈×C₇₉	—	2	2	no	—
86	C₈×C₇₉	—	2	1	no	—
87	C₈×C₇₉	#41	1	2	yes	—
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	C₂×D₇₉	—	4	2	no	—
93	C₂×D₇₉	—	4	2	no	—
94	C₂×D₇₉	—	4	2	no	—
95	C₄×C₇₉	—	2	1	no	—
96	C₄×C₇₉	—	2	2	no	—
97	C₄×C₇₉	—	2	1	no	—
98	C₄×C₇₉	—	2	2	no	—
99	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
100	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
101	Dic₇₉	—	4	2	no	—
102	Dic₇₉	—	4	2	no	—
103	C₂×C₇₉	#550	1	1	yes	characteristic
104	C₂×C₇₉	#575	1	1	yes	characteristic
105	C₂×C₇₉	#576	1	1	yes	characteristic
106	D₇₉	—	8	2	no	—
107	D₇₉	—	8	2	no	—
108	#3403	—	79	1	no	maximal, Sylow
109	C₇₉	#3403	1	1	yes	characteristic, complemented, Sylow
110	#559	—	79	2	no	—
111	#496	—	79	2	no	—
112	#495	—	79	1	no	—
113	#464	—	79	2	no	—
114	#463	—	79	2	no	—
115	#459	—	79	2	no	—
116	#446	D₇₉	1	1	yes	characteristic, complemented
117	#440	—	79	2	no	—
118	C₄×D₈	—	79	1	no	—
119	C₂×Q₁₆	—	79	1	no	—
120	C₂×SD₁₆	—	79	2	no	—
121	C₂×SD₁₆	—	79	2	no	—
122	C₂×D₈	—	79	1	no	—
123	C₂×D₈	—	79	2	no	—
124	C₂×D₈	—	79	1	no	—
125	#127	—	79	2	no	—
126	#124	—	79	2	no	—
127	#124	—	79	2	no	—
128	#124	—	158	2	no	—
129	#123	—	158	2	no	—
130	#123	—	158	2	no	—
131	C₄×Q₈	—	79	2	no	—
132	C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
133	C₄×D₄	—	79	2	no	—
134	C₄×D₄	—	79	2	no	—
135	C₄×D₄	—	158	2	no	—
136	#107	—	79	1	no	—
137	C₄⋊C₈	C₂×D₇₉	1	2	yes	—
138	C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
139	#103	—	79	2	no	—
140	#102	—	79	2	no	—
141	#102	—	79	2	no	—
142	#97	C₂×D₇₉	1	2	yes	—
143	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
144	C₂×Q₈	—	79	2	no	—
145	C₂×Q₈	D₃₁₆	1	1	yes	characteristic
146	C₂×D₄	—	79	2	no	—
147	C₂×D₄	—	79	2	no	—
148	C₂×D₄	—	158	2	no	—
149	C₂×D₄	—	158	2	no	—
150	C₂²×C₄	—	158	2	no	—
151	C₂²×C₄	—	158	2	no	—
152	Q₁₆	—	316	1	no	—
153	SD₁₆	—	316	2	no	—
154	SD₁₆	—	158	4	no	—
155	D₈	—	316	2	no	—
156	D₈	—	316	1	no	—
157	D₈	—	158	2	no	—
158	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
159	C₂×C₈	C₂²×D₇₉	1	2	yes	—
160	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
161	C₄⋊C₄	—	79	2	no	—
162	C₄⋊C₄	—	158	2	no	—
163	C₄⋊C₄	D₃₁₆	1	2	yes	—
164	C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
165	C₂²⋊C₄	—	158	2	no	—
166	C₂²⋊C₄	—	158	2	no	—
167	C₂²⋊C₄	—	158	2	no	—
168	C₂²⋊C₄	—	158	2	no	—
169	C₂²⋊C₄	—	158	2	no	—
170	C₂²⋊C₄	—	158	2	no	—
171	C₄²	—	158	2	no	—
172	C₄²	C₂²×D₇₉	1	1	yes	characteristic
173	C₄²	C₂²×D₇₉	1	2	yes	—
174	C₂³	—	158	2	no	—
175	C₂³	—	158	2	no	—
176	Q₈	—	4	1	no	—
177	Q₈	—	158	2	no	—
178	Q₈	—	158	2	no	—
179	D₄	—	316	4	no	—
180	D₄	—	632	2	no	—
181	D₄	—	158	2	no	—
182	D₄	—	158	2	no	—
183	D₄	—	158	2	no	—
184	D₄	—	158	2	no	—
185	C₂×C₄	—	632	2	no	—
186	C₂×C₄	—	632	2	no	—
187	C₂×C₄	—	158	2	no	—
188	C₂×C₄	—	158	2	no	—
189	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
190	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
191	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
192	C₂×C₄	C₂×D₃₁₆	1	2	yes	—
193	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	#11929625	1	2	yes	—
197	C₂²	—	316	2	no	—
198	C₂²	—	316	2	no	—
199	C₂²	—	316	2	no	—
200	C₂²	—	316	2	no	—
201	C₂²	—	316	2	no	—
202	C₂²	—	316	2	no	—
203	C₂²	C₂²×D₃₁₆	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	316	2	no	—
209	C₄	—	316	2	no	—
210	C₄	C₂×#11929625	1	1	yes	characteristic
211	C₄	#14579207	1	1	yes	characteristic
212	C₂	—	632	2	no	—
213	C₂	—	632	2	no	—
214	C₂	#19099235	1	1	yes	characteristic
215	C₂	#19099363	1	1	yes	characteristic
216	C₂	#19099368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099298

C₂

yes

maximal, complemented

#19098952

C₂

yes

maximal, complemented

#19098947

C₂

yes

maximal, characteristic, complemented

#19098786

C₂

yes

maximal, complemented

#19098779

C₂

yes

maximal, complemented

#19098759

C₂

yes

maximal, complemented

#19098665

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

#446×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929613

C₂²

yes

—

C₂×#11929613

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×D₆₃₂

C₂²

yes

—

#14579158

C₂²

yes

—

#14579150

—

#14579149

C₂²

yes

—

#14579149

C₂²

yes

—

#14579146

—

#14579146

—

C₄×Dic₁₅₈

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

C₄×D₃₁₆

C₂²

yes

—

#14579123

—

#14579095

C₂²

yes

characteristic

#14579086

C₂²

yes

—

#14579083

C₂²

yes

—

#14579083

C₂²

yes

—

C₄×Q₈×C₇₉

C₂²

yes

characteristic

C₄⋊C₈×C₇₉

C₂²

yes

—

C₄⋊C₈×C₇₉

C₂²

yes

characteristic

#97×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

C₂³

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₂³

yes

—

C₂×D₃₁₆

C₂³

yes

—

C₂×C₄×D₇₉

—

C₂×C₄×D₇₉

—

Dic₃₁₆

—

#11929613

—

#11929613

—

D₆₃₂

—

D₆₃₂

—

D₆₃₂

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

D₄

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

D₄

yes

characteristic

C₄²×C₇₉

C₂³

yes

characteristic

C₄²×C₇₉

C₂³

yes

—

C₂²×D₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

—

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×#41

yes

characteristic

100

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

101

Dic₇₉

—

102

Dic₇₉

—

103

C₂×C₇₉

#550

yes

characteristic

104

C₂×C₇₉

#575

yes

characteristic

105

C₂×C₇₉

#576

yes

characteristic

106

D₇₉

—

107

D₇₉

—

108

#3403

—

maximal, Sylow

109

C₇₉

#3403

yes

characteristic, complemented, Sylow

110

#559

—

111

#496

—

112

#495

—

113

#464

—

114

#463

—

115

#459

—

116

#446

D₇₉

yes

characteristic, complemented

117

#440

—

118

C₄×D₈

—

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

—

125

#127

—

126

#124

—

127

#124

—

128

#124

—

158

—

129

#123

—

158

—

130

#123

—

158

—

131

C₄×Q₈

—

132

C₄×Q₈

C₂×D₇₉

yes

characteristic

133

C₄×D₄

—

134

C₄×D₄

—

135

C₄×D₄

—

158

—

136

#107

—

137

C₄⋊C₈

C₂×D₇₉

yes

—

138

C₄⋊C₈

C₂×D₇₉

yes

characteristic

139

#103

—

140

#102

—

141

#102

—

142

#97

C₂×D₇₉

yes

—

143

C₄×C₈

C₂×D₇₉

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

D₃₁₆

yes

characteristic

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

158

—

149

C₂×D₄

—

158

—

150

C₂²×C₄

—

158

—

151

C₂²×C₄

—

158

—

152

Q₁₆

—

316

—

153

SD₁₆

—

316

—

154

SD₁₆

—

158

—

155

D₈

—

316

—

156

D₈

—

316

—

157

D₈

—

158

—

158

C₂×C₈

C₂²×D₇₉

yes

characteristic

159

C₂×C₈

C₂²×D₇₉

yes

—

160

C₂×C₈

C₂²×D₇₉

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

158

—

163

C₄⋊C₄

D₃₁₆

yes

—

164

C₄⋊C₄

D₃₁₆

yes

characteristic

165

C₂²⋊C₄

—

158

—

166

C₂²⋊C₄

—

158

—

167

C₂²⋊C₄

—

158

—

168

C₂²⋊C₄

—

158

—

169

C₂²⋊C₄

—

158

—

170

C₂²⋊C₄

—

158

—

171

C₄²

—

158

—

172

C₄²

C₂²×D₇₉

yes

characteristic

173

C₄²

C₂²×D₇₉

yes

—

174

C₂³

—

158

—

175

C₂³

—

158

—

176

Q₈

—

177

Q₈

—

158

—

178

Q₈

—

158

—

179

D₄

—

316

—

180

D₄

—

632

—

181

D₄

—

158

—

182

D₄

—

158

—

183

D₄

—

158

—

184

D₄

—

158

—

185

C₂×C₄

—

632

—

186

C₂×C₄

—

632

—

187

C₂×C₄

—

158

—

188

C₂×C₄

—

158

—

189

C₂×C₄

C₂×D₃₁₆

yes

—

190

C₂×C₄

C₂×D₃₁₆

yes

characteristic

191

C₂×C₄

C₂×D₃₁₆

yes

characteristic

192

C₂×C₄

C₂×D₃₁₆

yes

—

193

C₂×C₄

C₂³×D₇₉

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

#11929625

yes

—

197

C₂²

—

316

—

198

C₂²

—

316

—

199

C₂²

—

316

—

200

C₂²

—

316

—

201

C₂²

—

316

—

202

C₂²

—

316

—

203

C₂²

C₂²×D₃₁₆

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

316

—

209

C₄

—

316

—

210

C₄

C₂×#11929625

yes

characteristic

211

C₄

#14579207

yes

characteristic

212

C₂

—

632

—

213

C₂

—

632

—

214

C₂

#19099235

yes

characteristic

215

C₂

#19099363

yes

characteristic

216

C₂

#19099368

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#446×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#446	D₇₉
hypocenter	C₇₉	#3403
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₃₁₆

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#446×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #19099298	C₂	semidirect
2. #19098952	C₂	semidirect
3. #19098947	C₂	semidirect
4. #19098786	C₂	semidirect
5. #19098779	C₂	semidirect
6. #19098759	C₂	semidirect
7. #19098665	C₂	semidirect
8. C₄×D₆₃₂	C₂	semidirect
9. #446×C₇₉	C₂	semidirect
109. C₇₉	#3403	semidirect
116. #446	D₇₉	semidirect

H ≅ G/N

type

1. #19099298

C₂

semidirect

2. #19098952

C₂

semidirect

3. #19098947

C₂

semidirect

4. #19098786

C₂

semidirect

5. #19098779

C₂

semidirect

6. #19098759

C₂

semidirect

7. #19098665

C₂

semidirect

8. C₄×D₆₃₂

C₂

semidirect

9. #446×C₇₉

C₂

semidirect

109. C₇₉

#3403

semidirect

116. #446

D₇₉

semidirect

Subgroups — #23776490

Conjugacy classes of subgroups

Complements