Group #23776491 — subgroups

Subgroups — #23776491

Order 10112.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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	#19099295	C₂	1	2	yes	maximal, complemented
2	#19098949	C₂	1	2	yes	maximal, complemented
3	#19098787	C₂	1	2	yes	maximal, complemented
4	#19098775	C₂	1	2	yes	maximal, complemented
5	#19098756	C₂	1	2	yes	maximal, complemented
6	#19098693	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098664	C₂	1	2	yes	maximal, complemented
8	C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#495×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11929611	C₂²	1	2	yes	—
11	C₂×D₆₃₂	C₂²	1	1	yes	characteristic
12	C₂×#11929609	C₂²	1	2	yes	—
13	#14579157	C₂²	1	2	yes	—
14	#14579150	C₂²	1	2	yes	—
15	#14579147	—	2	2	no	—
16	#14579145	—	2	2	no	—
17	#14579143	—	2	2	no	—
18	Q₈×Dic₇₉	C₂²	1	2	yes	—
19	C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
20	C₄×D₃₁₆	C₂²	1	2	yes	—
21	D₄×Dic₇₉	C₂²	1	2	yes	—
22	(C₂²⋊D₇₉)⋊C₄	—	2	2	no	—
23	#14579095	C₂²	1	1	yes	characteristic
24	C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₇₉⋊C₈	C₂²	1	2	yes	—
26	#14579085	C₂²	1	2	yes	—
27	#14579083	C₂²	1	2	yes	—
28	#14579082	C₂²	1	2	yes	—
29	#14579071	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₇₉	C₂²	1	2	yes	—
32	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
33	#124×C₇₉	C₂²	1	2	yes	—
34	C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
35	C₂×Dic₁₅₈	D₄	1	1	yes	characteristic
36	C₂×D₃₁₆	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₇₉	—	2	2	no	—
38	C₂×C₂²⋊D₇₉	—	2	2	no	—
39	C₂²×Dic₇₉	—	2	2	no	—
40	C₂×C₄×D₇₉	—	2	2	no	—
41	#11929611	—	4	2	no	—
42	D₆₃₂	—	2	2	no	—
43	#11929609	—	4	2	no	—
44	C₂×C₇₉⋊C₈	C₂³	1	2	yes	—
45	C₄⋊Dic₇₉	—	2	2	no	—
46	C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
47	C₄⋊Dic₇₉	C₂³	1	2	yes	—
48	Dic₇₉⋊C₄	D₄	1	2	yes	—
49	C₂²⋊Dic₇₉	—	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	C₂×D₄×C₇₉	C₂³	1	2	yes	—
57	Q₁₆×C₇₉	—	4	1	no	—
58	SD₁₆×C₇₉	—	2	4	no	—
59	D₈×C₇₉	—	4	1	no	—
60	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₇₉	—	2	2	no	—
63	C₂²⋊C₄×C₇₉	—	2	2	no	—
64	C₄²×C₇₉	C₂³	1	1	yes	characteristic
65	C₂³×C₇₉	—	2	2	no	—
66	C₂²×D₇₉	—	2	2	no	—
67	Q₈×C₇₉	—	2	2	no	—
68	Q₈×C₇₉	—	2	2	no	—
69	D₄×C₇₉	—	2	2	no	—
70	D₄×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	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
75	C₂²⋊D₇₉	—	4	4	no	—
76	C₂²⋊D₇₉	—	8	2	no	—
77	C₂×Dic₇₉	—	2	2	no	—
78	C₂×Dic₇₉	—	8	2	no	—
79	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
80	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
81	D₃₁₆	—	2	2	no	—
82	D₃₁₆	—	2	2	no	—
83	C₄×D₇₉	—	8	2	no	—
84	Dic₁₅₈	—	4	1	no	—
85	C₈×C₇₉	—	2	1	no	—
86	C₈×C₇₉	#41	1	2	yes	—
87	C₇₉⋊C₈	—	2	2	no	—
88	C₂²×C₇₉	—	4	2	no	—
89	C₂²×C₇₉	—	4	2	no	—
90	C₂²×C₇₉	—	4	2	no	—
91	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
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	1	no	—
97	C₄×C₇₉	—	4	2	no	—
98	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
99	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
100	Dic₇₉	—	2	2	no	—
101	Dic₇₉	—	2	2	no	—
102	Dic₇₉	—	4	2	no	—
103	C₂×C₇₉	—	8	2	no	—
104	C₂×C₇₉	#550	1	1	yes	characteristic
105	C₂×C₇₉	#575	1	1	yes	characteristic
106	C₂×C₇₉	#576	1	1	yes	characteristic
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	D₇₉	1	1	yes	characteristic, complemented
113	#464	—	79	2	no	—
114	#463	—	79	2	no	—
115	#459	—	79	2	no	—
116	#446	—	79	1	no	—
117	#440	—	79	2	no	—
118	C₄×D₈	—	79	1	no	—
119	C₂×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
120	C₂×SD₁₆	—	79	2	no	—
121	C₂×SD₁₆	C₂×D₇₉	1	2	yes	—
122	C₂×D₈	—	79	2	no	—
123	C₂×D₈	—	79	1	no	—
124	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
125	#127	—	79	2	no	—
126	#124	—	79	2	no	—
127	#124	—	158	2	no	—
128	#124	C₂×D₇₉	1	2	yes	—
129	#123	—	158	2	no	—
130	#123	—	158	2	no	—
131	C₄×Q₈	—	79	1	no	—
132	C₄×Q₈	—	79	2	no	—
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	—	79	2	no	—
138	C₄⋊C₈	—	79	1	no	—
139	#103	—	79	2	no	—
140	#102	—	79	2	no	—
141	#102	—	79	2	no	—
142	#97	—	79	2	no	—
143	C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
144	C₂×Q₈	—	79	1	no	—
145	C₂×Q₈	C₂²×D₇₉	1	2	yes	—
146	C₂×D₄	—	79	2	no	—
147	C₂×D₄	—	158	2	no	—
148	C₂×D₄	—	158	2	no	—
149	C₂×D₄	C₂²×D₇₉	1	2	yes	—
150	C₂²×C₄	—	158	2	no	—
151	C₂²×C₄	—	158	2	no	—
152	Q₁₆	—	4	1	no	—
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	316	2	no	—
155	D₈	—	4	1	no	—
156	D₈	—	316	2	no	—
157	D₈	—	158	2	no	—
158	C₂×C₈	—	79	2	no	—
159	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
161	C₄⋊C₄	—	79	2	no	—
162	C₄⋊C₄	—	79	1	no	—
163	C₄⋊C₄	—	79	2	no	—
164	C₄⋊C₄	—	158	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	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	—	79	2	no	—
172	C₄²	—	158	2	no	—
173	C₄²	C₂²×D₇₉	1	1	yes	characteristic
174	C₂³	—	2	2	no	—
175	C₂³	—	158	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	316	1	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	316	4	no	—
182	D₄	—	632	2	no	—
183	D₄	—	158	2	no	—
184	D₄	—	158	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	79	2	no	—
187	C₂×C₄	—	79	2	no	—
188	C₂×C₄	—	632	2	no	—
189	C₂×C₄	—	632	2	no	—
190	C₂×C₄	—	158	2	no	—
191	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
192	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
193	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	158	2	no	—
196	C₈	#11929623	1	2	yes	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	316	2	no	—
201	C₂²	—	316	2	no	—
202	C₂²	—	316	2	no	—
203	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	4	2	no	—
207	C₄	—	316	2	no	—
208	C₄	—	158	2	no	—
209	C₄	—	158	2	no	—
210	C₄	C₂×#11929623	1	1	yes	characteristic
211	C₄	#14579206	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	—	632	2	no	—
214	C₂	#19099233	1	1	yes	characteristic
215	C₂	#19099360	1	1	yes	characteristic
216	C₂	#19099365	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099295

C₂

yes

maximal, complemented

#19098949

C₂

yes

maximal, complemented

#19098787

C₂

yes

maximal, complemented

#19098775

C₂

yes

maximal, complemented

#19098756

C₂

yes

maximal, complemented

#19098693

C₂

yes

maximal, characteristic, complemented

#19098664

C₂

yes

maximal, complemented

C₄×D₆₃₂

C₂

yes

maximal, characteristic, complemented

#495×C₇₉

C₂

yes

maximal, characteristic, complemented

C₂×#11929611

C₂²

yes

—

C₂×D₆₃₂

C₂²

yes

characteristic

C₂×#11929609

C₂²

yes

—

#14579157

C₂²

yes

—

#14579150

C₂²

yes

—

#14579147

—

#14579145

—

#14579143

—

Q₈×Dic₇₉

C₂²

yes

—

C₄×Dic₁₅₈

C₂²

yes

characteristic

C₄×D₃₁₆

C₂²

yes

—

D₄×Dic₇₉

C₂²

yes

—

(C₂²⋊D₇₉)⋊C₄

—

#14579095

C₂²

yes

characteristic

C₄⋊(C₇₉⋊C₈)

C₂²

yes

characteristic

Dic₇₉⋊C₈

C₂²

yes

—

#14579085

C₂²

yes

—

#14579083

C₂²

yes

—

#14579082

C₂²

yes

—

#14579071

C₂²

yes

—

C₂×Q₁₆×C₇₉

C₂²

yes

characteristic

C₂×SD₁₆×C₇₉

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

#124×C₇₉

C₂²

yes

—

C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

D₄

yes

characteristic

C₂×D₃₁₆

C₂³

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₂²⋊D₇₉

—

C₂²×Dic₇₉

—

C₂×C₄×D₇₉

—

#11929611

—

D₆₃₂

—

#11929609

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

characteristic

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

D₄

yes

—

C₂²⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

(C₂×D₇₉)⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

C₂³

yes

—

C₂×Q₈×C₇₉

C₂³

yes

—

C₂×D₄×C₇₉

C₂³

yes

—

Q₁₆×C₇₉

—

SD₁₆×C₇₉

—

D₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂³

yes

characteristic

C₂³×C₇₉

—

C₂²×D₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂²⋊D₇₉

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

#41

yes

—

C₇₉⋊C₈

—

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

C₂×#41

yes

characteristic

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

100

Dic₇₉

—

101

Dic₇₉

—

102

Dic₇₉

—

103

C₂×C₇₉

—

104

C₂×C₇₉

#550

yes

characteristic

105

C₂×C₇₉

#575

yes

characteristic

106

C₂×C₇₉

#576

yes

characteristic

107

D₇₉

—

108

#3403

—

maximal, Sylow

109

C₇₉

#3403

yes

characteristic, complemented, Sylow

110

#559

—

111

#496

—

112

#495

D₇₉

yes

characteristic, complemented

113

#464

—

114

#463

—

115

#459

—

116

#446

—

117

#440

—

118

C₄×D₈

—

119

C₂×Q₁₆

C₂×D₇₉

yes

characteristic

120

C₂×SD₁₆

—

121

C₂×SD₁₆

C₂×D₇₉

yes

—

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

C₂×D₇₉

yes

characteristic

125

#127

—

126

#124

—

127

#124

—

158

—

128

#124

C₂×D₇₉

yes

—

129

#123

—

158

—

130

#123

—

158

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

C₄×D₄

—

135

C₄×D₄

—

158

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

#103

—

140

#102

—

141

#102

—

142

#97

—

143

C₄×C₈

C₂×D₇₉

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

C₂²×D₇₉

yes

—

146

C₂×D₄

—

147

C₂×D₄

—

158

—

148

C₂×D₄

—

158

—

149

C₂×D₄

C₂²×D₇₉

yes

—

150

C₂²×C₄

—

158

—

151

C₂²×C₄

—

158

—

152

Q₁₆

—

153

SD₁₆

—

154

SD₁₆

—

316

—

155

D₈

—

156

D₈

—

316

—

157

D₈

—

158

—

158

C₂×C₈

—

159

C₂×C₈

C₂²×D₇₉

yes

characteristic

160

C₂×C₈

C₂²×D₇₉

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

158

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

158

—

168

C₂²⋊C₄

—

158

—

169

C₂²⋊C₄

—

158

—

170

C₂²⋊C₄

—

158

—

171

C₄²

—

172

C₄²

—

158

—

173

C₄²

C₂²×D₇₉

yes

characteristic

174

C₂³

—

175

C₂³

—

158

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

316

—

179

D₄

—

180

D₄

—

181

D₄

—

316

—

182

D₄

—

632

—

183

D₄

—

158

—

184

D₄

—

158

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

632

—

189

C₂×C₄

—

632

—

190

C₂×C₄

—

158

—

191

C₂×C₄

D₄×D₇₉

yes

characteristic

192

C₂×C₄

D₄×D₇₉

yes

characteristic

193

C₂×C₄

C₂³×D₇₉

yes

characteristic

194

C₈

—

195

C₈

—

158

—

196

C₈

#11929623

yes

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

316

—

201

C₂²

—

316

—

202

C₂²

—

316

—

203

C₂²

C₂×D₄×D₇₉

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

316

—

208

C₄

—

158

—

209

C₄

—

158

—

210

C₄

C₂×#11929623

yes

characteristic

211

C₄

#14579206

yes

characteristic

212

C₂

—

213

C₂

—

632

—

214

C₂

#19099233

yes

characteristic

215

C₂

#19099360

yes

characteristic

216

C₂

#19099365

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#495×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#495	D₇₉
hypocenter	C₇₉	#3403
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting

#495×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #19099295	C₂	semidirect
2. #19098949	C₂	semidirect
3. #19098787	C₂	semidirect
4. #19098775	C₂	semidirect
5. #19098756	C₂	semidirect
6. #19098693	C₂	semidirect
7. #19098664	C₂	semidirect
8. C₄×D₆₃₂	C₂	semidirect
9. #495×C₇₉	C₂	semidirect
109. C₇₉	#3403	semidirect
112. #495	D₇₉	semidirect

H ≅ G/N

type

1. #19099295

C₂

semidirect

2. #19098949

C₂

semidirect

3. #19098787

C₂

semidirect

4. #19098775

C₂

semidirect

5. #19098756

C₂

semidirect

6. #19098693

C₂

semidirect

7. #19098664

C₂

semidirect

8. C₄×D₆₃₂

C₂

semidirect

9. #495×C₇₉

C₂

semidirect

109. C₇₉

#3403

semidirect

112. #495

D₇₉

semidirect

Subgroups — #23776491

Conjugacy classes of subgroups

Complements