Group #23774746 — subgroups

Subgroups — #23774746

Order 10112.

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

normal subgroups only

This group has 360 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	#19099094	C₂	1	2	yes	maximal, complemented
2	#19098869	C₂	1	2	yes	maximal, complemented
3	#19098861	C₂	1	2	yes	maximal, complemented
4	#19098804	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098790	C₂	1	2	yes	maximal, complemented
6	#19098784	C₂	1	2	yes	maximal, complemented
7	#19098776	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#466×C₇₉	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₃₁₆	C₂²	1	1	yes	characteristic
11	C₂×#11929612	C₂²	1	2	yes	—
12	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#14579154	—	2	2	no	—
14	#14579152	C₂²	1	2	yes	—
15	#14579148	C₂²	1	2	yes	—
16	#14579142	—	2	2	no	—
17	#14579140	C₂²	1	2	yes	—
18	#14579126	C₂²	1	2	yes	—
19	D₄×Dic₇₉	—	2	2	no	—
20	D₄×Dic₇₉	C₂²	1	2	yes	—
21	#14579117	—	2	2	no	—
22	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
23	#14579095	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	#14579086	C₂²	1	2	yes	—
28	#14579084	C₂²	1	2	yes	—
29	#14579082	C₂²	1	2	yes	—
30	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	#121×C₇₉	C₂²	1	2	yes	—
33	#107×C₇₉	C₂²	1	1	yes	characteristic
34	#102×C₇₉	C₂²	1	2	yes	—
35	C₂×Dic₁₅₈	C₂³	1	2	yes	—
36	C₂²×Dic₇₉	—	2	2	no	—
37	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
39	Dic₃₁₆	—	4	1	no	—
40	#11929612	—	4	2	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	C₂³	1	2	yes	—
45	Dic₇₉⋊C₄	—	2	2	no	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	Dic₇₉⋊C₄	—	2	2	no	—
48	C₂²⋊Dic₇₉	—	2	2	no	—
49	C₂²⋊Dic₇₉	—	2	2	no	—
50	C₂²⋊Dic₇₉	—	2	2	no	—
51	C₂²⋊Dic₇₉	—	2	2	no	—
52	C₄×Dic₇₉	—	2	2	no	—
53	C₄×Dic₇₉	D₄	1	2	yes	—
54	C₄×Dic₇₉	C₂³	1	2	yes	—
55	C₂×D₄×C₇₉	—	2	2	no	—
56	C₂×D₄×C₇₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	D₈×C₇₉	—	4	1	no	—
59	C₂×C₈×C₇₉	—	4	1	no	—
60	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
63	C₂²⋊C₄×C₇₉	—	2	2	no	—
64	C₂³×C₇₉	—	2	2	no	—
65	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
66	D₄×C₇₉	—	2	2	no	—
67	D₄×C₇₉	—	2	2	no	—
68	D₄×C₇₉	—	4	4	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	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	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	2	2	no	—
76	C₂×Dic₇₉	—	2	2	no	—
77	C₂×Dic₇₉	—	4	1	no	—
78	C₂×Dic₇₉	—	4	4	no	—
79	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
80	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
81	Dic₁₅₈	—	2	2	no	—
82	Dic₁₅₈	—	2	2	no	—
83	C₈×C₇₉	—	2	1	no	—
84	C₈×C₇₉	—	2	1	no	—
85	C₇₉⋊C₈	—	2	2	no	—
86	C₂²×C₇₉	—	2	1	no	—
87	C₂²×C₇₉	—	2	1	no	—
88	C₂²×C₇₉	—	4	2	no	—
89	C₂²×C₇₉	—	4	2	no	—
90	C₂²×C₇₉	—	4	2	no	—
91	C₂²×C₇₉	#135	1	2	yes	—
92	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₇₉	—	2	1	no	—
94	C₄×C₇₉	—	4	2	no	—
95	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₇₉	—	2	2	no	—
98	Dic₇₉	—	2	2	no	—
99	Dic₇₉	—	4	2	no	—
100	Dic₇₉	—	4	2	no	—
101	C₂×C₇₉	—	2	2	no	—
102	C₂×C₇₉	—	8	2	no	—
103	C₂×C₇₉	#537	1	1	yes	characteristic
104	C₂×C₇₉	C₂×#135	1	1	yes	characteristic
105	C₂×C₇₉	#575	1	1	yes	characteristic
106	#3250	—	79	1	no	maximal, Sylow
107	C₇₉	#3250	1	1	yes	characteristic, complemented, Sylow
108	#529	—	79	2	no	—
109	#479	—	79	2	no	—
110	#478	—	79	2	no	—
111	#467	—	79	1	no	—
112	#466	D₇₉	1	1	yes	characteristic
113	#465	—	79	2	no	—
114	#464	—	79	2	no	—
115	#463	—	79	2	no	—
116	C₂×C₄⋊C₈	—	79	1	no	—
117	C₂×Q₁₆	—	79	1	no	—
118	C₂×SD₁₆	—	79	2	no	—
119	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
120	C₂²×C₈	—	79	1	no	—
121	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
122	#126	—	158	2	no	—
123	#125	—	79	2	no	—
124	#124	—	79	2	no	—
125	#123	—	158	2	no	—
126	C₂²⋊Q₈	—	79	2	no	—
127	#121	C₂×D₇₉	1	2	yes	—
128	C₄×Q₈	—	79	2	no	—
129	C₄×D₄	—	79	2	no	—
130	C₄×D₄	—	158	2	no	—
131	#117	—	158	2	no	—
132	C₂×C₄²	—	79	1	no	—
133	#107	—	79	1	no	—
134	#107	C₂×D₇₉	1	1	yes	characteristic
135	#106	—	79	2	no	—
136	C₄⋊C₈	—	79	2	no	—
137	C₄⋊C₈	—	79	2	no	—
138	#103	—	79	2	no	—
139	#103	—	79	2	no	—
140	#102	—	79	2	no	—
141	#102	C₂×D₇₉	1	2	yes	—
142	C₂×Q₈	—	79	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂²×D₇₉	1	2	yes	—
145	C₂²×C₄	—	79	1	no	—
146	C₂²×C₄	—	79	1	no	—
147	C₂²×C₄	—	158	2	no	—
148	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
149	Q₁₆	—	316	1	no	—
150	SD₁₆	—	316	2	no	—
151	D₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	316	1	no	—
154	C₂×C₈	—	79	2	no	—
155	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
157	C₄⋊C₄	—	79	2	no	—
158	C₄⋊C₄	—	158	2	no	—
159	C₄⋊C₄	—	158	2	no	—
160	C₄⋊C₄	—	158	2	no	—
161	C₄⋊C₄	—	158	2	no	—
162	C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	158	2	no	—
165	C₂²⋊C₄	—	158	2	no	—
166	C₂²⋊C₄	—	158	2	no	—
167	C₂²⋊C₄	—	158	2	no	—
168	C₄²	—	79	2	no	—
169	C₄²	—	79	2	no	—
170	C₄²	—	158	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	D₄×D₇₉	1	1	yes	characteristic
173	Q₈	—	158	2	no	—
174	Q₈	—	158	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	316	4	no	—
182	C₂×C₄	—	316	1	no	—
183	C₂×C₄	—	79	2	no	—
184	C₂×C₄	—	79	2	no	—
185	C₂×C₄	—	158	2	no	—
186	C₂×C₄	—	158	2	no	—
187	C₂×C₄	—	158	2	no	—
188	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	158	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	#14579180	1	2	yes	—
199	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	316	2	no	—
203	C₄	—	316	2	no	—
204	C₄	—	158	2	no	—
205	C₄	—	158	2	no	—
206	C₄	#14579206	1	1	yes	characteristic
207	C₄	#14579208	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	#19099140	1	1	yes	characteristic
211	C₂	C₂×#14579180	1	1	yes	characteristic
212	C₂	#19099360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19099094

C₂

yes

maximal, complemented

#19098869

C₂

yes

maximal, complemented

#19098861

C₂

yes

maximal, complemented

#19098804

C₂

yes

maximal, characteristic, complemented

#19098790

C₂

yes

maximal, complemented

#19098784

C₂

yes

maximal, complemented

#19098776

C₂

yes

maximal, complemented

C₂×Dic₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#466×C₇₉

C₂

yes

maximal, characteristic

C₂×Dic₃₁₆

C₂²

yes

characteristic

C₂×#11929612

C₂²

yes

—

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579154

—

#14579152

C₂²

yes

—

#14579148

C₂²

yes

—

#14579142

—

#14579140

C₂²

yes

—

#14579126

C₂²

yes

—

D₄×Dic₇₉

—

D₄×Dic₇₉

C₂²

yes

—

#14579117

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579095

C₂²

yes

characteristic

#14579092

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

Dic₇₉⋊C₈

C₂²

yes

—

#14579086

C₂²

yes

—

#14579084

C₂²

yes

—

#14579082

C₂²

yes

—

C₂×D₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

#121×C₇₉

C₂²

yes

—

#107×C₇₉

C₂²

yes

characteristic

#102×C₇₉

C₂²

yes

—

C₂×Dic₁₅₈

C₂³

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

Dic₃₁₆

—

#11929612

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂³

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

C₂³

yes

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

Dic₇₉⋊C₄

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₂²⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

D₄

yes

—

C₄×Dic₇₉

C₂³

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₂³

yes

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

D₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊C₄×C₇₉

C₂³

yes

—

C₂²⋊C₄×C₇₉

—

C₂³×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂⁴

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

C₂×D₄

yes

—

C₂×Dic₇₉

C₂×D₄

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#135

yes

—

C₂²×C₇₉

C₂²×D₄

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

H(5,𝔽₂)

yes

characteristic

C₄×C₇₉

2₋¹⁺⁴

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

100

Dic₇₉

—

101

C₂×C₇₉

—

102

C₂×C₇₉

—

103

C₂×C₇₉

#537

yes

characteristic

104

C₂×C₇₉

C₂×#135

yes

characteristic

105

C₂×C₇₉

yes

106

—

107

C₇₉

#3250

yes

characteristic, complemented, Sylow

108

#529

—

109

#479

—

110

#478

—

111

#467

—

112

#466

D₇₉

yes

characteristic

113

#465

—

114

#464

—

115

#463

—

116

C₂×C₄⋊C₈

—

117

C₂×Q₁₆

—

118

C₂×SD₁₆

—

119

C₂×D₈

C₂×D₇₉

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

C₂×D₇₉

yes

characteristic

122

#126

—

158

—

123

#125

—

124

#124

—

125

#123

—

158

—

126

C₂²⋊Q₈

—

127

#121

C₂×D₇₉

yes

—

128

C₄×Q₈

—

129

C₄×D₄

—

130

C₄×D₄

—

158

—

131

#117

—

158

—

132

C₂×C₄²

—

133

#107

—

134

#107

C₂×D₇₉

yes

characteristic

135

#106

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#103

—

140

#102

—

141

#102

C₂×D₇₉

yes

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

C₂²×D₇₉

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

158

—

148

C₂²×C₄

C₂²×D₇₉

yes

characteristic

149

Q₁₆

—

316

—

150

SD₁₆

—

316

—

151

D₈

—

152

C₂×C₈

—

153

C₂×C₈

—

316

—

154

C₂×C₈

—

155

C₂×C₈

C₂²×D₇₉

yes

characteristic

156

C₂×C₈

C₂²×D₇₉

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

158

—

159

C₄⋊C₄

—

158

—

160

C₄⋊C₄

—

158

—

161

C₄⋊C₄

—

158

—

162

C₄⋊C₄

C₂²×D₇₉

yes

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

158

—

165

C₂²⋊C₄

—

158

—

166

C₂²⋊C₄

—

158

—

167

C₂²⋊C₄

—

158

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

158

—

171

C₂³

—

172

C₂³

D₄×D₇₉

yes

characteristic

173

Q₈

—

158

—

174

Q₈

—

158

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

316

—

182

C₂×C₄

—

316

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

158

—

186

C₂×C₄

—

158

—

187

C₂×C₄

—

158

—

188

C₂×C₄

D₄×D₇₉

yes

characteristic

189

C₂×C₄

C₂³×D₇₉

yes

characteristic

190

C₈

—

191

C₈

—

192

C₈

—

158

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#14579180

yes

—

199

C₂²

C₂×D₄×D₇₉

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

316

—

203

C₄

—

316

—

204

C₄

—

158

—

205

C₄

—

158

—

206

C₄

#14579206

yes

characteristic

207

C₄

#14579208

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

#19099140

yes

characteristic

211

C₂

C₂×#14579180

yes

characteristic

212

C₂

#19099360

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting, prime radical	#466×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#466	D₇₉
hypocenter	C₇₉	#3250
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

subgroup

G/H

center

C₂²

C₂×D₄×D₇₉

commutator, cosocle

C₂×C₄×C₇₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₉

Fitting, prime radical

#466×C₇₉

C₂

socle

C₂²×C₇₉

C₂²×D₄

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

N	H ≅ G/N	type
1. #19099094	C₂	semidirect
2. #19098869	C₂	semidirect
3. #19098861	C₂	semidirect
4. #19098804	C₂	semidirect
5. #19098790	C₂	semidirect
6. #19098784	C₂	semidirect
7. #19098776	C₂	semidirect
8. C₂×Dic₇₉⋊C₈	C₂	semidirect
107. C₇₉	#3250	semidirect

H ≅ G/N

type

1. #19099094

C₂

semidirect

2. #19098869

C₂

semidirect

3. #19098861

C₂

semidirect

4. #19098804

C₂

semidirect

5. #19098790

C₂

semidirect

6. #19098784

C₂

semidirect

7. #19098776

C₂

semidirect

8. C₂×Dic₇₉⋊C₈

C₂

semidirect

107. C₇₉

#3250

semidirect

Subgroups — #23774746

Conjugacy classes of subgroups

Complements