Group #23764268 — subgroups

Subgroups — #23764268

Order 10112.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11929614	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579092	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579086	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579071	C₂	1	1	yes	maximal, characteristic
5	#19098470	C₂	1	1	yes	maximal, characteristic
6	#19098468	C₂	1	1	yes	maximal, characteristic
7	C₂×#103×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂×#11929614	—	2	2	no	—
10	C₂×#11929614	—	2	2	no	—
11	C₂×#11929614	C₄	1	4	yes	complemented
12	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
15	#14579092	—	2	2	no	—
16	#14579086	—	2	2	no	—
17	#14579071	—	2	2	no	—
18	C₇₉⋊#95	—	2	2	no	—
19	C₇₉⋊#95	—	2	2	no	—
20	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
23	#103×C₇₉	—	2	2	no	—
24	C₂×Dic₁₅₈	—	2	1	no	—
25	C₂×Dic₁₅₈	—	2	1	no	—
26	C₂×Dic₁₅₈	—	2	1	no	—
27	C₂×Dic₁₅₈	—	2	1	no	—
28	C₂×Dic₁₅₈	—	2	1	no	—
29	C₂×Dic₁₅₈	C₂×C₄	1	2	yes	—
30	C₂²×Dic₇₉	—	2	1	no	—
31	C₂²×Dic₇₉	—	2	1	no	—
32	C₂²×Dic₇₉	D₄	1	2	yes	—
33	#11929614	—	2	8	no	—
34	C₂×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
36	C₂×C₇₉⋊C₈	D₄	1	2	yes	—
37	C₄⋊Dic₇₉	—	2	1	no	—
38	C₄⋊Dic₇₉	—	2	1	no	—
39	C₄×Dic₇₉	—	2	1	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₂×Q₈×C₇₉	—	2	1	no	—
42	C₂×Q₈×C₇₉	—	2	1	no	—
43	C₂×Q₈×C₇₉	—	2	1	no	—
44	C₂×Q₈×C₇₉	—	2	1	no	—
45	C₂×Q₈×C₇₉	—	2	1	no	—
46	C₂×Q₈×C₇₉	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₇₉	—	2	1	no	—
48	C₂²×C₄×C₇₉	—	2	1	no	—
49	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₇₉	—	2	1	no	—
51	C₂×C₈×C₇₉	—	2	1	no	—
52	C₂×C₈×C₇₉	D₄	1	2	yes	—
53	C₄⋊C₄×C₇₉	—	2	1	no	—
54	C₄⋊C₄×C₇₉	—	2	1	no	—
55	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₇₉	—	2	2	no	—
57	Q₈×C₇₉	—	2	2	no	—
58	Q₈×C₇₉	—	4	1	no	—
59	Q₈×C₇₉	—	4	1	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	4	1	no	—
62	C₂×C₄×C₇₉	—	4	1	no	—
63	C₂×C₄×C₇₉	—	4	1	no	—
64	C₂×C₄×C₇₉	—	4	2	no	—
65	C₂×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
70	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
71	C₂×Dic₇₉	—	2	2	no	—
72	C₂×Dic₇₉	—	2	2	no	—
73	C₂×Dic₇₉	—	2	2	no	—
74	C₂×Dic₇₉	—	2	2	no	—
75	C₂×Dic₇₉	—	4	1	no	—
76	C₂×Dic₇₉	—	4	1	no	—
77	C₂×Dic₇₉	—	4	1	no	—
78	C₂×Dic₇₉	—	4	2	no	—
79	Dic₁₅₈	—	2	2	no	—
80	Dic₁₅₈	—	2	2	no	—
81	Dic₁₅₈	—	4	1	no	—
82	Dic₁₅₈	—	4	1	no	—
83	C₈×C₇₉	—	2	2	no	—
84	C₇₉⋊C₈	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₇₉	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₇₉	#121	1	1	yes	characteristic
87	C₂²×C₇₉	#127	1	1	yes	characteristic
88	C₂²×C₇₉	#137	1	2	yes	—
89	C₂²×C₇₉	#137	1	2	yes	—
90	C₄×C₇₉	—	4	2	no	—
91	C₄×C₇₉	—	8	1	no	—
92	C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
94	C₄×C₇₉	#121	1	1	yes	characteristic
95	C₄×C₇₉	#124	1	1	yes	characteristic
96	Dic₇₉	—	4	2	no	—
97	Dic₇₉	—	4	2	no	—
98	Dic₇₉	—	8	1	no	—
99	C₂×C₇₉	#390	1	1	yes	characteristic
100	C₂×C₇₉	#441	1	2	yes	—
101	C₂×C₇₉	#470	1	2	yes	—
102	C₂×C₇₉	#497	1	1	yes	characteristic
103	C₂×C₇₉	#497	1	1	yes	characteristic
104	#1971	—	79	1	no	maximal, Sylow
105	C₇₉	#1971	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₁₆	—	79	1	no	—
107	C₂×#106	—	79	1	no	—
108	C₂×#103	—	79	1	no	—
109	C₂×#103	D₇₉	1	1	yes	characteristic
110	C₂×#97	—	79	1	no	—
111	#391	—	79	1	no	—
112	#391	—	79	1	no	—
113	C₂²×Q₈	—	79	1	no	—
114	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
115	C₂×Q₁₆	—	79	4	no	—
116	C₂×Q₁₆	—	158	2	no	—
117	C₂×Q₁₆	—	158	2	no	—
118	C₂²×C₈	—	79	1	no	—
119	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	79	1	no	—
121	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
122	C₂×C₄²	—	79	1	no	—
123	#106	—	158	2	no	—
124	#103	—	2	2	no	—
125	#103	—	158	2	no	—
126	#97	—	158	2	no	—
127	#95	—	158	2	no	—
128	#95	—	158	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	79	2	no	—
135	C₂×Q₈	—	158	1	no	—
136	C₂×Q₈	—	158	1	no	—
137	C₂×Q₈	—	158	1	no	—
138	C₂×Q₈	—	158	1	no	—
139	C₂×Q₈	—	158	1	no	—
140	C₂×Q₈	C₄×D₇₉	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	79	2	no	—
144	C₂²×C₄	—	158	1	no	—
145	C₂²×C₄	—	158	1	no	—
146	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
147	Q₁₆	—	158	8	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	79	2	no	—
151	C₂×C₈	—	79	2	no	—
152	C₂×C₈	—	79	2	no	—
153	C₂×C₈	C₂²⋊D₇₉	1	2	yes	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	158	1	no	—
157	C₄⋊C₄	—	158	1	no	—
158	C₄²	—	158	1	no	—
159	C₄²	—	158	1	no	—
160	C₂³	D₄×D₇₉	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	316	1	no	—
166	Q₈	—	316	1	no	—
167	Q₈	—	158	2	no	—
168	Q₈	—	158	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	1	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	316	1	no	—
175	C₂×C₄	—	316	1	no	—
176	C₂×C₄	—	316	1	no	—
177	C₂×C₄	—	316	2	no	—
178	C₂×C₄	—	158	2	no	—
179	C₂×C₄	—	158	2	no	—
180	C₂×C₄	—	158	2	no	—
181	C₂×C₄	—	158	2	no	—
182	C₂×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
183	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
184	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
186	C₂×C₄	#11929623	1	1	yes	characteristic
187	C₂×C₄	#11929624	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	79	4	no	—
190	C₂²	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
191	C₂²	#14579134	1	1	yes	characteristic
192	C₂²	#14579157	1	1	yes	characteristic
193	C₂²	#14579190	1	2	yes	—
194	C₂²	#14579193	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	8	1	no	—
197	C₄	—	316	2	no	—
198	C₄	—	316	2	no	—
199	C₄	—	632	1	no	—
200	C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₄	#14579133	1	1	yes	characteristic
203	C₄	#14579147	1	1	yes	characteristic
204	C₂	#19098463	1	1	yes	characteristic
205	C₂	#19098667	1	2	yes	—
206	C₂	#19098820	1	2	yes	—
207	C₂	#19098954	1	1	yes	characteristic
208	C₂	#19098956	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11929614

C₂

yes

maximal, characteristic

C₂×#14579092

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic

C₂×#14579071

C₂

yes

maximal, characteristic

#19098470

C₂

yes

maximal, characteristic

#19098468

C₂

yes

maximal, characteristic

C₂×#103×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×C₄×Dic₇₉

C₂²

yes

—

—

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×Q₈×C₇₉

C₂²

yes

characteristic

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

#103×C₇₉

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

C₂×C₄

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

—

#11929614

—

C₂×C₇₉⋊C₈

C₂×C₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₂×C₇₉⋊C₈

D₄

yes

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

C₂×D₄

yes

C₂×C₄×C₇₉

yes

C₂×C₄×C₇₉

yes

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₇₉

C₄×D₄

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#137

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₉

C₄×D₄

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#390

yes

characteristic

100

C₂×C₇₉

#441

yes

—

101

C₂×C₇₉

#470

yes

—

102

C₂×C₇₉

#497

yes

characteristic

103

C₂×C₇₉

yes

104

—

105

C₇₉

#1971

yes

characteristic, complemented, Sylow

106

C₂²×Q₁₆

—

107

C₂×#106

—

108

C₂×#103

—

109

C₂×#103

D₇₉

yes

characteristic

110

C₂×#97

—

111

#391

—

112

#391

—

113

C₂²×Q₈

—

114

C₂²×Q₈

C₂×D₇₉

yes

characteristic

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

158

—

117

C₂×Q₁₆

—

158

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂×D₇₉

yes

characteristic

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

122

C₂×C₄²

—

123

#106

—

158

—

124

#103

—

125

#103

—

158

—

126

#97

—

158

—

127

#95

—

158

—

128

#95

—

158

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

158

—

136

C₂×Q₈

—

158

—

137

C₂×Q₈

—

158

—

138

C₂×Q₈

—

158

—

139

C₂×Q₈

—

158

—

140

C₂×Q₈

C₄×D₇₉

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

158

—

145

C₂²×C₄

—

158

—

146

C₂²×C₄

C₂²×D₇₉

yes

characteristic

147

Q₁₆

—

158

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂²⋊D₇₉

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

—

158

—

158

C₄²

—

158

—

159

C₄²

—

158

—

160

C₂³

D₄×D₇₉

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

316

—

166

Q₈

—

316

—

167

Q₈

—

158

—

168

Q₈

—

158

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

316

—

175

C₂×C₄

—

316

—

176

C₂×C₄

—

316

—

177

C₂×C₄

—

316

—

178

C₂×C₄

—

158

—

179

C₂×C₄

—

158

—

180

C₂×C₄

—

158

—

181

C₂×C₄

—

158

—

182

C₂×C₄

C₂×C₄×D₇₉

yes

characteristic

183

C₂×C₄

D₄×D₇₉

yes

characteristic

184

C₂×C₄

D₄×D₇₉

yes

characteristic

185

C₂×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

186

C₂×C₄

#11929623

yes

characteristic

187

C₂×C₄

#11929624

yes

characteristic

188

C₈

—

189

C₈

—

190

C₂²

(C₂²⋊D₇₉)⋊C₄

yes

characteristic

191

C₂²

#14579134

yes

characteristic

192

C₂²

#14579157

yes

characteristic

193

C₂²

#14579190

yes

—

194

C₂²

#14579193

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

316

—

198

C₄

—

316

—

199

C₄

—

632

—

200

C₄

C₂²⋊C₄×D₇₉

yes

characteristic

201

C₄

C₄×C₂²⋊D₇₉

yes

characteristic

202

C₄

#14579133

yes

characteristic

203

C₄

#14579147

yes

characteristic

204

C₂

#19098463

yes

characteristic

205

C₂

#19098667

yes

—

206

C₂

#19098820

yes

—

207

C₂

#19098954

yes

characteristic

208

C₂

#19098956

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	C₂×#103×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
cosocle	C₂²×C₄×C₇₉	C₂³
hypercenter	C₂×#103	D₇₉
hypocenter	C₇₉	#1971
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

C₂×#103×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

cosocle

C₂²×C₄×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

N	H ≅ G/N	type
11. C₂×#11929614	C₄	semidirect
105. C₇₉	#1971	semidirect

H ≅ G/N

type

11. C₂×#11929614

C₄

semidirect

105. C₇₉

#1971

semidirect

Subgroups — #23764268

Conjugacy classes of subgroups

Complements