Group #23764550 — subgroups

Subgroups — #23764550

Order 10112.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂×#14579152	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579086	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579084	C₂	1	1	yes	maximal, characteristic
4	#19098468	C₂	1	1	yes	maximal, characteristic
5	#19098325	C₂	1	1	yes	maximal, characteristic
6	#19098317	C₂	1	1	yes	maximal, characteristic
7	#340×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	#14579152	—	2	2	no	—
11	#14579152	—	2	2	no	—
12	C₂×C₄⋊Dic₇₉	—	2	1	no	—
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₇₉⋊C₄	—	2	1	no	—
15	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
16	#14579086	—	2	2	no	—
17	#14579084	—	2	2	no	—
18	C₇₉⋊#95	—	2	1	no	—
19	C₇₉⋊#95	—	2	1	no	—
20	C₂²×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₅₈	—	2	1	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	Q₈	1	2	yes	—
29	C₂²×Dic₇₉	—	2	1	no	—
30	C₂²×Dic₇₉	—	2	1	no	—
31	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
32	C₂²×Dic₇₉	D₄	1	1	yes	characteristic
33	C₂×C₇₉⋊C₈	—	2	1	no	—
34	C₂×C₇₉⋊C₈	—	2	1	no	—
35	C₂×C₇₉⋊C₈	—	2	1	no	—
36	C₄⋊Dic₇₉	—	2	4	no	—
37	C₄⋊Dic₇₉	—	2	1	no	—
38	C₄⋊Dic₇₉	D₄	1	2	yes	—
39	Dic₇₉⋊C₄	—	2	4	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₄×Dic₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂²×C₄×C₇₉	—	2	1	no	—
44	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₂×C₈×C₇₉	—	2	1	no	—
48	C₄⋊C₄×C₇₉	—	2	1	no	—
49	C₄⋊C₄×C₇₉	—	2	1	no	—
50	C₄⋊C₄×C₇₉	—	2	1	no	—
51	C₄⋊C₄×C₇₉	D₄	1	2	yes	—
52	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₉	—	2	2	no	—
54	C₂×C₄×C₇₉	—	4	1	no	—
55	C₂×C₄×C₇₉	—	4	1	no	—
56	C₂×C₄×C₇₉	—	4	1	no	—
57	C₂×C₄×C₇₉	—	4	1	no	—
58	C₂×C₄×C₇₉	—	4	1	no	—
59	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
64	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
65	C₂×Dic₇₉	—	2	1	no	—
66	C₂×Dic₇₉	—	2	1	no	—
67	C₂×Dic₇₉	—	2	2	no	—
68	C₂×Dic₇₉	—	2	2	no	—
69	C₂×Dic₇₉	—	2	1	no	—
70	C₂×Dic₇₉	—	2	1	no	—
71	C₂×Dic₇₉	—	2	1	no	—
72	C₂×Dic₇₉	—	2	1	no	—
73	C₂×Dic₇₉	—	4	1	no	—
74	C₂×Dic₇₉	—	4	1	no	—
75	C₂×Dic₇₉	—	4	1	no	—
76	C₂×Dic₇₉	—	4	1	no	—
77	Dic₁₅₈	—	2	2	no	—
78	Dic₁₅₈	—	2	2	no	—
79	Dic₁₅₈	—	4	1	no	—
80	Dic₁₅₈	—	4	1	no	—
81	C₈×C₇₉	—	4	1	no	—
82	C₇₉⋊C₈	—	4	1	no	—
83	C₂²×C₇₉	#120	1	1	yes	characteristic
84	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₇₉	#124	1	1	yes	characteristic
86	C₂²×C₇₉	#135	1	1	yes	characteristic
87	C₂²×C₇₉	#135	1	1	yes	characteristic
88	C₂²×C₇₉	#137	1	1	yes	characteristic
89	C₂²×C₇₉	#137	1	1	yes	characteristic
90	C₄×C₇₉	—	4	2	no	—
91	C₄×C₇₉	—	8	1	no	—
92	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₇₉	#124	1	1	yes	characteristic
95	C₄×C₇₉	#124	1	1	yes	characteristic
96	Dic₇₉	—	4	2	no	—
97	Dic₇₉	—	4	1	no	—
98	Dic₇₉	—	4	1	no	—
99	Dic₇₉	—	4	2	no	—
100	C₂×C₇₉	#393	1	1	yes	characteristic
101	C₂×C₇₉	#452	1	1	yes	characteristic
102	C₂×C₇₉	#452	1	1	yes	characteristic
103	C₂×C₇₉	#479	1	1	yes	characteristic
104	C₂×C₇₉	#479	1	1	yes	characteristic
105	C₂×C₇₉	#488	1	1	yes	characteristic
106	C₂×C₇₉	#491	1	1	yes	characteristic
107	#2026	—	79	1	no	maximal, Sylow
108	C₇₉	#2026	1	1	yes	characteristic, complemented, Sylow
109	C₂×#125	—	79	1	no	—
110	C₂×#103	—	79	1	no	—
111	C₂×#103	—	79	1	no	—
112	#391	—	79	1	no	—
113	#340	—	79	1	no	—
114	#340	D₇₉	1	1	yes	characteristic
115	#336	—	79	1	no	—
116	C₂²×Q₈	—	79	1	no	—
117	C₂²×C₈	—	79	1	no	—
118	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
119	#125	—	158	2	no	—
120	#125	—	158	2	no	—
121	C₂×C₄⋊C₄	—	79	1	no	—
122	C₂×C₄⋊C₄	—	158	1	no	—
123	C₂×C₄⋊C₄	—	158	1	no	—
124	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
126	C₂×C₄²	—	79	1	no	—
127	#103	—	158	2	no	—
128	#103	—	158	2	no	—
129	#95	—	158	1	no	—
130	#95	—	158	1	no	—
131	C₂×Q₈	—	79	2	no	—
132	C₂×Q₈	—	158	1	no	—
133	C₂×Q₈	—	158	1	no	—
134	C₂×Q₈	—	158	1	no	—
135	C₂×Q₈	—	158	1	no	—
136	C₂×Q₈	—	158	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	79	1	no	—
140	C₂²×C₄	—	79	1	no	—
141	C₂²×C₄	—	158	1	no	—
142	C₂²×C₄	—	158	1	no	—
143	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	158	1	no	—
148	C₂×C₈	—	158	1	no	—
149	C₂×C₈	—	158	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	79	2	no	—
154	C₄⋊C₄	—	158	4	no	—
155	C₄⋊C₄	—	158	4	no	—
156	C₄⋊C₄	—	158	1	no	—
157	C₄⋊C₄	D₃₁₆	1	2	yes	—
158	C₄²	—	158	1	no	—
159	C₄²	—	158	1	no	—
160	C₂³	D₄×D₇₉	1	1	yes	characteristic
161	Q₈	—	316	1	no	—
162	Q₈	—	316	1	no	—
163	Q₈	—	158	2	no	—
164	Q₈	—	158	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	316	1	no	—
172	C₂×C₄	—	316	1	no	—
173	C₂×C₄	—	316	1	no	—
174	C₂×C₄	—	316	1	no	—
175	C₂×C₄	—	158	1	no	—
176	C₂×C₄	—	158	1	no	—
177	C₂×C₄	—	158	2	no	—
178	C₂×C₄	—	158	2	no	—
179	C₂×C₄	—	158	1	no	—
180	C₂×C₄	—	158	1	no	—
181	C₂×C₄	—	158	1	no	—
182	C₂×C₄	—	158	1	no	—
183	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
184	C₂×C₄	C₂×D₃₁₆	1	1	yes	characteristic
185	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
186	C₂×C₄	#11929623	1	1	yes	characteristic
187	C₂×C₄	#11929624	1	1	yes	characteristic
188	C₂×C₄	#11929624	1	1	yes	characteristic
189	C₈	—	4	1	no	—
190	C₈	—	316	1	no	—
191	C₂²	#14579131	1	1	yes	characteristic
192	C₂²	#14579139	1	1	yes	characteristic
193	C₂²	#14579147	1	1	yes	characteristic
194	C₂²	#14579180	1	1	yes	characteristic
195	C₂²	#14579182	1	1	yes	characteristic
196	C₂²	#14579190	1	1	yes	characteristic
197	C₂²	#14579193	1	1	yes	characteristic
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	316	1	no	—
201	C₄	—	316	2	no	—
202	C₄	—	316	1	no	—
203	C₄	—	316	2	no	—
204	C₄	#14579139	1	1	yes	characteristic
205	C₄	#14579141	1	1	yes	characteristic
206	C₄	#14579147	1	1	yes	characteristic
207	C₄	#14579149	1	1	yes	characteristic
208	C₂	#19098476	1	1	yes	characteristic
209	C₂	#19098726	1	1	yes	characteristic
210	C₂	#19098728	1	1	yes	characteristic
211	C₂	#19098869	1	1	yes	characteristic
212	C₂	#19098873	1	1	yes	characteristic
213	C₂	#19098916	1	1	yes	characteristic
214	C₂	#19098930	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579152

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic

C₂×#14579084

C₂

yes

maximal, characteristic

#19098468

C₂

yes

maximal, characteristic

#19098325

C₂

yes

maximal, characteristic

#19098317

C₂

yes

maximal, characteristic

#340×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

#14579152

—

#14579152

—

C₂×C₄⋊Dic₇₉

—

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579086

—

#14579084

—

C₇₉⋊#95

—

C₇₉⋊#95

—

C₂²×C₈×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₉

C₂²

yes

characteristic

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

—

C₂×Dic₁₅₈

Q₈

yes

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

—

C₂²×Dic₇₉

D₄

yes

characteristic

C₂²×Dic₇₉

D₄

yes

characteristic

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

D₄

yes

—

Dic₇₉⋊C₄

—

C₄×Dic₇₉

—

C₄×Dic₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂³

yes

characteristic

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

D₄

yes

—

C₂³×C₇₉

C₂×D₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

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×Q₈

yes

C₂×C₄×C₇₉

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

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

#120

yes

characteristic

C₂²×C₇₉

C₂²⋊Q₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

100

C₂×C₇₉

#393

yes

characteristic

101

C₂×C₇₉

#452

yes

characteristic

102

C₂×C₇₉

#452

yes

characteristic

103

C₂×C₇₉

#479

yes

characteristic

104

C₂×C₇₉

#479

yes

characteristic

105

C₂×C₇₉

#488

yes

characteristic

106

C₂×C₇₉

yes

107

—

108

C₇₉

#2026

yes

characteristic, complemented, Sylow

109

C₂×#125

—

110

C₂×#103

—

111

C₂×#103

—

112

#391

—

113

#340

—

114

#340

D₇₉

yes

characteristic

115

#336

—

116

C₂²×Q₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₂×D₇₉

yes

characteristic

119

#125

—

158

—

120

#125

—

158

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

158

—

123

C₂×C₄⋊C₄

—

158

—

124

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

125

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

126

C₂×C₄²

—

127

#103

—

158

—

128

#103

—

158

—

129

#95

—

158

—

130

#95

—

158

—

131

C₂×Q₈

—

132

C₂×Q₈

—

158

—

133

C₂×Q₈

—

158

—

134

C₂×Q₈

—

158

—

135

C₂×Q₈

—

158

—

136

C₂×Q₈

—

158

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

158

—

142

C₂²×C₄

—

158

—

143

C₂²×C₄

C₂²×D₇₉

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

158

—

148

C₂×C₈

—

158

—

149

C₂×C₈

—

158

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

—

158

—

155

C₄⋊C₄

—

158

—

156

C₄⋊C₄

—

158

—

157

C₄⋊C₄

D₃₁₆

yes

—

158

C₄²

—

158

—

159

C₄²

—

158

—

160

C₂³

D₄×D₇₉

yes

characteristic

161

Q₈

—

316

—

162

Q₈

—

316

—

163

Q₈

—

158

—

164

Q₈

—

158

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

316

—

172

C₂×C₄

—

316

—

173

C₂×C₄

—

316

—

174

C₂×C₄

—

316

—

175

C₂×C₄

—

158

—

176

C₂×C₄

—

158

—

177

C₂×C₄

—

158

—

178

C₂×C₄

—

158

—

179

C₂×C₄

—

158

—

180

C₂×C₄

—

158

—

181

C₂×C₄

—

158

—

182

C₂×C₄

—

158

—

183

C₂×C₄

D₄×D₇₉

yes

characteristic

184

C₂×C₄

C₂×D₃₁₆

yes

characteristic

185

C₂×C₄

Q₈×D₇₉

yes

characteristic

186

C₂×C₄

#11929623

yes

characteristic

187

C₂×C₄

#11929624

yes

characteristic

188

C₂×C₄

#11929624

yes

characteristic

189

C₈

—

190

C₈

—

316

—

191

C₂²

#14579131

yes

characteristic

192

C₂²

#14579139

yes

characteristic

193

C₂²

#14579147

yes

characteristic

194

C₂²

#14579180

yes

characteristic

195

C₂²

#14579182

yes

characteristic

196

C₂²

#14579190

yes

characteristic

197

C₂²

#14579193

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

—

316

—

201

C₄

—

316

—

202

C₄

—

316

—

203

C₄

—

316

—

204

C₄

#14579139

yes

characteristic

205

C₄

#14579141

yes

characteristic

206

C₄

#14579147

yes

characteristic

207

C₄

#14579149

yes

characteristic

208

C₂

#19098476

yes

characteristic

209

C₂

#19098726

yes

characteristic

210

C₂

#19098728

yes

characteristic

211

C₂

#19098869

yes

characteristic

212

C₂

#19098873

yes

characteristic

213

C₂

#19098916

yes

characteristic

214

C₂

#19098930

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₉
commutator, cosocle	C₂²×C₄×C₇₉	C₂³
Frattini	C₂²×C₄	C₂²×D₇₉
Fitting	#340×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₄
hypercenter	#340	D₇₉
hypocenter	C₇₉	#2026
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₇₉

commutator, cosocle

C₂²×C₄×C₇₉

C₂³

Frattini

C₂²×C₄

C₂²×D₇₉

Fitting

#340×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₄

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
108. C₇₉	#2026	semidirect

H ≅ G/N

type

108. C₇₉

#2026

semidirect

Subgroups — #23764550

Conjugacy classes of subgroups

Complements