Group #23764796 — subgroups

Subgroups — #23764796

Order 10112.

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

normal subgroups only

This group has 248 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×#14579094	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579086	C₂	1	1	yes	maximal, characteristic
4	C₂×#14579084	C₂	1	1	yes	maximal, characteristic
5	#19098468	C₂	1	1	yes	maximal, characteristic
6	#19098459	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	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₇₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
13	#14579094	—	2	2	no	—
14	Dic₇₉⋊C₈	—	2	2	no	—
15	#14579086	—	2	2	no	—
16	#14579084	—	2	2	no	—
17	C₇₉⋊#95	—	2	1	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	D₄	1	2	yes	—
36	C₄⋊Dic₇₉	—	2	1	no	—
37	C₄⋊Dic₇₉	D₄	1	2	yes	—
38	Dic₇₉⋊C₄	—	4	2	no	—
39	C₄×Dic₇₉	—	2	1	no	—
40	C₄×Dic₇₉	—	2	1	no	—
41	C₂²×C₄×C₇₉	—	2	1	no	—
42	C₂²×C₄×C₇₉	—	2	1	no	—
43	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₉	—	2	1	no	—
45	C₂×C₈×C₇₉	—	2	1	no	—
46	C₂×C₈×C₇₉	—	2	1	no	—
47	C₄⋊C₄×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	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₉	—	4	1	no	—
53	C₂×C₄×C₇₉	—	4	1	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	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
63	C₂×C₄×C₇₉	#41	1	1	yes	characteristic
64	C₂×Dic₇₉	—	2	1	no	—
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	—	4	1	no	—
72	C₂×Dic₇₉	—	4	1	no	—
73	C₂×Dic₇₉	—	4	1	no	—
74	C₂×Dic₇₉	—	4	1	no	—
75	C₂×Dic₇₉	Q₁₆	1	2	yes	—
76	Dic₁₅₈	—	2	2	no	—
77	Dic₁₅₈	—	2	2	no	—
78	Dic₁₅₈	—	4	1	no	—
79	Dic₁₅₈	—	4	1	no	—
80	C₈×C₇₉	—	4	1	no	—
81	C₇₉⋊C₈	—	2	2	no	—
82	C₂²×C₇₉	#121	1	1	yes	characteristic
83	C₂²×C₇₉	#121	1	1	yes	characteristic
84	C₂²×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
85	C₂²×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₇₉	#135	1	1	yes	characteristic
87	C₂²×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂²×C₇₉	#137	1	1	yes	characteristic
89	C₄×C₇₉	—	8	1	no	—
90	C₄×C₇₉	—	8	1	no	—
91	C₄×C₇₉	#121	1	1	yes	characteristic
92	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
93	C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₄×C₇₉	#126	1	1	yes	characteristic
95	Dic₇₉	—	2	2	no	—
96	Dic₇₉	—	4	2	no	—
97	Dic₇₉	—	4	2	no	—
98	Dic₇₉	—	4	1	no	—
99	C₂×C₇₉	#396	1	1	yes	characteristic
100	C₂×C₇₉	#462	1	1	yes	characteristic
101	C₂×C₇₉	#463	1	1	yes	characteristic
102	C₂×C₇₉	#467	1	1	yes	characteristic
103	C₂×C₇₉	#468	1	1	yes	characteristic
104	C₂×C₇₉	#477	1	1	yes	characteristic
105	C₂×C₇₉	#479	1	1	yes	characteristic
106	#2068	—	79	1	no	maximal, Sylow
107	C₇₉	#2068	1	1	yes	characteristic, complemented, Sylow
108	C₂×#107	—	79	1	no	—
109	C₂×C₄⋊C₈	—	79	1	no	—
110	C₂×#103	—	79	1	no	—
111	C₂×#103	—	79	1	no	—
112	#391	—	79	1	no	—
113	#389	—	79	1	no	—
114	#340	D₇₉	1	1	yes	characteristic
115	C₂²×Q₈	—	79	1	no	—
116	C₂²×C₈	—	79	1	no	—
117	C₂²×C₈	C₂×D₇₉	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	79	1	no	—
119	C₂×C₄⋊C₄	—	158	1	no	—
120	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
122	C₂×C₄²	—	79	1	no	—
123	#107	—	158	2	no	—
124	C₄⋊C₈	—	158	2	no	—
125	#103	—	158	2	no	—
126	#103	—	158	2	no	—
127	#95	—	158	1	no	—
128	#95	—	158	1	no	—
129	#95	—	158	1	no	—
130	C₂×Q₈	—	79	2	no	—
131	C₂×Q₈	—	158	1	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	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	79	1	no	—
139	C₂²×C₄	—	79	1	no	—
140	C₂²×C₄	—	158	1	no	—
141	C₂²×C₄	—	158	1	no	—
142	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	79	2	no	—
147	C₂×C₈	—	158	1	no	—
148	C₂×C₈	—	158	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	—	2	1	no	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	316	2	no	—
154	C₄⋊C₄	—	79	2	no	—
155	C₄⋊C₄	—	158	1	no	—
156	C₄²	—	158	1	no	—
157	C₄²	—	158	1	no	—
158	C₂³	D₄×D₇₉	1	1	yes	characteristic
159	Q₈	—	316	1	no	—
160	Q₈	—	316	1	no	—
161	Q₈	—	158	2	no	—
162	Q₈	—	158	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	1	no	—
165	C₂×C₄	—	4	1	no	—
166	C₂×C₄	—	4	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	316	1	no	—
170	C₂×C₄	—	316	1	no	—
171	C₂×C₄	—	316	1	no	—
172	C₂×C₄	—	316	1	no	—
173	C₂×C₄	—	79	2	no	—
174	C₂×C₄	—	158	1	no	—
175	C₂×C₄	—	158	2	no	—
176	C₂×C₄	—	158	2	no	—
177	C₂×C₄	—	158	1	no	—
178	C₂×C₄	—	158	1	no	—
179	C₂×C₄	—	158	1	no	—
180	C₂×C₄	—	158	1	no	—
181	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
182	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
183	C₂×C₄	Q₈×D₇₉	1	1	yes	characteristic
184	C₂×C₄	#11929624	1	1	yes	characteristic
185	C₂×C₄	#11929624	1	1	yes	characteristic
186	C₂×C₄	#11929624	1	1	yes	characteristic
187	C₈	—	4	1	no	—
188	C₈	—	158	2	no	—
189	C₂²	#14579134	1	1	yes	characteristic
190	C₂²	#14579134	1	1	yes	characteristic
191	C₂²	#14579139	1	1	yes	characteristic
192	C₂²	Q₁₆×D₇₉	1	1	yes	characteristic
193	C₂²	#14579182	1	1	yes	characteristic
194	C₂²	#14579185	1	1	yes	characteristic
195	C₂²	#14579190	1	1	yes	characteristic
196	C₄	—	8	1	no	—
197	C₄	—	8	1	no	—
198	C₄	—	316	2	no	—
199	C₄	—	316	2	no	—
200	C₄	—	316	1	no	—
201	C₄	—	158	2	no	—
202	C₄	#14579134	1	1	yes	characteristic
203	C₄	#14579139	1	1	yes	characteristic
204	C₄	#14579139	1	1	yes	characteristic
205	C₄	#14579155	1	1	yes	characteristic
206	C₂	#19098487	1	1	yes	characteristic
207	C₂	#19098772	1	1	yes	characteristic
208	C₂	#19098777	1	1	yes	characteristic
209	C₂	#19098803	1	1	yes	characteristic
210	C₂	#19098808	1	1	yes	characteristic
211	C₂	#19098857	1	1	yes	characteristic
212	C₂	#19098873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14579094

C₂

yes

maximal, characteristic

C₂×Dic₇₉⋊C₈

C₂

yes

maximal, characteristic

C₂×#14579086

C₂

yes

maximal, characteristic

C₂×#14579084

C₂

yes

maximal, characteristic

#19098468

C₂

yes

maximal, characteristic

#19098459

C₂

yes

maximal, characteristic

#340×C₇₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₈

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₉

C₂²

yes

characteristic

C₂×Dic₇₉⋊C₄

—

C₂×C₄×Dic₇₉

C₂²

yes

characteristic

#14579094

—

Dic₇₉⋊C₈

—

#14579086

—

#14579084

—

C₇₉⋊#95

—

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

D₄

yes

—

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

—

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

Q₁₆

yes

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

Dic₁₅₈

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₉

C₂×Q₁₆

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#121

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

C₂²⋊Q₈

yes

characteristic

C₄×C₇₉

#126

yes

characteristic

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

Dic₇₉

—

C₂×C₇₉

#396

yes

characteristic

100

C₂×C₇₉

#462

yes

characteristic

101

C₂×C₇₉

#463

yes

characteristic

102

C₂×C₇₉

#467

yes

characteristic

103

C₂×C₇₉

#468

yes

characteristic

104

C₂×C₇₉

#477

yes

characteristic

105

C₂×C₇₉

yes

106

—

107

C₇₉

#2068

yes

characteristic, complemented, Sylow

108

C₂×#107

—

109

C₂×C₄⋊C₈

—

110

C₂×#103

—

111

C₂×#103

—

112

#391

—

113

#389

—

114

#340

D₇₉

yes

characteristic

115

C₂²×Q₈

—

116

C₂²×C₈

—

117

C₂²×C₈

C₂×D₇₉

yes

characteristic

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

158

—

120

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

121

C₂×C₄⋊C₄

C₂×D₇₉

yes

characteristic

122

C₂×C₄²

—

123

#107

—

158

—

124

C₄⋊C₈

—

158

—

125

#103

—

158

—

126

#103

—

158

—

127

#95

—

158

—

128

#95

—

158

—

129

#95

—

158

—

130

C₂×Q₈

—

131

C₂×Q₈

—

158

—

132

C₂×Q₈

—

158

—

133

C₂×Q₈

—

158

—

134

C₂×Q₈

—

158

—

135

C₂×Q₈

—

158

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

—

158

—

141

C₂²×C₄

—

158

—

142

C₂²×C₄

C₂²×D₇₉

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

158

—

148

C₂×C₈

—

158

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

316

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

158

—

156

C₄²

—

158

—

157

C₄²

—

158

—

158

C₂³

D₄×D₇₉

yes

characteristic

159

Q₈

—

316

—

160

Q₈

—

316

—

161

Q₈

—

158

—

162

Q₈

—

158

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

316

—

170

C₂×C₄

—

316

—

171

C₂×C₄

—

316

—

172

C₂×C₄

—

316

—

173

C₂×C₄

—

174

C₂×C₄

—

158

—

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

D₄×D₇₉

yes

characteristic

182

C₂×C₄

D₄×D₇₉

yes

characteristic

183

C₂×C₄

Q₈×D₇₉

yes

characteristic

184

C₂×C₄

#11929624

yes

characteristic

185

C₂×C₄

#11929624

yes

characteristic

186

C₂×C₄

#11929624

yes

characteristic

187

C₈

—

188

C₈

—

158

—

189

C₂²

#14579134

yes

characteristic

190

C₂²

#14579134

yes

characteristic

191

C₂²

#14579139

yes

characteristic

192

C₂²

Q₁₆×D₇₉

yes

characteristic

193

C₂²

#14579182

yes

characteristic

194

C₂²

#14579185

yes

characteristic

195

C₂²

#14579190

yes

characteristic

196

C₄

—

197

C₄

—

198

C₄

—

316

—

199

C₄

—

316

—

200

C₄

—

316

—

201

C₄

—

158

—

202

C₄

#14579134

yes

characteristic

203

C₄

#14579139

yes

characteristic

204

C₄

#14579139

yes

characteristic

205

C₄

#14579155

yes

characteristic

206

C₂

#19098487

yes

characteristic

207

C₂

#19098772

yes

characteristic

208

C₂

#19098777

yes

characteristic

209

C₂

#19098803

yes

characteristic

210

C₂

#19098808

yes

characteristic

211

C₂

#19098857

yes

characteristic

212

C₂

#19098873

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	#2068
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
107. C₇₉	#2068	semidirect

H ≅ G/N

type

107. C₇₉

#2068

semidirect

Subgroups — #23764796

Conjugacy classes of subgroups

Complements