Group #34176387 — subgroups

Subgroups — #34176387

Order 20224.

408 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 143 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	#23763517	C₂	1	2	yes	maximal
2	#23761223	C₂	1	4	yes	maximal
3	C₄×#125×C₇₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
5	#19098318	—	2	2	no	—
6	#19098318	C₂²	1	4	yes	—
7	C₂×#125×C₇₉	C₄	1	1	yes	characteristic
8	#389×C₇₉	C₂²	1	2	yes	—
9	#387×C₇₉	—	2	2	no	—
10	#382×C₇₉	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₇₉	—	2	2	no	—
12	C₄×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
13	C₄³×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₈	—	2	4	no	—
15	C₄⋊(C₇₉⋊C₈)	—	4	4	no	—
16	#125×C₇₉	—	2	2	no	—
17	#125×C₇₉	C₈	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
19	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
20	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₉	—	2	2	no	—
23	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₉	D₄	1	2	yes	—
26	C₂×C₄²×C₇₉	D₄	1	2	yes	—
27	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
28	#95×C₇₉	—	2	2	no	—
29	#95×C₇₉	—	2	2	no	—
30	#95×C₇₉	—	2	2	no	—
31	C₂×C₇₉⋊C₈	—	4	4	no	—
32	C₂×C₇₉⋊C₈	—	4	4	no	—
33	C₂×C₇₉⋊C₈	—	4	4	no	—
34	C₂²×C₄×C₇₉	—	2	2	no	—
35	C₂²×C₄×C₇₉	—	2	2	no	—
36	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₇₉	—	2	8	no	—
42	C₄⋊C₄×C₇₉	—	2	2	no	—
43	C₄⋊C₄×C₇₉	—	4	2	no	—
44	C₄⋊C₄×C₇₉	—	4	2	no	—
45	C₄⋊C₄×C₇₉	—	4	2	no	—
46	C₄⋊C₄×C₇₉	C₂×C₈	1	4	yes	—
47	C₄²×C₇₉	—	2	2	no	—
48	C₄²×C₇₉	—	2	2	no	—
49	C₄²×C₇₉	—	2	2	no	—
50	C₄²×C₇₉	—	4	2	no	—
51	C₄²×C₇₉	—	4	1	no	—
52	C₄²×C₇₉	—	4	4	no	—
53	C₄²×C₇₉	C₂×C₈	1	2	yes	—
54	C₄²×C₇₉	M₁₆	1	2	yes	—
55	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	1	no	—
58	C₂×C₄×C₇₉	—	2	1	no	—
59	C₂×C₄×C₇₉	—	2	2	no	—
60	C₂×C₄×C₇₉	—	2	2	no	—
61	C₂×C₄×C₇₉	—	2	2	no	—
62	C₂×C₄×C₇₉	—	2	2	no	—
63	C₂×C₄×C₇₉	—	2	4	no	—
64	C₂×C₄×C₇₉	—	2	1	no	—
65	C₂×C₄×C₇₉	—	2	2	no	—
66	C₂×C₄×C₇₉	—	4	2	no	—
67	C₂×C₄×C₇₉	—	4	2	no	—
68	C₂×C₄×C₇₉	—	4	2	no	—
69	C₂×C₄×C₇₉	—	4	2	no	—
70	C₂×C₄×C₇₉	—	4	2	no	—
71	C₂×C₄×C₇₉	C₂²⋊C₈	1	4	yes	—
72	C₂×C₄×C₇₉	#101	1	2	yes	—
73	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₇₉	#137	1	2	yes	—
77	C₇₉⋊C₈	—	8	4	no	—
78	C₂²×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₂²×C₇₉	#410	1	1	yes	characteristic
80	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
81	C₂²×C₇₉	#417	1	2	yes	—
82	C₂²×C₇₉	#421	1	2	yes	—
83	C₄×C₇₉	—	2	1	no	—
84	C₄×C₇₉	—	2	1	no	—
85	C₄×C₇₉	—	2	4	no	—
86	C₄×C₇₉	—	2	2	no	—
87	C₄×C₇₉	—	2	2	no	—
88	C₄×C₇₉	—	2	2	no	—
89	C₄×C₇₉	—	4	1	no	—
90	C₄×C₇₉	—	4	4	no	—
91	C₄×C₇₉	—	8	2	no	—
92	#7602	—	79	1	no	maximal, Sylow
93	C₂×C₇₉	#1461	1	1	yes	characteristic
94	C₂×C₇₉	#1477	1	2	yes	—
95	C₂×C₇₉	#1512	1	2	yes	—
96	C₂×C₇₉	#1534	1	1	yes	characteristic
97	C₂×C₇₉	#1561	1	1	yes	characteristic
98	C₄×#125	D₇₉	1	1	yes	characteristic
99	#1831	—	79	2	no	—
100	#1271	—	79	4	no	—
101	C₇₉	#7602	1	1	yes	characteristic, complemented, Sylow
102	C₂×#125	Dic₇₉	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	158	2	no	—
104	#389	C₂×D₇₉	1	2	yes	—
105	#387	—	2	2	no	—
106	#382	Dic₇₉	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	2	2	no	—
108	C₄×C₄⋊C₄	Dic₇₉	1	2	yes	—
109	C₄³	C₂×D₇₉	1	1	yes	characteristic
110	#336	—	79	4	no	—
111	#336	—	158	2	no	—
112	C₂²×C₈	—	158	4	no	—
113	#125	—	2	2	no	—
114	#125	C₇₉⋊C₈	1	4	yes	complemented
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
119	C₂×C₄²	—	2	2	no	—
120	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
124	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
125	C₄⋊C₈	—	316	4	no	—
126	#95	—	2	2	no	—
127	#95	—	2	2	no	—
128	#95	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
132	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
133	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
136	C₂×C₈	—	316	4	no	—
137	C₂×C₈	—	316	4	no	—
138	C₂×C₈	—	316	4	no	—
139	C₄⋊C₄	—	2	8	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	C₂×C₇₉⋊C₈	1	4	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	1	no	—
150	C₄²	—	4	4	no	—
151	C₄²	C₂×C₇₉⋊C₈	1	2	yes	—
152	C₄²	#11929608	1	2	yes	—
153	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₂²⋊(C₇₉⋊C₈)	1	4	yes	—
170	C₂×C₄	#14579080	1	2	yes	—
171	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
172	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
173	C₂×C₄	#14579186	1	2	yes	—
174	C₂×C₄	#14579191	1	2	yes	—
175	C₈	—	632	4	no	—
176	C₂²	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
177	C₂²	#19098537	1	1	yes	characteristic
178	C₂²	C₂×#14579080	1	1	yes	characteristic
179	C₂²	#19098564	1	2	yes	—
180	C₂²	#19098580	1	2	yes	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	4	4	no	—
188	C₄	—	4	1	no	—
189	C₄	—	8	2	no	—
190	C₂	#23761724	1	1	yes	characteristic
191	C₂	#23761787	1	2	yes	—
192	C₂	#23761938	1	2	yes	—
193	C₂	#23762027	1	1	yes	characteristic
194	C₂	#23762138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23763517

C₂

yes

maximal

#23761223

C₂

yes

maximal

C₄×#125×C₇₉

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₉⋊C₈)

—

#19098318

—

#19098318

C₂²

yes

—

C₂×#125×C₇₉

C₄

yes

characteristic

#389×C₇₉

C₂²

yes

—

#387×C₇₉

—

#382×C₇₉

C₄

yes

characteristic

C₄×C₄⋊C₄×C₇₉

—

C₄×C₄⋊C₄×C₇₉

C₄

yes

—

C₄³×C₇₉

C₂²

yes

characteristic

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

#125×C₇₉

—

#125×C₇₉

C₈

yes

complemented

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

—

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

D₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#95×C₇₉

—

#95×C₇₉

—

#95×C₇₉

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

C₂×D₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

C₂×C₈

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂×C₈

yes

—

C₄²×C₇₉

M₁₆

yes

—

C₂³×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂²⋊C₈

yes

—

C₂×C₄×C₇₉

#101

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₇₉

#137

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

C₂×C₂²⋊C₈

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#421

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

#1512

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

D₇₉

yes

characteristic

#1831

—

100

#1271

—

101

C₇₉

#7602

yes

characteristic, complemented, Sylow

102

C₂×#125

Dic₇₉

yes

characteristic

103

C₂×C₄⋊C₈

—

158

—

104

#389

C₂×D₇₉

yes

—

105

#387

—

106

#382

Dic₇₉

yes

characteristic

107

C₄×C₄⋊C₄

—

108

C₄×C₄⋊C₄

Dic₇₉

yes

—

109

C₄³

C₂×D₇₉

yes

characteristic

110

#336

—

111

#336

—

158

—

112

C₂²×C₈

—

158

—

113

#125

—

114

#125

C₇₉⋊C₈

yes

complemented

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×Dic₇₉

yes

—

118

C₂×C₄⋊C₄

C₂×Dic₇₉

yes

—

119

C₂×C₄²

—

120

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

121

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

122

C₂×C₄²

C₂²⋊D₇₉

yes

—

123

C₂×C₄²

C₂²⋊D₇₉

yes

—

124

C₂×C₄²

C₂²×D₇₉

yes

characteristic

125

C₄⋊C₈

—

316

—

126

#95

—

127

#95

—

128

#95

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

132

C₂²×C₄

C₂²⋊Dic₇₉

yes

—

133

C₂²×C₄

C₂²⋊Dic₇₉

yes

characteristic

134

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

—

136

C₂×C₈

—

316

—

137

C₂×C₈

—

316

—

138

C₂×C₈

—

316

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

141

C₄⋊C₄

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

C₂×C₇₉⋊C₈

yes

—

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

C₂×C₇₉⋊C₈

yes

—

152

C₄²

#11929608

yes

—

153

C₂³

C₂×C₂²⋊Dic₇₉

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂²⋊(C₇₉⋊C₈)

yes

—

170

C₂×C₄

#14579080

yes

—

171

C₂×C₄

C₂²×C₇₉⋊C₈

yes

characteristic

172

C₂×C₄

C₂×#11929608

yes

characteristic

173

C₂×C₄

#14579186

yes

—

174

C₂×C₄

#14579191

yes

—

175

C₈

—

632

—

176

C₂²

C₂×C₂²⋊(C₇₉⋊C₈)

yes

characteristic

177

C₂²

#19098537

yes

characteristic

178

C₂²

C₂×#14579080

yes

characteristic

179

C₂²

#19098564

yes

—

180

C₂²

#19098580

yes

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₂

#23761724

yes

characteristic

191

C₂

#23761787

yes

—

192

C₂

#23761938

yes

—

193

C₂

#23762027

yes

characteristic

194

C₂

#23762138

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#125×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#125	D₇₉
hypocenter	C₇₉	#7602
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₇₉

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₄×#125×C₇₉

C₂

socle, prime radical

C₂³×C₇₉

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
17. #125×C₇₉	C₈	semidirect
101. C₇₉	#7602	semidirect
114. #125	C₇₉⋊C₈	semidirect

H ≅ G/N

type

17. #125×C₇₉

C₈

semidirect

101. C₇₉

#7602

semidirect

114. #125

C₇₉⋊C₈

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23763517	2
#23761223	2
C₄×#125×C₇₉	2
#7602	79

subgroup

index

#23763517

#23761223

C₄×#125×C₇₉

#7602

Subgroups — #34176387

Conjugacy classes of subgroups

Complements

Maximal subgroups