Group #34204751 — subgroups

Subgroups — #34204751

Order 20224.

370 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 147 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	#23776503	C₂	1	1	yes	maximal, characteristic
2	#23765166	C₂	1	2	yes	maximal, complemented
3	#23765014	C₂	1	1	yes	maximal, characteristic
4	#23761489	C₂	1	2	yes	maximal, complemented
5	C₇₉×#1757	C₂	1	1	yes	maximal, characteristic
6	#19098980	—	2	4	no	—
7	C₂×#14579101	—	2	2	no	—
8	C₂×#14579101	C₂²	1	1	yes	characteristic
9	C₂×#14579101	C₂²	1	2	yes	—
10	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19098384	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
13	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
14	#413×C₇₉	C₄	1	1	yes	characteristic
15	#410×C₇₉	C₄	1	1	yes	characteristic
16	#405×C₇₉	C₂²	1	2	yes	—
17	#344×C₇₉	C₄	1	2	yes	—
18	#14579101	—	2	4	no	—
19	#14579101	—	2	2	no	—
20	#14579101	—	2	2	no	—
21	#14579101	D₄	1	4	yes	—
22	C₂×C₇₉⋊C₁₆	—	2	4	no	—
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	1	no	—
25	C₂×C₇₉⋊C₁₆	—	2	1	no	—
26	C₂×C₇₉⋊C₁₆	—	2	2	no	—
27	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
28	#131×C₇₉	—	2	2	no	—
29	#131×C₇₉	—	2	2	no	—
30	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
33	C₂²×C₈×C₇₉	D₄	1	2	yes	—
34	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
35	#117×C₇₉	C₂×C₄	1	2	yes	—
36	#101×C₇₉	C₈	1	2	yes	—
37	#100×C₇₉	C₈	1	2	yes	—
38	#99×C₇₉	—	2	2	no	—
39	#97×C₇₉	—	2	2	no	—
40	C₄×C₈×C₇₉	—	2	2	no	—
41	C₇₉⋊C₁₆	—	2	4	no	—
42	C₇₉⋊C₁₆	—	4	1	no	—
43	C₇₉⋊C₁₆	—	4	1	no	—
44	#41×C₇₉	—	2	2	no	—
45	#41×C₇₉	—	2	2	no	—
46	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
47	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
48	C₂×D₄×C₇₉	M₁₆	1	2	yes	—
49	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
51	M₁₆×C₇₉	—	2	4	no	—
52	M₁₆×C₇₉	—	2	2	no	—
53	M₁₆×C₇₉	—	2	2	no	—
54	M₁₆×C₇₉	C₂×C₈	1	4	yes	—
55	C₂×C₈×C₇₉	—	2	2	no	—
56	C₂×C₈×C₇₉	—	2	2	no	—
57	C₂×C₈×C₇₉	—	2	2	no	—
58	C₂×C₈×C₇₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₉	#41	1	2	yes	—
63	C₄⋊C₄×C₇₉	—	2	2	no	—
64	C₂²⋊C₄×C₇₉	—	2	2	no	—
65	C₄²×C₇₉	—	2	2	no	—
66	C₂³×C₇₉	C₄⋊C₈	1	2	yes	—
67	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
68	Q₈×C₇₉	—	2	2	no	—
69	D₄×C₇₉	—	2	4	no	—
70	D₄×C₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#131	1	2	yes	—
79	C₈×C₇₉	—	2	2	no	—
80	C₈×C₇₉	—	2	4	no	—
81	C₈×C₇₉	#120	1	1	yes	characteristic
82	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₇₉	#123	1	1	yes	characteristic
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	—	4	1	no	—
86	C₂²×C₇₉	—	4	2	no	—
87	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₉	#435	1	2	yes	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	8	2	no	—
91	C₄×C₇₉	#385	1	1	yes	characteristic
92	C₄×C₇₉	#424	1	1	yes	characteristic
93	C₄×C₇₉	C₈×D₄	1	2	yes	—
94	#12916	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	2	2	no	—
96	C₂×C₇₉	—	2	1	no	—
97	C₂×C₇₉	—	4	2	no	—
98	C₂×C₇₉	#1850	1	1	yes	characteristic
99	#3407	—	79	1	no	—
100	#2153	—	79	2	no	—
101	#2113	—	79	1	no	—
102	#1757	D₇₉	1	1	yes	characteristic
103	#1379	—	79	2	no	—
104	C₇₉	#12916	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
106	#504	—	158	4	no	—
107	C₂×M₃₂	—	79	2	no	—
108	C₂×M₃₂	—	79	1	no	—
109	C₂×M₃₂	—	158	2	no	—
110	C₂²×C₁₆	—	79	1	no	—
111	#413	Dic₇₉	1	1	yes	characteristic
112	#410	Dic₇₉	1	1	yes	characteristic
113	#405	C₂×D₇₉	1	2	yes	—
114	#364	—	158	4	no	—
115	C₂²⋊C₁₆	—	158	2	no	—
116	#344	Dic₇₉	1	2	yes	—
117	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
121	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
122	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
123	C₂²×C₈	D₃₁₆	1	2	yes	—
124	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
125	#117	C₂×Dic₇₉	1	2	yes	—
126	M₃₂	—	79	4	no	—
127	M₃₂	—	158	2	no	—
128	M₃₂	—	158	4	no	—
129	M₃₂	—	158	2	no	—
130	C₂×C₁₆	—	158	1	no	—
131	C₂×C₁₆	—	158	1	no	—
132	C₂×C₁₆	—	158	2	no	—
133	C₂×C₁₆	—	158	4	no	—
134	C₂×C₁₆	—	158	2	no	—
135	#101	C₇₉⋊C₈	1	2	yes	—
136	#100	C₇₉⋊C₈	1	2	yes	—
137	#99	—	2	2	no	—
138	#97	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
143	C₂×D₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
144	C₂×D₄	#11929608	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	4	no	—
150	M₁₆	C₂×C₇₉⋊C₈	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊Dic₇₉	1	2	yes	—
155	C₂×C₈	D₄×D₇₉	1	2	yes	—
156	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
157	C₂×C₈	C₂×Dic₁₅₈	1	1	yes	characteristic
158	C₂×C₈	#11929623	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	316	1	no	—
163	C₁₆	—	316	1	no	—
164	C₁₆	—	158	4	no	—
165	C₂³	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
166	C₂³	C₂×#11929608	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
175	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
176	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#14579168	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	#14579131	1	1	yes	characteristic
181	C₈	#14579138	1	2	yes	—
182	C₈	#14579145	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
187	C₂²	#19098635	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	8	2	no	—
190	C₄	#19098438	1	1	yes	characteristic
191	C₄	#19098590	1	1	yes	characteristic
192	C₄	D₄×C₇₉⋊C₈	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#23763580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23776503

C₂

yes

maximal, characteristic

#23765166

C₂

yes

maximal, complemented

#23765014

C₂

yes

maximal, characteristic

#23761489

C₂

yes

maximal, complemented

C₇₉×#1757

C₂

yes

maximal, characteristic

#19098980

—

C₂×#14579101

—

C₂×#14579101

C₂²

yes

characteristic

C₂×#14579101

C₂²

yes

—

C₂²×C₇₉⋊C₁₆

C₂²

yes

characteristic

#19098384

—

C₂²⋊(C₇₉⋊C₁₆)

—

C₂×#131×C₇₉

C₂²

yes

characteristic

#413×C₇₉

C₄

yes

characteristic

#410×C₇₉

C₄

yes

characteristic

#405×C₇₉

C₂²

yes

—

#344×C₇₉

C₄

yes

—

—

—

—

D₄

yes

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×#41×C₇₉

C₂×C₄

yes

characteristic

#131×C₇₉

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₉

Q₈

yes

—

C₂²×C₈×C₇₉

D₄

yes

—

C₂²×C₈×C₇₉

C₂³

yes

characteristic

#117×C₇₉

C₂×C₄

yes

—

#101×C₇₉

C₈

yes

—

#100×C₇₉

C₈

yes

—

#99×C₇₉

—

#97×C₇₉

—

C₄×C₈×C₇₉

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₂×C₈

yes

characteristic

C₂×D₄×C₇₉

C₂×C₈

yes

characteristic

C₂×D₄×C₇₉

M₁₆

yes

—

C₂²×C₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

M₁₆×C₇₉

C₂×C₈

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₄⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

—

C₂×C₈×C₇₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₂³×C₇₉

C₄⋊C₈

yes

—

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₄⋊C₈

yes

—

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#120

yes

characteristic

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#123

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₉

#435

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

C₈×D₄

yes

—

#12916

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1850

yes

characteristic

#3407

—

100

#2153

—

101

#2113

—

102

#1757

D₇₉

yes

characteristic

103

#1379

—

104

C₇₉

#12916

yes

characteristic, complemented, Sylow

105

C₂×#131

C₂×D₇₉

yes

characteristic

106

#504

—

158

—

107

C₂×M₃₂

—

108

C₂×M₃₂

—

109

C₂×M₃₂

—

158

—

110

C₂²×C₁₆

—

111

#413

Dic₇₉

yes

characteristic

112

#410

Dic₇₉

yes

characteristic

113

#405

C₂×D₇₉

yes

—

114

#364

—

158

—

115

C₂²⋊C₁₆

—

158

—

116

#344

Dic₇₉

yes

—

117

C₂×#41

C₂×Dic₇₉

yes

characteristic

118

#131

—

119

#131

—

120

C₂×M₁₆

Dic₁₅₈

yes

—

121

C₂×M₁₆

C₂×Dic₇₉

yes

—

122

C₂×M₁₆

C₂×Dic₇₉

yes

characteristic

123

C₂²×C₈

D₃₁₆

yes

—

124

C₂²×C₈

C₂²×D₇₉

yes

characteristic

125

#117

C₂×Dic₇₉

yes

—

126

M₃₂

—

127

M₃₂

—

158

—

128

M₃₂

—

158

—

129

M₃₂

—

158

—

130

C₂×C₁₆

—

158

—

131

C₂×C₁₆

—

158

—

132

C₂×C₁₆

—

158

—

133

C₂×C₁₆

—

158

—

134

C₂×C₁₆

—

158

—

135

#101

C₇₉⋊C₈

yes

—

136

#100

C₇₉⋊C₈

yes

—

137

#99

—

138

#97

—

139

C₄×C₈

—

140

#41

—

141

#41

—

142

C₂×Q₈

C₂×C₇₉⋊C₈

yes

characteristic

143

C₂×D₄

C₂×C₇₉⋊C₈

yes

characteristic

144

C₂×D₄

#11929608

yes

—

145

C₂²×C₄

C₄⋊Dic₇₉

yes

—

146

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

C₂×C₇₉⋊C₈

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₄⋊Dic₇₉

yes

—

155

C₂×C₈

D₄×D₇₉

yes

—

156

C₂×C₈

C₂×D₃₁₆

yes

characteristic

157

C₂×C₈

C₂×Dic₁₅₈

yes

characteristic

158

C₂×C₈

#11929623

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

316

—

163

C₁₆

—

316

—

164

C₁₆

—

158

—

165

C₂³

C₄⋊(C₇₉⋊C₈)

yes

—

166

C₂³

C₂×#11929608

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₄⋊(C₇₉⋊C₈)

yes

—

174

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

175

C₂×C₄

D₄×Dic₇₉

yes

—

176

C₂×C₄

C₂²×C₇₉⋊C₈

yes

characteristic

177

C₂×C₄

#14579168

yes

—

178

C₈

—

179

C₈

—

180

C₈

#14579131

yes

characteristic

181

C₈

#14579138

yes

—

182

C₈

#14579145

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×C₄⋊(C₇₉⋊C₈)

yes

characteristic

187

C₂²

#19098635

yes

—

188

C₄

—

189

C₄

—

190

C₄

#19098438

yes

characteristic

191

C₄

#19098590

yes

characteristic

192

C₄

D₄×C₇₉⋊C₈

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#23763580

yes

characteristic

	subgroup	G/H
center	C₈	#14579131
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1757	C₂
socle	C₂×C₇₉	#1850
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1757	D₇₉
hypocenter	C₇₉	#12916
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

subgroup

G/H

center

C₈

#14579131

commutator

C₂×C₄×C₇₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1757

C₂

socle

C₂×C₇₉

#1850

prime radical

C₂×#41×C₇₉

C₂×C₄

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

N	H ≅ G/N	type
2. #23765166	C₂	semidirect
4. #23761489	C₂	semidirect
104. C₇₉	#12916	semidirect

H ≅ G/N

type

2. #23765166

C₂

semidirect

4. #23761489

C₂

semidirect

104. C₇₉

#12916

semidirect

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

subgroup	index
#23776503	2
#23765166	2
#23765014	2
#23761489	2
C₇₉×#1757	2
#12916	79

subgroup

index

C₇₉×#1757

Subgroups — #34204751

Conjugacy classes of subgroups

Complements

Maximal subgroups