Group #34181287 — subgroups

Subgroups — #34181287

Order 20224.

404 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 127 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	#23762156	C₂	1	2	yes	maximal, complemented
2	#23762135	C₂	1	2	yes	maximal
3	C₇₉⋊#1309	C₂	1	2	yes	maximal, complemented
4	C₇₉×#2931	C₂	1	1	yes	maximal, characteristic
5	#19098587	C₂²	1	4	yes	—
6	#19098314	—	2	2	no	—
7	#19098314	—	2	2	no	—
8	#19098312	—	2	2	no	—
9	#19098307	—	2	2	no	—
10	C₂×#131×C₇₉	C₄	1	2	yes	—
11	#532×C₇₉	C₂²	1	1	yes	characteristic
12	#436×C₇₉	—	2	2	no	—
13	C₈×D₄×C₇₉	—	2	2	no	—
14	#423×C₇₉	C₂²	1	1	yes	characteristic
15	#408×C₇₉	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#11929608	—	2	4	no	—
18	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
20	C₂×#41×C₇₉	C₂×C₄	1	2	yes	—
21	#131×C₇₉	—	4	4	no	—
22	C₂×M₁₆×C₇₉	—	2	2	no	—
23	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₉	—	2	2	no	—
25	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
28	#127×C₇₉	D₄	1	1	yes	characteristic
29	#124×C₇₉	Q₈	1	2	yes	—
30	#121×C₇₉	—	2	2	no	—
31	C₄×D₄×C₇₉	—	2	2	no	—
32	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₉	—	2	2	no	—
34	C₂²⋊C₈×C₇₉	—	2	4	no	—
35	C₄×C₈×C₇₉	D₄	1	2	yes	—
36	C₄×C₈×C₇₉	D₄	1	2	yes	—
37	#11929608	—	8	4	no	—
38	C₂×C₇₉⋊C₈	—	4	4	no	—
39	#41×C₇₉	—	4	2	no	—
40	#41×C₇₉	—	4	2	no	—
41	C₂×Q₈×C₇₉	C₄⋊C₄	1	2	yes	—
42	C₂×D₄×C₇₉	—	2	2	no	—
43	C₂×D₄×C₇₉	—	2	2	no	—
44	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇₉	—	2	2	no	—
46	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₉	#41	1	2	yes	—
48	M₁₆×C₇₉	—	4	2	no	—
49	M₁₆×C₇₉	—	4	4	no	—
50	C₂×C₈×C₇₉	—	2	2	no	—
51	C₂×C₈×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	4	1	no	—
53	C₂×C₈×C₇₉	—	4	4	no	—
54	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₇₉	—	2	2	no	—
58	C₂²⋊C₄×C₇₉	—	2	4	no	—
59	C₄²×C₇₉	C₂×D₄	1	2	yes	—
60	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
62	C₂³×C₇₉	—	2	2	no	—
63	C₂³×C₇₉	#117	1	1	yes	characteristic
64	Q₈×C₇₉	—	4	2	no	—
65	D₄×C₇₉	—	4	2	no	—
66	D₄×C₇₉	—	4	4	no	—
67	D₄×C₇₉	—	4	4	no	—
68	C₂×C₄×C₇₉	—	2	1	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	4	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	4	2	no	—
73	C₂×C₄×C₇₉	—	4	2	no	—
74	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	#121	1	2	yes	—
77	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
78	C₈×C₇₉	—	2	1	no	—
79	C₈×C₇₉	—	2	2	no	—
80	C₈×C₇₉	—	4	2	no	—
81	C₈×C₇₉	#100	1	2	yes	—
82	C₇₉⋊C₈	—	8	4	no	—
83	C₂²×C₇₉	—	2	1	no	—
84	C₂²×C₇₉	—	2	2	no	—
85	C₂²×C₇₉	—	4	2	no	—
86	C₂²×C₇₉	—	4	4	no	—
87	C₂²×C₇₉	#380	1	1	yes	characteristic
88	C₄×C₇₉	—	2	1	no	—
89	C₄×C₇₉	—	2	2	no	—
90	C₄×C₇₉	—	4	2	no	—
91	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
92	C₄×C₇₉	#413	1	1	yes	characteristic
93	C₄×C₇₉	#471	1	4	yes	—
94	#8636	—	79	1	no	maximal, Sylow
95	C₂×C₇₉	—	4	1	no	—
96	C₂×C₇₉	—	8	2	no	—
97	C₂×C₇₉	#1800	1	1	yes	characteristic
98	C₂×C₇₉	#1815	1	2	yes	—
99	#2931	D₇₉	1	1	yes	characteristic
100	#1566	—	79	2	no	—
101	#1560	—	79	2	no	—
102	#1309	—	79	2	no	—
103	C₇₉	#8636	1	1	yes	characteristic, complemented, Sylow
104	C₂×#131	Dic₇₉	1	2	yes	—
105	#532	C₂×D₇₉	1	1	yes	characteristic
106	#436	—	2	2	no	—
107	C₈×D₄	—	2	2	no	—
108	#423	—	79	4	no	—
109	#423	C₂×D₇₉	1	1	yes	characteristic
110	#408	Dic₇₉	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
112	#334	—	158	2	no	—
113	#334	—	158	2	no	—
114	#333	—	158	2	no	—
115	#331	—	158	2	no	—
116	C₂×#41	C₂×Dic₇₉	1	2	yes	—
117	#131	—	4	4	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	158	4	no	—
120	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
124	C₄⋊Q₈	D₃₁₆	1	1	yes	characteristic
125	#127	D₃₁₆	1	1	yes	characteristic
126	#124	Dic₁₅₈	1	2	yes	—
127	#121	—	2	2	no	—
128	C₄×D₄	—	2	2	no	—
129	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	158	4	no	—
132	C₄⋊C₈	—	158	4	no	—
133	C₂²⋊C₈	—	2	4	no	—
134	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
135	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
136	#41	—	4	2	no	—
137	#41	—	4	2	no	—
138	C₂×Q₈	C₄⋊Dic₇₉	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
144	C₂²×C₄	#11929624	1	2	yes	—
145	M₁₆	—	4	4	no	—
146	M₁₆	—	4	2	no	—
147	M₁₆	—	632	4	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	316	4	no	—
153	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
154	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
155	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
156	C₄⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	4	no	—
158	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
159	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
160	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
161	C₂³	—	2	2	no	—
162	C₂³	#14579120	1	1	yes	characteristic
163	Q₈	—	4	2	no	—
164	D₄	—	4	2	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
175	C₂×C₄	#14579135	1	2	yes	—
176	C₂×C₄	#14579140	1	2	yes	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	4	2	no	—
180	C₈	—	632	4	no	—
181	C₈	#14579078	1	2	yes	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	4	4	no	—
185	C₂²	—	4	2	no	—
186	C₂²	#19098423	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	1	no	—
189	C₄	—	4	2	no	—
190	C₄	C₂×#14579078	1	1	yes	characteristic
191	C₄	#19098547	1	1	yes	characteristic
192	C₄	#19098826	1	4	yes	—
193	C₂	—	4	1	no	—
194	C₂	—	8	2	no	—
195	C₂	#23763392	1	1	yes	characteristic
196	C₂	#23763459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762156

C₂

yes

maximal, complemented

#23762135

C₂

yes

maximal

C₇₉⋊#1309

C₂

yes

maximal, complemented

C₇₉×#2931

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

—

—

—

—

C₂×#131×C₇₉

C₄

yes

—

#532×C₇₉

C₂²

yes

characteristic

#436×C₇₉

—

C₈×D₄×C₇₉

—

#423×C₇₉

C₂²

yes

characteristic

#408×C₇₉

C₄

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₂×#41×C₇₉

C₂×C₄

yes

—

#131×C₇₉

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂²×C₈×C₇₉

—

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₇₉

D₄

yes

characteristic

#127×C₇₉

D₄

yes

characteristic

#124×C₇₉

Q₈

yes

—

#121×C₇₉

—

C₄×D₄×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₂²⋊C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

#41×C₇₉

—

C₂×Q₈×C₇₉

C₄⋊C₄

yes

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

C₄⋊C₄

yes

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

—

C₂³×C₇₉

#117

yes

characteristic

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

#100

yes

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#380

yes

characteristic

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#8636

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

C₂×C₇₉

#1800

yes

characteristic

C₂×C₇₉

#1815

yes

—

#2931

D₇₉

yes

characteristic

100

#1566

—

101

#1560

—

102

#1309

—

103

C₇₉

#8636

yes

characteristic, complemented, Sylow

104

C₂×#131

Dic₇₉

yes

—

105

#532

C₂×D₇₉

yes

characteristic

106

#436

—

107

C₈×D₄

—

108

#423

—

109

#423

C₂×D₇₉

yes

characteristic

110

#408

Dic₇₉

yes

—

111

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

112

#334

—

158

—

113

#334

—

158

—

114

#333

—

158

—

115

#331

—

158

—

116

C₂×#41

C₂×Dic₇₉

yes

—

117

#131

—

118

C₂×M₁₆

—

119

C₂×M₁₆

—

158

—

120

C₂×M₁₆

C₂×Dic₇₉

yes

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

123

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

124

C₄⋊Q₈

D₃₁₆

yes

characteristic

125

#127

D₃₁₆

yes

characteristic

126

#124

Dic₁₅₈

yes

—

127

#121

—

128

C₄×D₄

—

129

C₂×C₄²

C₂²×D₇₉

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

158

—

132

C₄⋊C₈

—

158

—

133

C₂²⋊C₈

—

134

C₄×C₈

C₂²⋊D₇₉

yes

—

135

C₄×C₈

C₂²⋊D₇₉

yes

—

136

#41

—

137

#41

—

138

C₂×Q₈

C₄⋊Dic₇₉

yes

—

139

C₂×D₄

—

140

C₂×D₄

—

141

C₂×D₄

C₄⋊Dic₇₉

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

144

C₂²×C₄

#11929624

yes

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

632

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

316

—

153

C₂×C₈

C₂²⋊Dic₇₉

yes

—

154

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

155

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

156

C₄⋊C₄

—

157

C₂²⋊C₄

—

158

C₄²

C₂×C₂²⋊D₇₉

yes

—

159

C₄²

C₂×D₃₁₆

yes

characteristic

160

C₄²

C₂×Dic₁₅₈

yes

characteristic

161

C₂³

—

162

C₂³

#14579120

yes

characteristic

163

Q₈

—

164

D₄

—

165

D₄

—

166

D₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

174

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

175

C₂×C₄

#14579135

yes

—

176

C₂×C₄

#14579140

yes

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

—

632

—

181

C₈

#14579078

yes

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#19098423

yes

characteristic

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×#14579078

yes

characteristic

191

C₄

#19098547

yes

characteristic

192

C₄

#19098826

yes

—

193

C₂

—

194

C₂

—

195

C₂

#23763392

yes

characteristic

196

C₂

#23763459

yes

—

	subgroup	G/H
center	C₂²	#19098423
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2931	C₂
socle	C₂²×C₇₉	#380
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2931	D₇₉
hypocenter	C₇₉	#8636
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#19098423

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#2931

C₂

socle

C₂²×C₇₉

#380

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #23762156	C₂	semidirect
3. C₇₉⋊#1309	C₂	semidirect
103. C₇₉	#8636	semidirect

H ≅ G/N

type

1. #23762156

C₂

semidirect

3. C₇₉⋊#1309

C₂

semidirect

103. C₇₉

#8636

semidirect

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

subgroup	index
#23762156	2
#23762135	2
C₇₉⋊#1309	2
C₇₉×#2931	2
#8636	79

subgroup

index

#23762156

#23762135

C₇₉⋊#1309

C₇₉×#2931

#8636

Subgroups — #34181287

Conjugacy classes of subgroups

Complements

Maximal subgroups