Group #34181295 — subgroups

Subgroups — #34181295

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	#23762159	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	#19098316	—	2	2	no	—
7	#19098316	—	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	2	no	—
68	D₄×C₇₉	—	4	2	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	2	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	—	4	2	no	—
75	C₂×C₄×C₇₉	—	4	2	no	—
76	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₉	#121	1	2	yes	—
79	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
80	C₈×C₇₉	—	2	1	no	—
81	C₈×C₇₉	—	2	2	no	—
82	C₈×C₇₉	—	4	2	no	—
83	C₈×C₇₉	#100	1	2	yes	—
84	C₇₉⋊C₈	—	8	4	no	—
85	C₂²×C₇₉	—	2	1	no	—
86	C₂²×C₇₉	—	2	2	no	—
87	C₂²×C₇₉	—	4	2	no	—
88	C₂²×C₇₉	—	4	4	no	—
89	C₂²×C₇₉	#380	1	1	yes	characteristic
90	C₄×C₇₉	—	2	1	no	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	2	no	—
93	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
94	C₄×C₇₉	#413	1	1	yes	characteristic
95	C₄×C₇₉	#472	1	2	yes	—
96	C₄×C₇₉	#473	1	2	yes	—
97	#8638	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	4	1	no	—
99	C₂×C₇₉	—	8	2	no	—
100	C₂×C₇₉	#1800	1	1	yes	characteristic
101	C₂×C₇₉	#1816	1	2	yes	—
102	#2931	D₇₉	1	1	yes	characteristic
103	#1567	—	79	2	no	—
104	#1560	—	79	2	no	—
105	#1309	—	79	2	no	—
106	C₇₉	#8638	1	1	yes	characteristic, complemented, Sylow
107	C₂×#131	Dic₇₉	1	2	yes	—
108	#532	C₂×D₇₉	1	1	yes	characteristic
109	#436	—	2	2	no	—
110	C₈×D₄	—	2	2	no	—
111	#423	—	79	4	no	—
112	#423	C₂×D₇₉	1	1	yes	characteristic
113	#408	Dic₇₉	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
115	#335	—	158	2	no	—
116	#335	—	158	2	no	—
117	#333	—	158	2	no	—
118	#331	—	158	2	no	—
119	C₂×#41	C₂×Dic₇₉	1	2	yes	—
120	#131	—	4	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	158	4	no	—
123	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
127	C₄⋊Q₈	D₃₁₆	1	1	yes	characteristic
128	#127	D₃₁₆	1	1	yes	characteristic
129	#124	Dic₁₅₈	1	2	yes	—
130	#121	—	2	2	no	—
131	C₄×D₄	—	2	2	no	—
132	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	158	4	no	—
135	C₄⋊C₈	—	158	4	no	—
136	C₂²⋊C₈	—	2	4	no	—
137	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
138	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₄⋊Dic₇₉	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₄⋊Dic₇₉	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
147	C₂²×C₄	#11929624	1	2	yes	—
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	632	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	316	4	no	—
156	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
162	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
163	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
164	C₂³	—	2	2	no	—
165	C₂³	#14579120	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	2	no	—
170	D₄	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
180	C₂×C₄	#14579135	1	2	yes	—
181	C₂×C₄	#14579140	1	2	yes	—
182	C₈	—	2	1	no	—
183	C₈	—	2	2	no	—
184	C₈	—	4	2	no	—
185	C₈	—	632	4	no	—
186	C₈	#14579078	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	4	4	no	—
190	C₂²	—	4	2	no	—
191	C₂²	#19098423	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	1	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×#14579078	1	1	yes	characteristic
196	C₄	#19098547	1	1	yes	characteristic
197	C₄	#19098833	1	2	yes	—
198	C₄	#19098838	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	#23763392	1	1	yes	characteristic
202	C₂	#23763464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23762159

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

—

D₄×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×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

—

C₄×C₇₉

#473

yes

—

#8638

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

100

C₂×C₇₉

#1800

yes

characteristic

101

C₂×C₇₉

#1816

yes

—

102

#2931

D₇₉

yes

characteristic

103

#1567

—

104

#1560

—

105

#1309

—

106

C₇₉

#8638

yes

characteristic, complemented, Sylow

107

C₂×#131

Dic₇₉

yes

—

108

#532

C₂×D₇₉

yes

characteristic

109

#436

—

110

C₈×D₄

—

111

#423

—

112

#423

C₂×D₇₉

yes

characteristic

113

#408

Dic₇₉

yes

—

114

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

115

#335

—

158

—

116

#335

—

158

—

117

#333

—

158

—

118

#331

—

158

—

119

C₂×#41

C₂×Dic₇₉

yes

—

120

#131

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

158

—

123

C₂×M₁₆

C₂×Dic₇₉

yes

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

126

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

127

C₄⋊Q₈

D₃₁₆

yes

characteristic

128

#127

D₃₁₆

yes

characteristic

129

#124

Dic₁₅₈

yes

—

130

#121

—

131

C₄×D₄

—

132

C₂×C₄²

C₂²×D₇₉

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

158

—

135

C₄⋊C₈

—

158

—

136

C₂²⋊C₈

—

137

C₄×C₈

C₂²⋊D₇₉

yes

—

138

C₄×C₈

C₂²⋊D₇₉

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

C₄⋊Dic₇₉

yes

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

C₄⋊Dic₇₉

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

147

C₂²×C₄

#11929624

yes

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

632

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

316

—

156

C₂×C₈

C₂²⋊Dic₇₉

yes

—

157

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

158

C₂×C₈

C₂²⋊Dic₇₉

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂×C₂²⋊D₇₉

yes

—

162

C₄²

C₂×D₃₁₆

yes

characteristic

163

C₄²

C₂×Dic₁₅₈

yes

characteristic

164

C₂³

—

165

C₂³

#14579120

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

179

C₂×C₄

C₂×C₄⋊Dic₇₉

yes

characteristic

180

C₂×C₄

#14579135

yes

—

181

C₂×C₄

#14579140

yes

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

632

—

186

C₈

#14579078

yes

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#19098423

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₂×#14579078

yes

characteristic

196

C₄

#19098547

yes

characteristic

197

C₄

#19098833

yes

—

198

C₄

#19098838

yes

—

199

C₂

—

200

C₂

—

201

C₂

#23763392

yes

characteristic

202

C₂

#23763464

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	#8638
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. #23762159	C₂	semidirect
3. C₇₉⋊#1309	C₂	semidirect
106. C₇₉	#8638	semidirect

H ≅ G/N

type

1. #23762159

C₂

semidirect

3. C₇₉⋊#1309

C₂

semidirect

106. C₇₉

#8638

semidirect

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

subgroup	index
#23762159	2
#23762135	2
C₇₉⋊#1309	2
C₇₉×#2931	2
#8638	79

subgroup

index

#23762159

#23762135

C₇₉⋊#1309

C₇₉×#2931

#8638

Subgroups — #34181295

Conjugacy classes of subgroups

Complements

Maximal subgroups