Group #34181286 — subgroups

Subgroups — #34181286

Order 20224.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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	#23772111	C₂	1	1	yes	maximal, characteristic
2	#23762155	C₂	1	2	yes	maximal, complemented
3	#23762133	C₂	1	2	yes	maximal
4	C₇₉⋊#1309	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14579169	C₄	1	2	yes	—
7	#19099113	C₂²	1	1	yes	characteristic
8	#19098640	—	2	2	no	—
9	#19098628	—	2	2	no	—
10	#19098587	C₂²	1	2	yes	—
11	#19098531	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
13	#19098313	—	2	2	no	—
14	#19098313	—	2	2	no	—
15	#19098311	—	2	2	no	—
16	#19098306	—	2	2	no	—
17	#423×C₇₉	C₂²	1	2	yes	—
18	#423×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#11929624	C₂×C₄	1	2	yes	—
20	#14579169	—	4	4	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929607	—	2	2	no	—
23	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₇₉	—	2	2	no	—
26	#14579161	D₄	1	1	yes	characteristic
27	#14579158	D₄	1	1	yes	characteristic
28	#14579149	Q₈	1	2	yes	—
29	#14579135	—	2	2	no	—
30	C₄×D₃₁₆	—	2	2	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
33	(C₂×D₇₉)⋊C₈	—	2	4	no	—
34	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
35	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
36	C₂×M₁₆×C₇₉	—	2	2	no	—
37	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₉	—	2	2	no	—
40	C₄⋊C₈×C₇₉	—	2	2	no	—
41	C₄⋊C₈×C₇₉	—	2	2	no	—
42	#11929624	—	4	2	no	—
43	#11929624	—	4	2	no	—
44	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
45	C₂×D₃₁₆	—	2	2	no	—
46	C₂×D₃₁₆	C₄⋊C₄	1	2	yes	—
47	C₂×C₂²⋊D₇₉	—	2	2	no	—
48	C₂×C₄×D₇₉	—	2	2	no	—
49	#11929608	—	8	2	no	—
50	#11929607	—	4	4	no	—
51	C₂×C₇₉⋊C₈	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	4	1	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
55	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
57	C₈×D₇₉	—	4	4	no	—
58	C₄⋊Dic₇₉	—	2	2	no	—
59	(C₂×D₇₉)⋊C₄	—	2	4	no	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	#41	1	2	yes	—
62	M₁₆×C₇₉	—	4	2	no	—
63	M₁₆×C₇₉	—	8	2	no	—
64	C₂×C₈×C₇₉	—	2	2	no	—
65	C₂×C₈×C₇₉	—	4	2	no	—
66	C₄²×C₇₉	C₂×D₄	1	2	yes	—
67	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
69	C₂³×C₇₉	#117	1	1	yes	characteristic
70	C₂²×D₇₉	—	2	2	no	—
71	C₂×C₄×C₇₉	—	2	1	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₉	#121	1	2	yes	—
78	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
79	C₂²⋊D₇₉	—	4	4	no	—
80	C₂×Dic₇₉	—	2	2	no	—
81	D₃₁₆	—	4	2	no	—
82	D₃₁₆	—	4	2	no	—
83	D₃₁₆	—	4	2	no	—
84	C₄×D₇₉	—	4	2	no	—
85	C₄×D₇₉	—	4	2	no	—
86	Dic₁₅₈	—	4	2	no	—
87	C₈×C₇₉	—	4	2	no	—
88	C₈×C₇₉	—	8	2	no	—
89	C₇₉⋊C₈	—	2	1	no	—
90	C₇₉⋊C₈	—	2	2	no	—
91	C₇₉⋊C₈	—	8	2	no	—
92	C₇₉⋊C₈	#100	1	2	yes	—
93	C₂²×C₇₉	—	2	1	no	—
94	C₂²×C₇₉	—	2	2	no	—
95	C₂²×C₇₉	#380	1	1	yes	characteristic
96	C₂×D₇₉	—	4	2	no	—
97	C₂×D₇₉	—	4	4	no	—
98	C₄×C₇₉	—	2	1	no	—
99	C₄×C₇₉	—	2	2	no	—
100	C₄×C₇₉	C₂×#100	1	1	yes	characteristic
101	C₄×C₇₉	#413	1	1	yes	characteristic
102	C₄×C₇₉	#471	1	2	yes	—
103	C₄×C₇₉	#471	1	2	yes	—
104	Dic₇₉	—	4	2	no	—
105	#8636	—	79	1	no	maximal, Sylow
106	C₂×C₇₉	—	4	1	no	—
107	C₂×C₇₉	#1800	1	1	yes	characteristic
108	C₂×C₇₉	#1815	1	2	yes	—
109	D₇₉	—	8	2	no	—
110	#2931	—	79	1	no	—
111	#1566	—	79	2	no	—
112	#1560	—	79	2	no	—
113	#1309	—	79	1	no	—
114	#1309	D₇₉	1	1	yes	characteristic, complemented
115	C₇₉	#8636	1	1	yes	characteristic, complemented, Sylow
116	C₂×#131	—	79	2	no	—
117	#532	—	79	1	no	—
118	#436	—	158	2	no	—
119	C₈×D₄	—	158	2	no	—
120	#423	—	79	2	no	—
121	#423	C₂×D₇₉	1	2	yes	—
122	#423	C₂×D₇₉	1	1	yes	characteristic
123	#408	—	79	2	no	—
124	C₂×C₄×C₈	—	79	1	no	—
125	#334	—	158	2	no	—
126	#334	—	158	2	no	—
127	#333	—	158	2	no	—
128	#331	—	158	2	no	—
129	C₂×#41	—	79	2	no	—
130	#131	—	316	4	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	158	2	no	—
133	C₂×M₁₆	—	158	2	no	—
134	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
135	C₂²×C₈	—	79	1	no	—
136	C₂²×C₈	—	79	1	no	—
137	C₂²×C₈	—	158	2	no	—
138	C₄⋊Q₈	—	79	1	no	—
139	#127	—	79	1	no	—
140	#124	—	79	2	no	—
141	#121	—	158	2	no	—
142	C₄×D₄	—	158	2	no	—
143	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	158	2	no	—
148	C₄⋊C₈	—	158	2	no	—
149	C₂²⋊C₈	—	158	4	no	—
150	C₄×C₈	—	79	2	no	—
151	C₄×C₈	—	79	2	no	—
152	#41	—	316	2	no	—
153	#41	—	316	2	no	—
154	C₂×Q₈	—	79	2	no	—
155	C₂×D₄	—	79	2	no	—
156	C₂×D₄	—	158	2	no	—
157	C₂×D₄	—	158	2	no	—
158	C₂²×C₄	—	158	2	no	—
159	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
160	C₂²×C₄	#11929624	1	2	yes	—
161	M₁₆	—	4	2	no	—
162	M₁₆	—	8	2	no	—
163	M₁₆	—	316	4	no	—
164	M₁₆	—	632	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	316	4	no	—
168	C₂×C₈	—	316	2	no	—
169	C₂×C₈	—	316	1	no	—
170	C₂×C₈	—	79	1	no	—
171	C₂×C₈	—	79	2	no	—
172	C₂×C₈	—	79	1	no	—
173	C₂×C₈	—	158	2	no	—
174	C₄⋊C₄	—	158	2	no	—
175	C₂²⋊C₄	—	158	4	no	—
176	C₄²	D₄×D₇₉	1	2	yes	—
177	C₄²	D₄×D₇₉	1	1	yes	characteristic
178	C₄²	Q₈×D₇₉	1	1	yes	characteristic
179	C₂³	—	158	2	no	—
180	C₂³	#14579118	1	1	yes	characteristic
181	Q₈	—	316	2	no	—
182	D₄	—	316	2	no	—
183	D₄	—	316	4	no	—
184	D₄	—	316	2	no	—
185	D₄	—	316	2	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	316	2	no	—
191	C₂×C₄	—	316	2	no	—
192	C₂×C₄	—	158	2	no	—
193	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
194	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
195	C₂×C₄	#14579134	1	2	yes	—
196	C₂×C₄	#14579139	1	2	yes	—
197	C₈	—	4	2	no	—
198	C₈	—	8	2	no	—
199	C₈	—	79	2	no	—
200	C₈	—	632	2	no	—
201	C₈	—	158	1	no	—
202	C₈	—	158	2	no	—
203	C₂²	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	316	4	no	—
206	C₂²	—	316	2	no	—
207	C₂²	#19098422	1	1	yes	characteristic
208	C₄	—	2	2	no	—
209	C₄	—	2	1	no	—
210	C₄	—	316	2	no	—
211	C₄	#100×D₇₉	1	1	yes	characteristic
212	C₄	#19098546	1	1	yes	characteristic
213	C₄	#19098825	1	2	yes	—
214	C₄	#19098829	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	632	2	no	—
217	C₂	#23763391	1	1	yes	characteristic
218	C₂	#23763458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23772111

C₂

yes

maximal, characteristic

#23762155

C₂

yes

maximal, complemented

#23762133

C₂

yes

maximal

C₇₉⋊#1309

C₂

yes

maximal, characteristic, complemented

C₇₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#14579169

C₄

yes

—

#19099113

C₂²

yes

—

—

C₂²

yes

—

#19098531

C₄

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

—

—

—

#423×C₇₉

C₂²

yes

—

#423×C₇₉

C₂²

yes

characteristic

C₂×#11929624

C₂×C₄

yes

—

#14579169

—

C₂×#11929608

—

C₂×#11929607

—

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₇₉⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈×D₇₉

—

#14579161

D₄

yes

characteristic

#14579158

D₄

yes

characteristic

#14579149

Q₈

yes

—

#14579135

—

C₄×D₃₁₆

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

(C₂×D₇₉)⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×M₁₆×C₇₉

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#11929624

—

#11929624

—

C₂×Dic₁₅₈

C₄⋊C₄

yes

—

C₂×D₃₁₆

—

C₂×D₃₁₆

C₄⋊C₄

yes

—

C₂×C₂²⋊D₇₉

—

C₂×C₄×D₇₉

—

#11929608

—

#11929607

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

—

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₇₉⋊C₈

C₂²⋊C₄

yes

characteristic

C₈×D₇₉

—

C₄⋊Dic₇₉

—

(C₂×D₇₉)⋊C₄

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₄²×C₇₉

C₂×D₄

yes

—

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×Q₈

yes

characteristic

C₂³×C₇₉

#117

yes

characteristic

C₂²×D₇₉

—

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

—

C₂²⋊D₇₉

—

C₂×Dic₇₉

—

D₃₁₆

—

D₃₁₆

—

D₃₁₆

—

C₄×D₇₉

—

C₄×D₇₉

—

Dic₁₅₈

—

C₈×C₇₉

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₇₉⋊C₈

#100

yes

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#380

yes

characteristic

C₂×D₇₉

—

C₂×D₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

100

C₄×C₇₉

C₂×#100

yes

characteristic

101

C₄×C₇₉

#413

yes

characteristic

102

C₄×C₇₉

#471

yes

—

103

C₄×C₇₉

#471

yes

—

104

Dic₇₉

—

105

#8636

—

maximal, Sylow

106

C₂×C₇₉

—

107

C₂×C₇₉

#1800

yes

characteristic

108

C₂×C₇₉

#1815

yes

—

109

D₇₉

—

110

#2931

—

111

#1566

—

112

#1560

—

113

#1309

—

114

#1309

D₇₉

yes

characteristic, complemented

115

C₇₉

#8636

yes

characteristic, complemented, Sylow

116

C₂×#131

—

117

#532

—

118

#436

—

158

—

119

C₈×D₄

—

158

—

120

#423

—

121

#423

C₂×D₇₉

yes

—

122

#423

C₂×D₇₉

yes

characteristic

123

#408

—

124

C₂×C₄×C₈

—

125

#334

—

158

—

126

#334

—

158

—

127

#333

—

158

—

128

#331

—

158

—

129

C₂×#41

—

130

#131

—

316

—

131

C₂×M₁₆

—

132

C₂×M₁₆

—

158

—

133

C₂×M₁₆

—

158

—

134

C₂×M₁₆

C₄×D₇₉

yes

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

158

—

138

C₄⋊Q₈

—

139

#127

—

140

#124

—

141

#121

—

158

—

142

C₄×D₄

—

158

—

143

C₂×C₄²

C₂²×D₇₉

yes

characteristic

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

158

—

148

C₄⋊C₈

—

158

—

149

C₂²⋊C₈

—

158

—

150

C₄×C₈

—

151

C₄×C₈

—

152

#41

—

316

—

153

#41

—

316

—

154

C₂×Q₈

—

155

C₂×D₄

—

156

C₂×D₄

—

158

—

157

C₂×D₄

—

158

—

158

C₂²×C₄

—

158

—

159

C₂²×C₄

C₂×C₄×D₇₉

yes

characteristic

160

C₂²×C₄

#11929624

yes

—

161

M₁₆

—

162

M₁₆

—

163

M₁₆

—

316

—

164

M₁₆

—

632

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

316

—

168

C₂×C₈

—

316

—

169

C₂×C₈

—

316

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

158

—

174

C₄⋊C₄

—

158

—

175

C₂²⋊C₄

—

158

—

176

C₄²

D₄×D₇₉

yes

—

177

C₄²

D₄×D₇₉

yes

characteristic

178

C₄²

Q₈×D₇₉

yes

characteristic

179

C₂³

—

158

—

180

C₂³

#14579118

yes

characteristic

181

Q₈

—

316

—

182

D₄

—

316

—

183

D₄

—

316

—

184

D₄

—

316

—

185

D₄

—

316

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

316

—

191

C₂×C₄

—

316

—

192

C₂×C₄

—

158

—

193

C₂×C₄

C₂²⋊C₄×D₇₉

yes

characteristic

194

C₂×C₄

C₄⋊C₄×D₇₉

yes

characteristic

195

C₂×C₄

#14579134

yes

—

196

C₂×C₄

#14579139

yes

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

632

—

201

C₈

—

158

—

202

C₈

—

158

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

316

—

206

C₂²

—

316

—

207

C₂²

#19098422

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

—

316

—

211

C₄

#100×D₇₉

yes

characteristic

212

C₄

#19098546

yes

characteristic

213

C₄

#19098825

yes

—

214

C₄

#19098829

yes

—

215

C₂

—

216

C₂

—

632

—

217

C₂

#23763391

yes

characteristic

218

C₂

#23763458

yes

—

	subgroup	G/H
center	C₂²	#19098422
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1309	C₂
socle	C₂²×C₇₉	#380
prime radical	#19099113	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1309	D₇₉
hypocenter	C₇₉	#8636
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#19098422

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#1309

C₂

C₂²×C₇₉

C₂²

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
2. #23762155	C₂	semidirect
4. C₇₉⋊#1309	C₂	semidirect
5. C₇₉×#1309	C₂	semidirect
114. #1309	D₇₉	semidirect
115. C₇₉	#8636	semidirect

H ≅ G/N

type

2. #23762155

C₂

semidirect

4. C₇₉⋊#1309

C₂

semidirect

5. C₇₉×#1309

C₂

semidirect

114. #1309

D₇₉

semidirect

115. C₇₉

#8636

semidirect

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

subgroup	index
#23772111	2
#23762155	2
#23762133	2
C₇₉⋊#1309	2
C₇₉×#1309	2
#8636	79

subgroup

index

#23772111

#23762155

#23762133

C₇₉⋊#1309

C₇₉×#1309

#8636

Subgroups — #34181286

Conjugacy classes of subgroups

Complements

Maximal subgroups