Group #34188217 — subgroups

Subgroups — #34188217

Order 20224.

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

normal subgroups only

This group has 428 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	#23763538	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761855	C₂	1	2	yes	maximal, complemented
3	#23761851	C₂	1	2	yes	maximal
4	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₉×#3131	C₂	1	1	yes	maximal, characteristic
6	#19098587	C₂²	1	2	yes	—
7	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
8	#19098328	—	2	2	no	—
9	Q₈⋊(C₇₉⋊C₈)	—	2	4	no	—
10	D₄⋊(C₇₉⋊C₈)	—	2	4	no	—
11	C₈×C₇₉⋊C₈	—	2	2	no	—
12	#533×C₇₉	C₂²	1	1	yes	characteristic
13	#532×C₇₉	C₂²	1	1	yes	characteristic
14	#488×C₇₉	—	2	2	no	—
15	#487×C₇₉	C₄	1	2	yes	—
16	#486×C₇₉	C₄	1	2	yes	—
17	#416×C₇₉	—	2	2	no	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#11929608	—	2	2	no	—
20	C₂²×C₇₉⋊C₈	—	2	2	no	—
21	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
23	C₄×C₇₉⋊C₈	—	2	2	no	—
24	C₄×C₇₉⋊C₈	D₄	1	4	yes	—
25	C₂×#41×C₇₉	—	2	1	no	—
26	C₂²×C₈×C₇₉	D₄	1	2	yes	—
27	C₄⋊Q₈×C₇₉	C₂×C₄	1	2	yes	—
28	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
29	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
30	#125×C₇₉	—	2	1	no	—
31	#124×C₇₉	—	2	1	no	—
32	C₂²⋊Q₈×C₇₉	—	2	2	no	—
33	#121×C₇₉	—	2	2	no	—
34	C₄×Q₈×C₇₉	—	2	2	no	—
35	C₄×D₄×C₇₉	—	2	2	no	—
36	#117×C₇₉	—	2	1	no	—
37	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
38	#103×C₇₉	—	2	4	no	—
39	#102×C₇₉	—	2	4	no	—
40	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₇₉	D₄	1	2	yes	—
42	#11929608	—	4	4	no	—
43	C₂×C₇₉⋊C₈	—	2	8	no	—
44	C₂×C₇₉⋊C₈	—	2	4	no	—
45	C₂×C₇₉⋊C₈	—	4	2	no	—
46	#41×C₇₉	—	4	4	no	—
47	C₂×Q₈×C₇₉	—	2	1	no	—
48	C₂×Q₈×C₇₉	—	2	2	no	—
49	C₂×D₄×C₇₉	—	2	1	no	—
50	C₂×D₄×C₇₉	—	2	2	no	—
51	C₂×D₄×C₇₉	—	4	1	no	—
52	C₂²×C₄×C₇₉	—	4	1	no	—
53	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₉	#41	1	2	yes	—
55	C₂×C₈×C₇₉	—	2	4	no	—
56	C₂×C₈×C₇₉	C₂²⋊C₄	1	4	yes	—
57	C₄⋊C₄×C₇₉	—	2	2	no	—
58	C₄⋊C₄×C₇₉	—	2	2	no	—
59	C₄⋊C₄×C₇₉	—	2	2	no	—
60	C₄⋊C₄×C₇₉	—	4	2	no	—
61	C₂²⋊C₄×C₇₉	—	4	1	no	—
62	C₂²⋊C₄×C₇₉	—	4	2	no	—
63	C₄²×C₇₉	—	4	1	no	—
64	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₇₉	C₂×D₄	1	2	yes	—
67	C₂³×C₇₉	—	4	1	no	—
68	C₂³×C₇₉	#124	1	1	yes	characteristic
69	Q₈×C₇₉	—	4	2	no	—
70	Q₈×C₇₉	—	4	4	no	—
71	D₄×C₇₉	—	4	2	no	—
72	D₄×C₇₉	—	4	4	no	—
73	D₄×C₇₉	—	8	2	no	—
74	C₂×C₄×C₇₉	—	2	4	no	—
75	C₂×C₄×C₇₉	—	4	2	no	—
76	C₂×C₄×C₇₉	—	4	1	no	—
77	C₂×C₄×C₇₉	—	8	2	no	—
78	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₇₉	#121	1	2	yes	—
81	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
82	C₂×C₄×C₇₉	#135	1	4	yes	—
83	C₈×C₇₉	—	2	4	no	—
84	C₇₉⋊C₈	—	2	8	no	—
85	C₇₉⋊C₈	—	8	2	no	—
86	C₂²×C₇₉	—	2	2	no	—
87	C₂²×C₇₉	—	8	1	no	—
88	C₂²×C₇₉	—	8	2	no	—
89	C₂²×C₇₉	#390	1	1	yes	characteristic
90	C₂²×C₇₉	#488	1	2	yes	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	8	1	no	—
93	C₄×C₇₉	—	8	2	no	—
94	C₄×C₇₉	#416	1	2	yes	—
95	C₄×C₇₉	#443	1	4	yes	—
96	C₄×C₇₉	#495	1	2	yes	—
97	#9857	—	79	1	no	maximal, Sylow
98	C₂×C₇₉	—	2	2	no	—
99	C₂×C₇₉	—	16	1	no	—
100	C₂×C₇₉	#1958	1	1	yes	characteristic
101	C₂×C₇₉	#1974	1	2	yes	—
102	#3131	D₇₉	1	1	yes	characteristic
103	#1837	—	79	1	no	—
104	#1493	—	79	2	no	—
105	#1492	—	79	2	no	—
106	C₂×C₈²	—	79	1	no	—
107	C₇₉	#9857	1	1	yes	characteristic, complemented, Sylow
108	#533	C₂×D₇₉	1	1	yes	characteristic
109	#532	C₂×D₇₉	1	1	yes	characteristic
110	#488	—	2	2	no	—
111	#487	Dic₇₉	1	2	yes	—
112	#486	Dic₇₉	1	2	yes	—
113	#423	—	79	2	no	—
114	#416	—	2	2	no	—
115	C₂×C₄×C₈	—	79	2	no	—
116	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
117	#341	—	158	2	no	—
118	Q₈⋊C₈	—	158	4	no	—
119	D₄⋊C₈	—	158	4	no	—
120	C₈²	—	158	2	no	—
121	C₂×#41	—	2	1	no	—
122	C₂×M₁₆	—	158	2	no	—
123	C₂²×C₈	—	158	2	no	—
124	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
125	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
126	C₄⋊Q₈	C₂×Dic₇₉	1	2	yes	—
127	#127	C₂×Dic₇₉	1	1	yes	characteristic
128	#125	—	2	1	no	—
129	#124	—	2	1	no	—
130	C₂²⋊Q₈	—	2	2	no	—
131	#121	—	2	2	no	—
132	C₄×Q₈	—	2	2	no	—
133	C₄×D₄	—	2	2	no	—
134	#117	—	2	1	no	—
135	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
136	C₄⋊C₈	—	158	2	no	—
137	C₄⋊C₈	—	158	2	no	—
138	#103	—	2	4	no	—
139	#102	—	2	4	no	—
140	C₄×C₈	—	79	4	no	—
141	C₄×C₈	—	158	2	no	—
142	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
143	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
144	#41	—	4	4	no	—
145	C₂×Q₈	—	2	1	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×D₄	—	2	1	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	4	1	no	—
150	C₂²×C₄	—	4	1	no	—
151	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
152	C₂²×C₄	#11929623	1	2	yes	—
153	M₁₆	—	316	4	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	316	2	no	—
156	C₂×C₈	—	158	8	no	—
157	C₂×C₈	—	158	4	no	—
158	C₂×C₈	C₂²⋊Dic₇₉	1	4	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	1	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
167	C₄²	D₄×D₇₉	1	2	yes	—
168	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
169	C₂³	—	4	1	no	—
170	C₂³	#14579148	1	1	yes	characteristic
171	Q₈	—	4	4	no	—
172	Q₈	—	4	2	no	—
173	D₄	—	4	4	no	—
174	D₄	—	4	2	no	—
175	D₄	—	8	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
181	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
182	C₂×C₄	#14579132	1	2	yes	—
183	C₂×C₄	#14579157	1	1	yes	characteristic
184	C₂×C₄	#14579181	1	4	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	632	2	no	—
187	C₈	—	158	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	8	2	no	—
190	C₂²	—	8	1	no	—
191	C₂²	#19098466	1	1	yes	characteristic
192	C₂²	#19098917	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	8	2	no	—
195	C₄	—	8	1	no	—
196	C₄	#19098559	1	2	yes	—
197	C₄	#19098680	1	4	yes	—
198	C₄	#19098945	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	16	1	no	—
201	C₂	#23764212	1	1	yes	characteristic
202	C₂	#23764289	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23763538

C₂

yes

maximal, characteristic, complemented

#23761855

C₂

yes

maximal, complemented

#23761851

C₂

yes

maximal

C₂×C₈×C₇₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₇₉×#3131

C₂

yes

maximal, characteristic

#19098587

C₂²

yes

—

C₂×C₄×C₇₉⋊C₈

C₂²

yes

—

#19098328

—

Q₈⋊(C₇₉⋊C₈)

—

D₄⋊(C₇₉⋊C₈)

—

C₈×C₇₉⋊C₈

—

#533×C₇₉

C₂²

yes

characteristic

#532×C₇₉

C₂²

yes

characteristic

#488×C₇₉

—

#487×C₇₉

C₄

yes

—

#486×C₇₉

C₄

yes

—

#416×C₇₉

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

C₂×#11929608

—

C₂²×C₇₉⋊C₈

—

C₄⋊(C₇₉⋊C₈)

—

C₄⋊(C₇₉⋊C₈)

—

C₄×C₇₉⋊C₈

—

C₄×C₇₉⋊C₈

D₄

yes

—

C₂×#41×C₇₉

—

C₂²×C₈×C₇₉

D₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

—

C₄⋊Q₈×C₇₉

C₂×C₄

yes

characteristic

#127×C₇₉

C₂×C₄

yes

characteristic

#125×C₇₉

—

#124×C₇₉

—

C₂²⋊Q₈×C₇₉

—

#121×C₇₉

—

C₄×Q₈×C₇₉

—

C₄×D₄×C₇₉

—

#117×C₇₉

—

C₂×C₄²×C₇₉

C₂³

yes

characteristic

#103×C₇₉

—

#102×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

#11929608

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

C₂×C₇₉⋊C₈

—

#41×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×Q₈×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂×D₄×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₉

#41

yes

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²×C₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

characteristic

C₄²×C₇₉

C₂×D₄

yes

—

C₂³×C₇₉

—

C₂³×C₇₉

#124

yes

characteristic

Q₈×C₇₉

—

Q₈×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

D₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×D₄

yes

—

C₂×C₄×C₇₉

#121

yes

—

C₂×C₄×C₇₉

#127

yes

characteristic

C₂×C₄×C₇₉

#135

yes

—

C₈×C₇₉

—

C₇₉⋊C₈

—

C₇₉⋊C₈

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

—

C₂²×C₇₉

#390

yes

characteristic

C₂²×C₇₉

#488

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

#416

yes

—

C₄×C₇₉

#443

yes

—

C₄×C₇₉

#495

yes

—

#9857

—

maximal, Sylow

C₂×C₇₉

—

C₂×C₇₉

—

100

C₂×C₇₉

#1958

yes

characteristic

101

C₂×C₇₉

#1974

yes

—

102

#3131

D₇₉

yes

characteristic

103

#1837

—

104

#1493

—

105

#1492

—

106

C₂×C₈²

—

107

C₇₉

#9857

yes

characteristic, complemented, Sylow

108

#533

C₂×D₇₉

yes

characteristic

109

#532

C₂×D₇₉

yes

characteristic

110

#488

—

111

#487

Dic₇₉

yes

—

112

#486

Dic₇₉

yes

—

113

#423

—

114

#416

—

115

C₂×C₄×C₈

—

116

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

117

#341

—

158

—

118

Q₈⋊C₈

—

158

—

119

D₄⋊C₈

—

158

—

120

C₈²

—

158

—

121

C₂×#41

—

122

C₂×M₁₆

—

158

—

123

C₂²×C₈

—

158

—

124

C₂²×C₈

C₂²⋊D₇₉

yes

—

125

C₄⋊Q₈

C₂×Dic₇₉

yes

characteristic

126

C₄⋊Q₈

C₂×Dic₇₉

yes

—

127

#127

C₂×Dic₇₉

yes

characteristic

128

#125

—

129

#124

—

130

C₂²⋊Q₈

—

131

#121

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

#117

—

135

C₂×C₄²

C₂²×D₇₉

yes

characteristic

136

C₄⋊C₈

—

158

—

137

C₄⋊C₈

—

158

—

138

#103

—

139

#102

—

140

C₄×C₈

—

141

C₄×C₈

—

158

—

142

C₄×C₈

C₂×Dic₇₉

yes

—

143

C₄×C₈

C₂²⋊D₇₉

yes

—

144

#41

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

152

C₂²×C₄

#11929623

yes

—

153

M₁₆

—

316

—

154

C₂×C₈

—

155

C₂×C₈

—

316

—

156

C₂×C₈

—

158

—

157

C₂×C₈

—

158

—

158

C₂×C₈

C₂²⋊Dic₇₉

yes

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

C₂²×Dic₇₉

yes

characteristic

167

C₄²

D₄×D₇₉

yes

—

168

C₄²

C₂×C₂²⋊D₇₉

yes

characteristic

169

C₂³

—

170

C₂³

#14579148

yes

characteristic

171

Q₈

—

172

Q₈

—

173

D₄

—

174

D₄

—

175

D₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

181

C₂×C₄

D₄×Dic₇₉

yes

—

182

C₂×C₄

#14579132

yes

—

183

C₂×C₄

#14579157

yes

characteristic

184

C₂×C₄

#14579181

yes

—

185

C₈

—

186

C₈

—

632

—

187

C₈

—

158

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#19098466

yes

characteristic

192

C₂²

#19098917

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

#19098559

yes

—

197

C₄

#19098680

yes

—

198

C₄

#19098945

yes

—

199

C₂

—

200

C₂

—

201

C₂

#23764212

yes

characteristic

202

C₂

#23764289

yes

—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#3131	C₂
socle	C₂²×C₇₉	#390
prime radical	#532×C₇₉	C₂²
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3131	D₇₉
hypocenter	C₇₉	#9857
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

subgroup

G/H

center

C₂×C₄

#14579157

commutator

C₄²×C₇₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₉

Fitting

C₇₉×#3131

C₂

socle

C₂²×C₇₉

#390

prime radical

#532×C₇₉

C₂²

cosocle

C₂×C₄²×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

N	H ≅ G/N	type
1. #23763538	C₂	semidirect
2. #23761855	C₂	semidirect
4. C₂×C₈×C₇₉⋊C₈	C₂	semidirect
107. C₇₉	#9857	semidirect

H ≅ G/N

type

1. #23763538

C₂

semidirect

2. #23761855

C₂

semidirect

4. C₂×C₈×C₇₉⋊C₈

C₂

semidirect

107. C₇₉

#9857

semidirect

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

subgroup	index
#23763538	2
#23761855	2
#23761851	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#3131	2
#9857	79

subgroup

index

#23763538

#23761855

#23761851

C₂×C₈×C₇₉⋊C₈

C₇₉×#3131

#9857

Subgroups — #34188217

Conjugacy classes of subgroups

Complements

Maximal subgroups