Group #72223021 — subgroups

Subgroups — #72223021

Order 3200.

880 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 187 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	C₅²×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	#13102952	C₂	1	1	yes	maximal, characteristic, complemented
3	#13102942	C₂	1	1	yes	maximal, characteristic, complemented
4	#13102937	C₂	1	1	yes	maximal, characteristic, complemented
5	#13102935	C₂	1	1	yes	maximal, characteristic
6	#13102557	C₂	1	1	yes	maximal, characteristic, complemented
7	#13099488	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅²×SD₁₆	—	2	2	no	—
9	C₅²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₅²×#127	C₂²	1	1	yes	characteristic
11	C₂×#87810	—	2	2	no	—
12	C₂×#87809	—	2	2	no	—
13	#11717194	C₂²	1	1	yes	characteristic
14	#11717189	C₂²	1	1	yes	characteristic
15	C₄×C₅²×C₈	C₂²	1	1	yes	characteristic
16	#11716688	—	2	2	no	—
17	#11716676	—	2	2	no	—
18	#11716675	—	2	2	no	—
19	C₄×#6029	C₂²	1	1	yes	characteristic
20	C₄×#6029	C₂²	1	1	yes	characteristic
21	#10602052	—	5	6	no	maximal
22	C₂×C₅²×Q₈	—	2	1	no	—
23	C₂×C₅²×D₄	—	2	1	no	—
24	C₂×C₅²×D₄	—	2	1	no	—
25	C₂×C₅²×D₄	—	2	1	no	—
26	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
27	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
28	C₂×C₄⋊(C₅⋊D₅)	—	2	1	no	—
29	C₂×#6045	—	2	1	no	—
30	C₅²×SD₁₆	—	4	4	no	—
31	C₂×C₅²×C₈	D₄	1	2	yes	—
32	C₅²×C₄⋊C₄	—	2	1	no	—
33	C₅²×C₄⋊C₄	—	2	1	no	—
34	C₄²×C₅²	C₂³	1	1	yes	characteristic
35	#87810	—	4	4	no	—
36	#87809	—	4	4	no	—
37	C₄⋊#1039	—	2	1	no	—
38	C₄⋊#1039	—	2	1	no	—
39	C₂×#6029	D₄	1	2	yes	—
40	C₂×#6029	D₄	1	2	yes	—
41	C₅×#492	D₅	1	6	yes	complemented
42	#65377	—	5	6	no	—
43	#65368	—	5	6	no	—
44	#65365	—	5	6	no	—
45	#65362	—	5	6	no	—
46	#64983	—	5	6	no	—
47	#64674	—	5	6	no	—
48	C₂³×C₅²	—	4	1	no	—
49	C₂²×C₅⋊D₅	—	4	1	no	—
50	C₅²×Q₈	—	4	2	no	—
51	C₅²×D₄	—	4	2	no	—
52	C₅²×D₄	—	4	2	no	—
53	C₅²×D₄	—	4	2	no	—
54	C₂×C₄×C₅²	—	4	1	no	—
55	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
58	C₂×#1039	—	4	1	no	—
59	C₄⋊(C₅⋊D₅)	—	4	2	no	—
60	C₄⋊(C₅⋊D₅)	—	4	2	no	—
61	C₄⋊(C₅⋊D₅)	—	4	2	no	—
62	#6045	—	4	2	no	—
63	C₅²×C₈	—	2	2	no	—
64	#6029	—	2	2	no	—
65	#6029	SD₁₆	1	4	yes	complemented
66	C₂×C₅×SD₁₆	—	2	12	no	—
67	C₅×C₄⋊Q₈	C₂×D₅	1	6	yes	—
68	C₅×#127	C₂×D₅	1	6	yes	—
69	C₂×#687	—	10	12	no	—
70	C₂×#686	—	10	12	no	—
71	#4020	—	5	6	no	—
72	#4015	—	5	6	no	—
73	C₄×C₅×C₈	C₂×D₅	1	6	yes	—
74	#3953	—	10	12	no	—
75	#3941	—	10	12	no	—
76	#3940	—	10	12	no	—
77	C₄×#168	—	5	6	no	—
78	C₄×#168	—	5	6	no	—
79	#1715	—	25	1	no	Sylow
80	C₂²×C₅²	—	8	1	no	—
81	C₂²×C₅²	—	8	1	no	—
82	C₂²×C₅²	—	8	1	no	—
83	C₂²×C₅²	#127	1	1	yes	characteristic
84	C₂×C₅⋊D₅	—	8	1	no	—
85	C₂×C₅⋊D₅	—	8	1	no	—
86	C₂×C₅⋊D₅	—	8	1	no	—
87	C₄×C₅²	—	8	1	no	—
88	C₄×C₅²	C₂×SD₁₆	1	2	yes	—
89	C₄×C₅²	AGL(1,ℤ/8)	1	2	yes	—
90	C₄×C₅²	AGL(1,ℤ/8)	1	2	yes	—
91	#1039	—	8	1	no	—
92	C₂×C₅×Q₈	—	2	6	no	—
93	C₂×C₅×D₄	—	2	6	no	—
94	C₂×C₅×D₄	—	2	6	no	—
95	C₂×C₅×D₄	—	2	6	no	—
96	C₂×D₂₀	—	10	6	no	—
97	C₂×D₂₀	—	10	6	no	—
98	C₂×D₂₀	—	10	6	no	—
99	C₂×Dic₁₀	—	10	6	no	—
100	C₅×SD₁₆	—	4	24	no	—
101	C₂×C₅×C₈	C₂²⋊D₅	1	12	yes	—
102	C₅×C₄⋊C₄	—	2	6	no	—
103	C₅×C₄⋊C₄	—	2	6	no	—
104	C₄²×C₅	C₂²×D₅	1	6	yes	—
105	#687	—	20	24	no	—
106	#686	—	20	24	no	—
107	C₄⋊Dic₅	—	10	6	no	—
108	C₄⋊Dic₅	—	10	6	no	—
109	C₂×#168	—	5	12	no	—
110	C₂×#168	—	5	12	no	—
111	#498	—	25	1	no	—
112	#493	—	25	1	no	—
113	#492	—	25	1	no	—
114	#492	C₅⋊D₅	1	1	yes	characteristic, complemented
115	#486	—	25	1	no	—
116	#486	—	25	1	no	—
117	#322	—	25	1	no	—
118	C₂×C₅²	—	16	1	no	—
119	C₂×C₅²	#492	1	1	yes	characteristic
120	C₂×C₅²	#496	1	1	yes	characteristic
121	C₂×C₅²	#496	1	1	yes	characteristic
122	C₅⋊D₅	—	16	1	no	—
123	C₂³×C₅	—	4	6	no	—
124	C₂²×D₅	—	20	6	no	—
125	C₅×Q₈	—	4	12	no	—
126	C₅×D₄	—	4	12	no	—
127	C₅×D₄	—	4	12	no	—
128	C₅×D₄	—	4	12	no	—
129	C₂×C₄×C₅	—	4	6	no	—
130	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
131	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
132	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
133	C₂×Dic₅	—	20	6	no	—
134	D₂₀	—	20	12	no	—
135	D₂₀	—	20	12	no	—
136	D₂₀	—	20	12	no	—
137	Dic₁₀	—	20	12	no	—
138	C₅×C₈	—	2	12	no	—
139	#168	—	5	24	no	—
140	#168	—	10	12	no	—
141	C₂×SD₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	50	2	no	—
143	C₂×D₈	—	50	2	no	—
144	C₄⋊Q₈	—	25	1	no	—
145	C₄⋊Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
146	#127	—	25	1	no	—
147	#127	C₂×C₅⋊D₅	1	1	yes	characteristic
148	#106	—	50	2	no	—
149	#102	—	50	2	no	—
150	#102	—	50	2	no	—
151	C₄×C₈	—	25	1	no	—
152	C₄×C₈	—	25	1	no	—
153	C₄×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
154	C₅²	#1715	1	1	yes	characteristic, complemented, Sylow
155	C₂²×C₅	—	8	6	no	—
156	C₂²×C₅	—	8	6	no	—
157	C₂²×C₅	—	8	6	no	—
158	C₂²×C₅	#4086	1	6	yes	—
159	C₂×D₅	—	40	6	no	—
160	C₂×D₅	—	40	6	no	—
161	C₂×D₅	—	40	6	no	—
162	C₄×C₅	—	8	6	no	—
163	C₄×C₅	D₅×SD₁₆	1	12	yes	—
164	C₄×C₅	#4060	1	12	yes	—
165	C₄×C₅	#4078	1	12	yes	—
166	Dic₅	—	40	6	no	—
167	C₂×Q₈	—	2	1	no	—
168	C₂×Q₈	—	50	1	no	—
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	50	1	no	—
173	C₂×D₄	—	50	1	no	—
174	C₂×D₄	—	50	1	no	—
175	SD₁₆	—	4	4	no	—
176	SD₁₆	—	100	4	no	—
177	D₈	—	100	4	no	—
178	C₂×C₈	—	25	2	no	—
179	C₂×C₈	—	25	2	no	—
180	C₂×C₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	50	1	no	—
184	C₄⋊C₄	—	50	1	no	—
185	C₄²	C₂²×C₅⋊D₅	1	1	yes	characteristic
186	C₂×C₅	—	16	6	no	—
187	C₂×C₅	#65363	1	6	yes	—
188	C₂×C₅	#65464	1	6	yes	—
189	C₂×C₅	#65465	1	6	yes	—
190	D₅	—	80	6	no	—
191	C₂³	—	4	1	no	—
192	C₂³	—	100	1	no	—
193	Q₈	—	4	2	no	—
194	Q₈	—	100	2	no	—
195	D₄	—	4	2	no	—
196	D₄	—	4	2	no	—
197	D₄	—	4	2	no	—
198	D₄	—	100	2	no	—
199	D₄	—	100	2	no	—
200	D₄	—	100	2	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	100	1	no	—
203	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
204	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
205	C₂×C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	25	4	no	—
208	C₈	—	50	2	no	—
209	C₅	#10602052	1	6	yes	complemented
210	C₂²	—	8	1	no	—
211	C₂²	—	8	1	no	—
212	C₂²	—	8	1	no	—
213	C₂²	—	200	1	no	—
214	C₂²	—	200	1	no	—
215	C₂²	—	200	1	no	—
216	C₂²	#11717260	1	1	yes	characteristic
217	C₄	—	8	1	no	—
218	C₄	—	200	1	no	—
219	C₄	SD₁₆×C₅⋊D₅	1	2	yes	—
220	C₄	#11717234	1	2	yes	—
221	C₄	#11717252	1	2	yes	—
222	C₂	—	16	1	no	—
223	C₂	—	400	1	no	—
224	C₂	#13102936	1	1	yes	characteristic
225	C₂	#13103027	1	1	yes	characteristic
226	C₂	#13103030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#492

C₂

yes

maximal, characteristic, complemented

#13102952

C₂

yes

maximal, characteristic, complemented

#13102942

C₂

yes

maximal, characteristic, complemented

#13102937

C₂

yes

maximal, characteristic, complemented

#13102935

C₂

yes

maximal, characteristic

#13102557

C₂

yes

maximal, characteristic, complemented

#13099488

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×SD₁₆

—

C₅²×C₄⋊Q₈

C₂²

yes

characteristic

C₅²×#127

C₂²

yes

—

—

C₂²

yes

characteristic

#11717189

C₂²

yes

characteristic

C₄×C₅²×C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#6029

C₂²

yes

characteristic

#10602052

—

maximal

C₂×C₅²×Q₈

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×C₄⋊(C₅⋊D₅)

—

C₂×#6045

—

C₅²×SD₁₆

—

C₂×C₅²×C₈

D₄

yes

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂³

yes

—

—

—

—

D₄

yes

—

C₂×#6029

D₄

yes

—

C₅×#492

D₅

yes

—

—

—

—

—

—

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×D₄

yes

characteristic

C₂×C₄×C₅²

C₂×D₄

yes

characteristic

C₂×C₄×C₅²

C₂×D₄

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

#6045

—

C₅²×C₈

—

#6029

—

#6029

SD₁₆

yes

complemented

C₂×C₅×SD₁₆

—

C₅×C₄⋊Q₈

C₂×D₅

yes

—

C₅×#127

C₂×D₅

yes

—

—

—

—

—

C₄×C₅×C₈

C₂×D₅

yes

—

—

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

#127

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×SD₁₆

yes

—

C₄×C₅²

AGL(1,ℤ/8)

yes

—

C₄×C₅²

AGL(1,ℤ/8)

yes

—

#1039

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

100

C₅×SD₁₆

—

101

C₂×C₅×C₈

C₂²⋊D₅

yes

—

102

C₅×C₄⋊C₄

—

103

C₅×C₄⋊C₄

—

104

C₄²×C₅

C₂²×D₅

yes

—

105

#687

—

106

#686

—

107

C₄⋊Dic₅

—

108

C₄⋊Dic₅

—

109

C₂×#168

—

110

C₂×#168

—

111

#498

—

112

#493

—

113

#492

—

114

#492

C₅⋊D₅

yes

characteristic, complemented

115

#486

—

116

#486

—

117

#322

—

118

C₂×C₅²

—

119

C₂×C₅²

#492

yes

characteristic

120

C₂×C₅²

#496

yes

characteristic

121

C₂×C₅²

#496

yes

characteristic

122

C₅⋊D₅

—

123

C₂³×C₅

—

124

C₂²×D₅

—

125

C₅×Q₈

—

126

C₅×D₄

—

127

C₅×D₄

—

128

C₅×D₄

—

129

C₂×C₄×C₅

—

130

C₂×C₄×C₅

D₄×D₅

yes

—

131

C₂×C₄×C₅

D₄×D₅

yes

—

132

C₂×C₄×C₅

C₂×C₂²⋊D₅

yes

—

133

C₂×Dic₅

—

134

D₂₀

—

135

D₂₀

—

136

D₂₀

—

137

Dic₁₀

—

138

C₅×C₈

—

139

#168

—

140

#168

—

141

C₂×SD₁₆

—

142

C₂×SD₁₆

—

143

C₂×D₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

C₂×C₅⋊D₅

yes

characteristic

146

#127

—

147

#127

C₂×C₅⋊D₅

yes

characteristic

148

#106

—

149

#102

—

150

#102

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

C₂×C₅⋊D₅

yes

characteristic

154

C₅²

#1715

yes

characteristic, complemented, Sylow

155

C₂²×C₅

—

156

C₂²×C₅

—

157

C₂²×C₅

—

158

C₂²×C₅

#4086

yes

—

159

C₂×D₅

—

160

C₂×D₅

—

161

C₂×D₅

—

162

C₄×C₅

—

163

C₄×C₅

D₅×SD₁₆

yes

—

164

C₄×C₅

#4060

yes

—

165

C₄×C₅

#4078

yes

—

166

Dic₅

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×D₄

—

170

C₂×D₄

—

171

C₂×D₄

—

172

C₂×D₄

—

173

C₂×D₄

—

174

C₂×D₄

—

175

SD₁₆

—

176

SD₁₆

—

100

—

177

D₈

—

100

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

C₂²⋊(C₅⋊D₅)

yes

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄²

C₂²×C₅⋊D₅

yes

characteristic

186

C₂×C₅

—

187

C₂×C₅

#65363

yes

—

188

C₂×C₅

#65464

yes

—

189

C₂×C₅

#65465

yes

—

190

D₅

—

191

C₂³

—

192

C₂³

—

100

—

193

Q₈

—

194

Q₈

—

100

—

195

D₄

—

196

D₄

—

197

D₄

—

198

D₄

—

100

—

199

D₄

—

100

—

200

D₄

—

100

—

201

C₂×C₄

—

202

C₂×C₄

—

100

—

203

C₂×C₄

D₄×C₅⋊D₅

yes

characteristic

204

C₂×C₄

D₄×C₅⋊D₅

yes

characteristic

205

C₂×C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

206

C₈

—

207

C₈

—

208

C₈

—

209

C₅

#10602052

yes

complemented

210

C₂²

—

211

C₂²

—

212

C₂²

—

213

C₂²

—

200

—

214

C₂²

—

200

—

215

C₂²

—

200

—

216

C₂²

#11717260

yes

characteristic

217

C₄

—

218

C₄

—

200

—

219

C₄

SD₁₆×C₅⋊D₅

yes

—

220

C₄

#11717234

yes

—

221

C₄

#11717252

yes

—

222

C₂

—

223

C₂

—

400

—

224

C₂

#13102936

yes

characteristic

225

C₂

#13103027

yes

characteristic

226

C₂

#13103030

yes

characteristic

	subgroup	G/H
center	C₂²	#11717260
commutator, cosocle	C₄²×C₅²	C₂³
Frattini	C₄²	C₂²×C₅⋊D₅
Fitting	C₅²×#492	C₂
socle	C₂²×C₅²	#127
prime radical	#13102935	C₂
hypercenter	#492	C₅⋊D₅
hypocenter	C₅²	#1715
LCS	G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#11717260

commutator, cosocle

C₄²×C₅²

C₂³

Frattini

C₄²

C₂²×C₅⋊D₅

Fitting

C₅²×#492

C₂

C₂²×C₅²

C₂

C₅⋊D₅

C₅²

G ▷ C₄²×C₅² ▷ C₂²×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. C₅²×#492	C₂	semidirect
2. #13102952	C₂	semidirect
3. #13102942	C₂	semidirect
4. #13102937	C₂	semidirect
6. #13102557	C₂	semidirect
7. #13099488	C₂	semidirect
41. C₅×#492	D₅	semidirect
65. #6029	SD₁₆	semidirect
114. #492	C₅⋊D₅	semidirect
154. C₅²	#1715	semidirect
209. C₅	#10602052	semidirect

H ≅ G/N

type

1. C₅²×#492

C₂

semidirect

2. #13102952

C₂

semidirect

3. #13102942

C₂

semidirect

4. #13102937

C₂

semidirect

6. #13102557

C₂

semidirect

7. #13099488

C₂

semidirect

41. C₅×#492

D₅

semidirect

65. #6029

SD₁₆

semidirect

114. #492

C₅⋊D₅

semidirect

154. C₅²

#1715

semidirect

209. C₅

#10602052

semidirect

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

subgroup	index
C₅²×#492	2
#13102952	2
#13102942	2
#13102937	2
#13102935	2
#13102557	2
#13099488	2
#10602052	5

subgroup

index

C₅²×#492

Subgroups — #72223021

Conjugacy classes of subgroups

Complements

Maximal subgroups