Group #13020868 — subgroups

Subgroups — #13020868

Order 1280.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#10604016	C₂	1	2	yes	maximal
2	#10603858	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
4	#10599320	C₂	1	1	yes	maximal, characteristic
5	#10598594	C₂	1	2	yes	maximal
6	C₂×#3995	—	2	1	no	—
7	C₂×#3995	—	2	1	no	—
8	C₂×C₈×AGL(1,𝔽₅)	—	2	1	no	—
9	C₄×C₄⋊AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	#65681	C₂²	1	2	yes	—
11	D₅⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
12	#65122	C₂²	1	2	yes	—
13	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
14	(C₂×AGL(1,𝔽₅))⋊C₈	—	2	1	no	—
15	#64884	—	2	2	no	—
16	#64882	—	2	2	no	—
17	C₅×#334	C₄	1	1	yes	characteristic, complemented
18	#64699	C₄	1	2	yes	complemented
19	#64677	C₄	1	1	yes	characteristic, complemented
20	#9627	—	5	1	no	maximal, Sylow
21	C₂×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
22	C₂×C₄×AGL(1,𝔽₅)	—	2	1	no	—
23	C₂×#700	—	2	2	no	—
24	C₂×D₅⋊C₈	—	2	1	no	—
25	C₂×C₈×D₅	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅	D₄	1	1	yes	characteristic
27	C₄²×D₅	C₂³	1	1	yes	characteristic
28	(C₂×AGL(1,𝔽₅))⋊C₄	Q₈	1	2	yes	—
29	#4004	—	2	1	no	—
30	C₄⋊#170	—	2	2	no	—
31	C₄⋊#170	—	2	2	no	—
32	C₄⋊#170	D₄	1	2	yes	—
33	#3995	—	4	2	no	—
34	#3995	—	4	2	no	—
35	C₈×AGL(1,𝔽₅)	—	4	2	no	—
36	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	#3952	—	2	2	no	—
39	C₈×Dic₅	—	2	2	no	—
40	C₈×Dic₅	Q₈	1	1	yes	characteristic
41	C₈×Dic₅	Q₈	1	1	yes	characteristic
42	C₄⋊#168	C₂×C₄	1	2	yes	—
43	C₄×#168	C₂×C₄	1	1	yes	characteristic
44	#1848	—	5	1	no	—
45	C₈×C₄⋊C₄	—	5	1	no	—
46	#1571	—	5	2	no	—
47	#1564	—	5	1	no	—
48	#1303	—	5	2	no	—
49	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
50	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₅	#41	1	1	yes	characteristic
53	C₂×#170	—	2	2	no	—
54	C₂×#170	—	4	2	no	—
55	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
56	C₄×AGL(1,𝔽₅)	—	4	1	no	—
57	C₄×AGL(1,𝔽₅)	—	4	1	no	—
58	#700	—	4	2	no	—
59	#700	—	4	2	no	—
60	D₅⋊C₈	—	4	1	no	—
61	D₅⋊C₈	—	4	1	no	—
62	C₂×C₅×C₈	—	2	2	no	—
63	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
65	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₅	#41	1	1	yes	characteristic
69	C₂×#168	—	2	2	no	—
70	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
71	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
72	C₈×D₅	—	2	1	no	—
73	C₈×D₅	—	2	1	no	—
74	C₈×D₅	D₈	1	2	yes	—
75	C₈×D₅	Q₁₆	1	2	yes	—
76	#431	—	5	2	no	—
77	C₂×#108	—	10	1	no	—
78	C₂×#108	—	10	1	no	—
79	#423	—	5	2	no	—
80	C₂×C₄⋊C₈	—	5	1	no	—
81	C₂×C₄×C₈	—	5	1	no	—
82	C₂×C₄×C₈	—	10	1	no	—
83	C₄×C₄⋊C₄	—	5	1	no	—
84	#336	—	10	1	no	—
85	#335	—	5	2	no	—
86	#335	—	10	2	no	—
87	#335	—	10	2	no	—
88	#334	—	5	1	no	—
89	#334	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
90	C₂²×D₅	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×AGL(1,𝔽₅)	—	2	4	no	—
92	C₂×AGL(1,𝔽₅)	—	4	2	no	—
93	C₂×AGL(1,𝔽₅)	—	4	2	no	—
94	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
97	C₂×Dic₅	#121	1	1	yes	characteristic
98	C₂×Dic₅	#125	1	1	yes	characteristic
99	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
100	C₄×D₅	—	2	2	no	—
101	C₄×D₅	—	4	1	no	—
102	C₄×D₅	C₂×D₈	1	1	yes	characteristic
103	C₄×D₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₄×D₅	AGL(1,ℤ/8)	1	1	yes	characteristic
105	C₄×D₅	#137	1	1	yes	characteristic
106	#170	—	4	2	no	—
107	#170	—	8	2	no	—
108	C₅×C₈	—	2	1	no	—
109	C₅×C₈	—	4	2	no	—
110	C₅×C₈	#107	1	2	yes	—
111	#168	—	2	1	no	—
112	#168	—	4	2	no	—
113	#168	#107	1	2	yes	—
114	C₂×M₁₆	—	10	2	no	—
115	C₂²×C₈	—	5	1	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂²×C₈	—	10	1	no	—
118	C₂×C₄⋊C₄	—	10	1	no	—
119	C₂×C₄²	—	5	1	no	—
120	C₂×C₄²	—	10	1	no	—
121	#108	—	20	2	no	—
122	#108	—	20	2	no	—
123	C₄⋊C₈	—	5	2	no	—
124	C₄⋊C₈	—	5	2	no	—
125	C₄⋊C₈	—	10	1	no	—
126	C₄⋊C₈	—	10	2	no	—
127	C₄⋊C₈	—	10	2	no	—
128	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
129	C₂²⋊C₈	—	10	2	no	—
130	C₄×C₈	—	5	1	no	—
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	—	10	2	no	—
134	C₄×C₈	—	20	2	no	—
135	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
136	#95	—	5	2	no	—
137	C₂²×C₅	#389	1	1	yes	characteristic
138	C₂×D₅	—	2	1	no	—
139	C₂×D₅	—	2	1	no	—
140	C₂×D₅	#474	1	1	yes	characteristic
141	C₂×D₅	#477	1	1	yes	characteristic
142	AGL(1,𝔽₅)	—	4	4	no	—
143	C₄×C₅	—	2	1	no	—
144	C₄×C₅	C₂×#107	1	1	yes	characteristic
145	C₄×C₅	#428	1	1	yes	characteristic
146	C₄×C₅	#443	1	2	yes	—
147	Dic₅	—	2	1	no	—
148	Dic₅	—	2	1	no	—
149	Dic₅	#459	1	1	yes	characteristic
150	Dic₅	#462	1	1	yes	characteristic
151	C₂²×C₄	—	5	1	no	—
152	C₂²×C₄	—	5	1	no	—
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	10	2	no	—
155	M₁₆	—	20	2	no	—
156	M₁₆	—	20	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	2	no	—
160	C₂×C₈	—	5	2	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	10	1	no	—
163	C₂×C₈	—	10	1	no	—
164	C₂×C₈	—	10	2	no	—
165	C₂×C₈	—	10	2	no	—
166	C₂×C₈	—	20	1	no	—
167	C₂×C₈	—	20	1	no	—
168	C₂×C₈	—	20	2	no	—
169	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₄⋊C₄	—	10	4	no	—
172	C₄²	—	5	1	no	—
173	C₄²	—	5	1	no	—
174	C₄²	—	5	1	no	—
175	C₄²	—	20	1	no	—
176	C₄²	—	20	1	no	—
177	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₂×C₅	#1945	1	1	yes	characteristic
179	C₂×C₅	#1947	1	1	yes	characteristic
180	C₂×C₅	#1953	1	1	yes	characteristic
181	D₅	—	2	1	no	—
182	D₅	—	2	1	no	—
183	C₂³	—	5	1	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	5	1	no	—
187	C₂×C₄	—	5	1	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	—	5	1	no	—
191	C₂×C₄	—	10	2	no	—
192	C₂×C₄	—	10	4	no	—
193	C₂×C₄	—	20	1	no	—
194	C₂×C₄	—	20	2	no	—
195	C₂×C₄	—	20	2	no	—
196	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	4	2	no	—
201	C₈	—	5	2	no	—
202	C₈	—	10	1	no	—
203	C₈	—	20	2	no	—
204	C₈	—	20	2	no	—
205	C₈	—	40	2	no	—
206	C₈	#3994	1	2	yes	—
207	C₅	#9627	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	5	1	no	—
209	C₂²	—	5	1	no	—
210	C₂²	—	10	1	no	—
211	C₂²	—	10	1	no	—
212	C₂²	#65712	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	5	1	no	—
215	C₄	—	5	1	no	—
216	C₄	—	10	1	no	—
217	C₄	—	10	1	no	—
218	C₄	—	20	4	no	—
219	C₄	C₂×#3994	1	1	yes	characteristic
220	C₄	#65727	1	1	yes	characteristic
221	C₄	#65736	1	2	yes	—
222	C₂	—	10	1	no	—
223	C₂	—	10	1	no	—
224	C₂	#10604030	1	1	yes	characteristic
225	C₂	#10604057	1	1	yes	characteristic
226	C₂	#10604091	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604016

C₂

yes

maximal

#10603858

C₂

yes

maximal, characteristic

C₈×C₄⋊AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

#10599320

C₂

yes

maximal, characteristic

#10598594

C₂

yes

maximal

C₂×#3995

—

C₂×#3995

—

C₂×C₈×AGL(1,𝔽₅)

—

C₄×C₄⋊AGL(1,𝔽₅)

C₂²

yes

characteristic

#65681

C₂²

yes

—

D₅⋊(C₄⋊C₈)

C₂²

yes

characteristic

#65122

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₈

—

#64884

—

#64882

—

C₅×#334

C₄

yes

characteristic, complemented

#64699

C₄

yes

complemented

#64677

C₄

yes

characteristic, complemented

#9627

—

maximal, Sylow

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

—

C₂×#700

—

C₂×D₅⋊C₈

—

C₂×C₈×D₅

D₄

yes

characteristic

C₂×C₈×D₅

D₄

yes

characteristic

C₄²×D₅

C₂³

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

Q₈

yes

—

—

—

—

D₄

yes

—

#3995

—

#3995

—

C₈×AGL(1,𝔽₅)

—

C₅×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

C₈×Dic₅

—

C₈×Dic₅

Q₈

yes

characteristic

C₈×Dic₅

Q₈

yes

characteristic

C₄⋊#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

#1848

—

C₈×C₄⋊C₄

—

#1571

—

#1564

—

#1303

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

C₂×C₄×D₅

yes

—

—

—

—

—

—

—

D₅⋊C₈

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

C₄×Dic₅

yes

—

C₄⋊C₄

yes

characteristic

C₂×#168

C₄⋊C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₈×D₅

D₈

yes

—

C₈×D₅

Q₁₆

yes

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

—

—

—

—

—

yes

characteristic, complemented

C₂²×D₅

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

C₂×Dic₅

yes

C₂×Dic₅

yes

C₂×Dic₅

C₄⋊Q₈

yes

characteristic

100

C₄×D₅

—

101

C₄×D₅

—

102

C₄×D₅

C₂×D₈

yes

characteristic

103

C₄×D₅

C₂×Q₁₆

yes

characteristic

104

C₄×D₅

AGL(1,ℤ/8)

yes

characteristic

105

C₄×D₅

#137

yes

characteristic

106

#170

—

107

#170

—

108

C₅×C₈

—

109

C₅×C₈

—

110

C₅×C₈

#107

yes

—

111

#168

—

112

#168

—

113

#168

#107

yes

—

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄²

—

120

C₂×C₄²

—

121

#108

—

122

#108

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

129

C₂²⋊C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

136

#95

—

137

C₂²×C₅

#389

yes

characteristic

138

C₂×D₅

—

139

C₂×D₅

—

140

C₂×D₅

#474

yes

characteristic

141

C₂×D₅

#477

yes

characteristic

142

AGL(1,𝔽₅)

—

143

C₄×C₅

—

144

C₄×C₅

C₂×#107

yes

characteristic

145

C₄×C₅

#428

yes

characteristic

146

C₄×C₅

#443

yes

—

147

Dic₅

—

148

Dic₅

—

149

Dic₅

#459

yes

characteristic

150

Dic₅

#462

yes

characteristic

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

M₁₆

—

156

M₁₆

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

170

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

171

C₄⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

—

176

C₄²

—

177

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

178

C₂×C₅

#1945

yes

characteristic

179

C₂×C₅

#1947

yes

characteristic

180

C₂×C₅

#1953

yes

characteristic

181

D₅

—

182

D₅

—

183

C₂³

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

197

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

198

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

#3994

yes

—

207

C₅

#9627

yes

characteristic, complemented, Sylow

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#65712

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

C₂×#3994

yes

characteristic

220

C₄

#65727

yes

characteristic

221

C₄

#65736

yes

—

222

C₂

—

223

C₂

—

224

C₂

#10604030

yes

characteristic

225

C₂

#10604057

yes

characteristic

226

C₂

#10604091

yes

characteristic

	subgroup	G/H
center	C₂²	#65712
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#334	C₄
socle	C₂²×C₅	#389
prime radical	C₂²×D₅	C₂²⋊Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	#334	AGL(1,𝔽₅)
hypocenter	C₅	#9627
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#65712

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂²×D₅

C₂²⋊Q₈

cosocle

C₄²×D₅

C₂³

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
17. C₅×#334	C₄	semidirect
18. #64699	C₄	semidirect
19. #64677	C₄	semidirect
89. #334	AGL(1,𝔽₅)	semidirect
207. C₅	#9627	semidirect

H ≅ G/N

type

17. C₅×#334

C₄

semidirect

18. #64699

C₄

semidirect

19. #64677

C₄

semidirect

89. #334

AGL(1,𝔽₅)

semidirect

207. C₅

#9627

semidirect

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

subgroup	index
#10604016	2
#10603858	2
C₈×C₄⋊AGL(1,𝔽₅)	2
#10599320	2
#10598594	2
#9627	5

subgroup

index

C₈×C₄⋊AGL(1,𝔽₅)

Subgroups — #13020868

Conjugacy classes of subgroups

Complements

Maximal subgroups