Group #13017347 — subgroups

Subgroups — #13017347

Order 1280.

398 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₂×#64883	C₂	1	1	yes	maximal, characteristic
2	C₂×#64879	C₂	1	1	yes	maximal, characteristic
3	#10604315	C₂	1	2	yes	maximal
4	#10599578	C₂	1	1	yes	maximal, characteristic
5	#10598721	C₂	1	2	yes	maximal
6	C₂×#4002	—	2	2	no	—
7	C₂×C₄⋊#170	C₂²	1	2	yes	—
8	C₂×C₄×#170	C₂²	1	1	yes	characteristic
9	C₂×C₄×#170	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₅	C₂²	1	2	yes	—
12	C₂×#3939	C₄	1	2	yes	—
13	#64918	—	2	2	no	—
14	#64883	C₄	1	4	yes	complemented
15	#64879	C₄	1	4	yes	complemented
16	C₅×#340	C₄	1	1	yes	characteristic
17	#64771	C₄	1	1	yes	characteristic
18	#8287	—	5	1	no	maximal, Sylow
19	C₂²×#170	—	2	2	no	—
20	C₂²×#170	—	2	1	no	—
21	C₂²×#170	D₄	1	2	yes	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅	—	2	2	no	—
26	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
27	C₂²×#168	C₂×C₄	1	1	yes	characteristic
28	#4002	—	4	4	no	—
29	C₄⋊#170	—	2	2	no	—
30	C₄⋊#170	C₂×C₄	1	4	yes	—
31	C₄×#170	—	2	1	no	—
32	C₄×#170	C₂×C₄	1	2	yes	—
33	C₄×#170	C₂×C₄	1	2	yes	—
34	C₄×#170	Q₈	1	2	yes	—
35	#3963	—	2	2	no	—
36	C₈×Dic₅	C₂×C₄	1	2	yes	—
37	C₈×Dic₅	C₂×C₄	1	2	yes	—
38	#3939	—	2	4	no	—
39	C₄×#107	—	5	1	no	—
40	#1770	—	5	2	no	—
41	C₂×#334	—	5	1	no	—
42	C₂×#322	—	5	1	no	—
43	#1279	—	5	2	no	—
44	C₂²×C₄×C₅	—	2	2	no	—
45	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅	—	2	2	no	—
47	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅	#41	1	1	yes	characteristic
49	C₂×#170	—	2	2	no	—
50	C₂×#170	—	2	2	no	—
51	C₂×#170	—	2	2	no	—
52	C₂×#170	—	2	4	no	—
53	C₂×#170	—	4	2	no	—
54	C₂×#170	—	4	2	no	—
55	C₂×#170	—	4	2	no	—
56	C₂×#170	C₄⋊C₄	1	4	yes	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₄²	1	2	yes	—
59	C₅×C₄⋊C₄	—	2	2	no	—
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₄⋊Dic₅	—	2	2	no	—
62	C₄⋊Dic₅	—	2	2	no	—
63	C₄×Dic₅	—	4	2	no	—
64	C₄×Dic₅	—	4	2	no	—
65	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₅	#41	1	1	yes	characteristic
69	C₂×#168	C₄²	1	2	yes	—
70	C₂×#168	M₁₆	1	4	yes	—
71	C₂×#107	—	5	2	no	—
72	C₂×C₄⋊C₈	—	5	2	no	—
73	C₂×#97	—	10	2	no	—
74	C₂×C₄×C₈	—	5	1	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₂×C₄×C₈	—	5	1	no	—
77	C₄×C₄⋊C₄	—	5	2	no	—
78	#340	—	5	1	no	—
79	#340	AGL(1,𝔽₅)	1	1	yes	characteristic
80	#336	—	10	2	no	—
81	#334	—	5	4	no	—
82	#322	—	5	4	no	—
83	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
90	C₂×C₄×C₅	#131	1	2	yes	—
91	C₂×Dic₅	—	2	2	no	—
92	C₂×Dic₅	—	4	2	no	—
93	C₂×Dic₅	—	4	2	no	—
94	C₂×Dic₅	—	4	2	no	—
95	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×Dic₅	#117	1	1	yes	characteristic
97	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
98	C₂×Dic₅	C₄×Q₈	1	1	yes	characteristic
99	C₂×Dic₅	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×Dic₅	#137	1	2	yes	—
101	#170	—	2	4	no	—
102	#170	—	4	2	no	—
103	#170	—	4	4	no	—
104	C₅×C₈	—	2	2	no	—
105	#168	#97	1	4	yes	complemented
106	C₂²×C₈	—	5	1	no	—
107	C₂²×C₈	—	5	2	no	—
108	C₂²×C₈	—	10	2	no	—
109	C₂²×C₈	—	10	1	no	—
110	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	5	2	no	—
112	C₂×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
113	C₂×C₄²	—	5	1	no	—
114	C₂×C₄²	—	10	2	no	—
115	#107	—	10	4	no	—
116	C₄⋊C₈	—	5	4	no	—
117	C₄⋊C₈	—	10	2	no	—
118	#97	—	20	4	no	—
119	C₄×C₈	—	5	2	no	—
120	C₄×C₈	—	5	2	no	—
121	C₄×C₈	—	5	2	no	—
122	C₄×C₈	—	5	2	no	—
123	C₄×C₈	—	5	2	no	—
124	C₄×C₈	—	10	1	no	—
125	#95	—	10	2	no	—
126	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₅	#436	1	2	yes	—
128	C₂²×C₅	#441	1	1	yes	characteristic
129	C₂²×C₅	#442	1	1	yes	characteristic
130	C₂²×C₅	#444	1	2	yes	—
131	C₄×C₅	—	8	2	no	—
132	C₄×C₅	C₂×#97	1	1	yes	characteristic
133	C₄×C₅	#405	1	1	yes	characteristic
134	C₄×C₅	#446	1	2	yes	—
135	Dic₅	—	2	1	no	—
136	Dic₅	—	2	1	no	—
137	Dic₅	—	8	2	no	—
138	Dic₅	#428	1	2	yes	—
139	Dic₅	#440	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	5	1	no	—
142	C₂²×C₄	—	5	1	no	—
143	C₂²×C₄	—	10	2	no	—
144	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	5	4	no	—
147	C₂×C₈	—	5	2	no	—
148	C₂×C₈	—	5	4	no	—
149	C₂×C₈	—	10	2	no	—
150	C₂×C₈	—	10	2	no	—
151	C₂×C₈	—	10	2	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	20	2	no	—
154	C₂×C₈	—	20	2	no	—
155	C₂×C₈	—	20	2	no	—
156	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	10	2	no	—
160	C₄⋊C₄	—	10	2	no	—
161	C₄²	—	5	1	no	—
162	C₄²	—	5	1	no	—
163	C₄²	—	5	1	no	—
164	C₄²	—	5	1	no	—
165	C₄²	—	20	2	no	—
166	C₄²	—	20	2	no	—
167	C₂×C₅	#1593	1	2	yes	—
168	C₂×C₅	#1594	1	2	yes	—
169	C₂×C₅	#1770	1	1	yes	characteristic
170	C₂×C₅	#1773	1	1	yes	characteristic
171	C₂×C₅	#1776	1	1	yes	characteristic
172	C₂³	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	5	1	no	—
177	C₂×C₄	—	5	1	no	—
178	C₂×C₄	—	5	2	no	—
179	C₂×C₄	—	5	2	no	—
180	C₂×C₄	—	5	1	no	—
181	C₂×C₄	—	5	1	no	—
182	C₂×C₄	—	10	2	no	—
183	C₂×C₄	—	20	2	no	—
184	C₂×C₄	—	20	2	no	—
185	C₂×C₄	—	20	2	no	—
186	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
187	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×C₄	#4131	1	2	yes	—
189	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	5	4	no	—
192	C₈	—	10	4	no	—
193	C₈	—	20	4	no	—
194	C₈	—	20	2	no	—
195	C₅	#8287	1	1	yes	characteristic, complemented, Sylow
196	C₂²	#65695	1	2	yes	—
197	C₂²	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂²	#65731	1	1	yes	characteristic
199	C₂²	#65737	1	2	yes	—
200	C₂²	#65739	1	1	yes	characteristic
201	C₄	—	5	2	no	—
202	C₄	—	5	2	no	—
203	C₄	—	8	2	no	—
204	C₄	—	10	1	no	—
205	C₄	—	10	1	no	—
206	C₄	—	40	2	no	—
207	C₄	#65676	1	1	yes	characteristic
208	C₄	#65691	1	1	yes	characteristic
209	C₄	#65707	1	2	yes	—
210	C₂	#10603780	1	1	yes	characteristic
211	C₂	#10603950	1	2	yes	—
212	C₂	#10603982	1	2	yes	—
213	C₂	#10604054	1	1	yes	characteristic
214	C₂	#10604072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64883

C₂

yes

maximal, characteristic

C₂×#64879

C₂

yes

maximal, characteristic

#10604315

C₂

yes

maximal

#10599578

C₂

yes

maximal, characteristic

#10598721

C₂

yes

maximal

C₂×#4002

—

C₂×C₄⋊#170

C₂²

yes

—

C₂×C₄×#170

C₂²

yes

characteristic

C₂×C₄×#170

C₂²

yes

characteristic

C₂×C₈×Dic₅

C₂²

yes

characteristic

C₄⋊C₄×Dic₅

C₂²

yes

—

C₂×#3939

C₄

yes

—

#64918

—

#64883

C₄

yes

complemented

#64879

C₄

yes

complemented

C₅×#340

C₄

yes

characteristic

#64771

C₄

yes

characteristic

#8287

—

maximal, Sylow

C₂²×#170

—

C₂²×#170

—

C₂²×#170

D₄

yes

—

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊Dic₅

C₂×C₄

yes

—

C₂×C₄×Dic₅

—

C₂×C₄×Dic₅

C₂³

yes

characteristic

C₂²×#168

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×#170

—

C₄×#170

C₂×C₄

yes

—

C₄×#170

C₂×C₄

yes

—

C₄×#170

Q₈

yes

—

#3963

—

C₈×Dic₅

C₂×C₄

yes

—

C₈×Dic₅

C₂×C₄

yes

—

—

—

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

C₂×D₄

yes

C₂²×Dic₅

yes

—

—

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

C₄×Dic₅

yes

C₄²

yes

—

C₂×#168

M₁₆

yes

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

—

yes

—

—

—

C₂³×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅

#117

yes

characteristic

C₂×Dic₅

C₄×D₄

yes

characteristic

C₂×Dic₅

C₄×Q₈

yes

characteristic

C₂×Dic₅

AGL(1,ℤ/8)

yes

—

100

C₂×Dic₅

#137

yes

—

101

#170

—

102

#170

—

103

#170

—

104

C₅×C₈

—

105

#168

#97

yes

complemented

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

113

C₂×C₄²

—

114

C₂×C₄²

—

115

#107

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

#97

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

#95

—

126

C₂²×C₅

C₄×C₄⋊C₄

yes

characteristic

127

C₂²×C₅

#436

yes

—

128

C₂²×C₅

#441

yes

characteristic

129

C₂²×C₅

#442

yes

characteristic

130

C₂²×C₅

#444

yes

—

131

C₄×C₅

—

132

C₄×C₅

C₂×#97

yes

characteristic

133

C₄×C₅

#405

yes

characteristic

134

C₄×C₅

#446

yes

—

135

Dic₅

—

136

Dic₅

—

137

Dic₅

—

138

Dic₅

#428

yes

—

139

Dic₅

#440

yes

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄²

—

162

C₄²

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₂×C₅

#1593

yes

—

168

C₂×C₅

#1594

yes

—

169

C₂×C₅

#1770

yes

characteristic

170

C₂×C₅

#1773

yes

characteristic

171

C₂×C₅

#1776

yes

characteristic

172

C₂³

D₄×AGL(1,𝔽₅)

yes

characteristic

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

D₅⋊M₁₆

yes

—

187

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

188

C₂×C₄

#4131

yes

—

189

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₅

#8287

yes

characteristic, complemented, Sylow

196

C₂²

#65695

yes

—

197

C₂²

C₄⋊C₄×AGL(1,𝔽₅)

yes

characteristic

198

C₂²

#65731

yes

characteristic

199

C₂²

#65737

yes

—

200

C₂²

#65739

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

#65676

yes

characteristic

208

C₄

#65691

yes

characteristic

209

C₄

#65707

yes

—

210

C₂

#10603780

yes

characteristic

211

C₂

#10603950

yes

—

212

C₂

#10603982

yes

—

213

C₂

#10604054

yes

characteristic

214

C₂

#10604072

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#340	C₄
socle, prime radical	C₂³×C₅	C₄×D₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	#340	AGL(1,𝔽₅)
hypocenter	C₅	#8287
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₂²×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#340

C₄

socle, prime radical

C₂³×C₅

C₄×D₄

cosocle

C₂×C₄×Dic₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
14. #64883	C₄	semidirect
15. #64879	C₄	semidirect
105. #168	#97	semidirect
195. C₅	#8287	semidirect

H ≅ G/N

type

14. #64883

C₄

semidirect

15. #64879

C₄

semidirect

105. #168

#97

semidirect

195. C₅

#8287

semidirect

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

subgroup	index
C₂×#64883	2
C₂×#64879	2
#10604315	2
#10599578	2
#10598721	2
#8287	5

subgroup

index

Subgroups — #13017347

Conjugacy classes of subgroups

Complements

Maximal subgroups