Group #13020801 — subgroups

Subgroups — #13020801

Order 1280.

378 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 124 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×#64881	C₂	1	1	yes	maximal, characteristic
2	C₂×#170⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#10604325	C₂	1	2	yes	maximal
4	#10599577	C₂	1	1	yes	maximal, characteristic
5	#10598721	C₂	1	2	yes	maximal
6	C₂×#4004	—	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×#3940	C₄	1	2	yes	—
13	#64918	—	2	2	no	—
14	#64881	—	2	2	no	—
15	#170⋊C₈	—	2	2	no	—
16	C₅×#340	C₄	1	1	yes	characteristic
17	#64771	C₄	1	1	yes	characteristic
18	#9597	—	5	1	no	maximal, Sylow
19	C₂²×#170	—	2	1	no	—
20	C₂²×#170	—	2	2	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	#4004	—	4	4	no	—
29	C₄⋊#170	—	2	2	no	—
30	C₄⋊#170	—	2	2	no	—
31	C₄×#170	—	2	1	no	—
32	C₄×#170	—	2	1	no	—
33	C₄×#170	—	2	1	no	—
34	C₄×#170	Q₈	1	2	yes	—
35	#3963	—	2	2	no	—
36	C₈×Dic₅	D₄	1	2	yes	—
37	C₈×Dic₅	Q₈	1	2	yes	—
38	#3940	—	2	4	no	—
39	#1939	—	5	2	no	—
40	C₄×#106	—	5	1	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	—	4	1	no	—
53	C₂×#170	—	4	1	no	—
54	C₂×#170	—	4	2	no	—
55	C₂×#170	—	4	2	no	—
56	C₂×#170	—	4	2	no	—
57	C₂×#170	SD₁₆	1	4	yes	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₅×C₄⋊C₄	—	2	2	no	—
62	C₄⋊Dic₅	—	2	2	no	—
63	C₄⋊Dic₅	M₁₆	1	4	yes	—
64	C₄×Dic₅	—	4	2	no	—
65	C₄×Dic₅	—	4	2	no	—
66	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
69	C₄×Dic₅	#41	1	1	yes	characteristic
70	C₂×#168	—	2	2	no	—
71	C₂×#168	C₄⋊C₄	1	2	yes	—
72	C₂×#106	—	5	2	no	—
73	C₂×C₄⋊C₈	—	5	2	no	—
74	C₂×C₄⋊C₈	—	10	2	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₂×C₄×C₈	—	5	1	no	—
77	C₂×C₄×C₈	—	5	1	no	—
78	C₄×C₄⋊C₄	—	5	2	no	—
79	#340	—	5	1	no	—
80	#340	AGL(1,𝔽₅)	1	1	yes	characteristic
81	#336	—	10	2	no	—
82	#334	—	10	2	no	—
83	#322	—	10	2	no	—
84	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅	#131	1	2	yes	—
92	C₂×Dic₅	—	2	1	no	—
93	C₂×Dic₅	—	2	1	no	—
94	C₂×Dic₅	—	2	4	no	—
95	C₂×Dic₅	—	4	2	no	—
96	C₂×Dic₅	—	4	2	no	—
97	C₂×Dic₅	#121	1	1	yes	characteristic
98	C₂×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×Dic₅	#125	1	1	yes	characteristic
100	C₂×Dic₅	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×Dic₅	C₂×SD₁₆	1	2	yes	—
102	C₂×Dic₅	#135	1	2	yes	—
103	#170	—	2	4	no	—
104	#170	—	4	2	no	—
105	#170	—	8	2	no	—
106	C₅×C₈	—	2	2	no	—
107	#168	—	2	2	no	—
108	C₂²×C₈	—	5	2	no	—
109	C₂²×C₈	—	5	1	no	—
110	C₂²×C₈	—	10	1	no	—
111	C₂²×C₈	—	10	2	no	—
112	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	5	2	no	—
114	C₂×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
115	C₂×C₄²	—	5	1	no	—
116	C₂×C₄²	—	10	2	no	—
117	#106	—	10	4	no	—
118	C₄⋊C₈	—	10	2	no	—
119	C₄⋊C₈	—	10	2	no	—
120	C₄⋊C₈	—	20	4	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	C₄×C₈	—	10	1	no	—
126	C₄×C₈	—	10	1	no	—
127	#95	—	10	2	no	—
128	C₂²×C₅	#389	1	1	yes	characteristic
129	C₂²×C₅	#435	1	2	yes	—
130	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
131	C₂²×C₅	#444	1	1	yes	characteristic
132	C₂²×C₅	#444	1	1	yes	characteristic
133	C₄×C₅	—	8	2	no	—
134	C₄×C₅	#423	1	1	yes	characteristic
135	C₄×C₅	#424	1	1	yes	characteristic
136	C₄×C₅	#446	1	2	yes	—
137	Dic₅	—	2	1	no	—
138	Dic₅	—	2	1	no	—
139	Dic₅	—	4	4	no	—
140	Dic₅	#465	1	2	yes	—
141	Dic₅	#498	1	1	yes	characteristic
142	Dic₅	#499	1	1	yes	characteristic
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	5	1	no	—
145	C₂²×C₄	—	5	1	no	—
146	C₂²×C₄	—	10	2	no	—
147	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	5	4	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	10	2	no	—
152	C₂×C₈	—	10	2	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	—	10	2	no	—
155	C₂×C₈	—	20	1	no	—
156	C₂×C₈	—	20	1	no	—
157	C₂×C₈	—	20	2	no	—
158	C₂×C₈	—	20	2	no	—
159	C₂×C₈	—	20	2	no	—
160	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	5	4	no	—
164	C₄⋊C₄	—	10	2	no	—
165	C₄²	—	5	1	no	—
166	C₄²	—	5	1	no	—
167	C₄²	—	5	1	no	—
168	C₄²	—	5	1	no	—
169	C₄²	—	20	2	no	—
170	C₄²	—	20	2	no	—
171	C₂×C₅	#1582	1	2	yes	—
172	C₂×C₅	#1583	1	2	yes	—
173	C₂×C₅	#1940	1	1	yes	characteristic
174	C₂×C₅	#1941	1	1	yes	characteristic
175	C₂×C₅	#1946	1	1	yes	characteristic
176	C₂³	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	5	1	no	—
181	C₂×C₄	—	5	1	no	—
182	C₂×C₄	—	5	2	no	—
183	C₂×C₄	—	5	2	no	—
184	C₂×C₄	—	5	1	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	10	1	no	—
187	C₂×C₄	—	10	1	no	—
188	C₂×C₄	—	10	4	no	—
189	C₂×C₄	—	20	2	no	—
190	C₂×C₄	—	20	2	no	—
191	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
192	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₄	#4131	1	2	yes	—
194	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₈	—	2	2	no	—
196	C₈	—	10	4	no	—
197	C₈	—	10	2	no	—
198	C₈	—	20	2	no	—
199	C₈	—	40	2	no	—
200	C₅	#9597	1	1	yes	characteristic, complemented, Sylow
201	C₂²	#65694	1	2	yes	—
202	C₂²	#65712	1	1	yes	characteristic
203	C₂²	#65733	1	1	yes	characteristic
204	C₂²	SD₁₆×AGL(1,𝔽₅)	1	2	yes	—
205	C₂²	#65741	1	1	yes	characteristic
206	C₄	—	5	2	no	—
207	C₄	—	5	1	no	—
208	C₄	—	5	1	no	—
209	C₄	—	8	2	no	—
210	C₄	—	10	1	no	—
211	C₄	—	10	1	no	—
212	C₄	—	20	4	no	—
213	C₄	#65682	1	1	yes	characteristic
214	C₄	#65696	1	1	yes	characteristic
215	C₄	#65707	1	2	yes	—
216	C₂	#10603797	1	1	yes	characteristic
217	C₂	#10603942	1	2	yes	—
218	C₂	#10603971	1	2	yes	—
219	C₂	#10604051	1	1	yes	characteristic
220	C₂	#10604082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64881

C₂

yes

maximal, characteristic

C₂×#170⋊C₈

C₂

yes

maximal, characteristic

#10604325

C₂

yes

maximal

#10599577

C₂

yes

maximal, characteristic

#10598721

C₂

yes

maximal

C₂×#4004

—

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×#3940

C₄

yes

—

—

—

—

C₄

yes

characteristic

#64771

C₄

yes

characteristic

#9597

—

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

—

—

—

—

—

—

Q₈

yes

—

#3963

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

—

—

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

C₂×D₄

yes

C₂²×Dic₅

yes

—

—

—

—

—

—

—

—

SD₁₆

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄⋊Dic₅

—

C₄⋊Dic₅

M₁₆

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₄

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₄×Dic₅

C₂×Q₈

yes

C₄×Dic₅

yes

—

C₄⋊C₄

yes

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

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×C₄⋊C₄

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×Dic₅

#121

yes

characteristic

C₂×Dic₅

C₂²⋊Q₈

yes

characteristic

C₂×Dic₅

#125

yes

characteristic

100

C₂×Dic₅

C₄⋊Q₈

yes

characteristic

101

C₂×Dic₅

C₂×SD₁₆

yes

—

102

C₂×Dic₅

#135

yes

—

103

#170

—

104

#170

—

105

#170

—

106

C₅×C₈

—

107

#168

—

108

C₂²×C₈

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

115

C₂×C₄²

—

116

C₂×C₄²

—

117

#106

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

#95

—

128

C₂²×C₅

#389

yes

characteristic

129

C₂²×C₅

#435

yes

—

130

C₂²×C₅

C₄×SD₁₆

yes

—

131

C₂²×C₅

#444

yes

characteristic

132

C₂²×C₅

#444

yes

characteristic

133

C₄×C₅

—

134

C₄×C₅

#423

yes

characteristic

135

C₄×C₅

#424

yes

characteristic

136

C₄×C₅

#446

yes

—

137

Dic₅

—

138

Dic₅

—

139

Dic₅

—

140

Dic₅

#465

yes

—

141

Dic₅

#498

yes

characteristic

142

Dic₅

#499

yes

characteristic

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

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

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₂×C₅

#1582

yes

—

172

C₂×C₅

#1583

yes

—

173

C₂×C₅

#1940

yes

characteristic

174

C₂×C₅

#1941

yes

characteristic

175

C₂×C₅

#1946

yes

characteristic

176

C₂³

D₄×AGL(1,𝔽₅)

yes

characteristic

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

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

D₅⋊M₁₆

yes

—

192

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

193

C₂×C₄

#4131

yes

—

194

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₅

#9597

yes

characteristic, complemented, Sylow

201

C₂²

#65694

yes

—

202

C₂²

#65712

yes

characteristic

203

C₂²

#65733

yes

characteristic

204

C₂²

SD₁₆×AGL(1,𝔽₅)

yes

—

205

C₂²

#65741

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

#65682

yes

characteristic

214

C₄

#65696

yes

characteristic

215

C₄

#65707

yes

—

216

C₂

#10603797

yes

characteristic

217

C₂

#10603942

yes

—

218

C₂

#10603971

yes

—

219

C₂

#10604051

yes

characteristic

220

C₂

#10604082

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×AGL(1,𝔽₅)
commutator	C₂²×C₄×C₅	C₂²×C₄
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	#9597
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×AGL(1,𝔽₅)

commutator

C₂²×C₄×C₅

C₂²×C₄

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 ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
200. C₅	#9597	semidirect

H ≅ G/N

type

200. C₅

#9597

semidirect

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

subgroup	index
C₂×#64881	2
C₂×#170⋊C₈	2
#10604325	2
#10599577	2
#10598721	2
#9597	5

subgroup

index

C₂×#170⋊C₈

Subgroups — #13020801

Conjugacy classes of subgroups

Complements

Maximal subgroups