Group #13026000 — subgroups

Subgroups — #13026000

Order 1280.

334 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 128 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	#10604113	C₂	1	2	yes	maximal
2	#10603851	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	D₅×#364	C₂	1	1	yes	maximal, characteristic
5	#10598673	C₂	1	2	yes	maximal
6	D₅⋊#108	C₄	1	1	yes	characteristic
7	#65721	C₄	1	1	yes	characteristic
8	#65718	C₂²	1	2	yes	—
9	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×D₅⋊C₁₆	C₂²	1	1	yes	characteristic
11	D₅×M₃₂	C₂²	1	2	yes	—
12	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
13	#64885	C₄	1	2	yes	complemented
14	#64862	C₄	1	2	yes	—
15	C₄×#674	C₄	1	2	yes	complemented
16	#64844	—	2	2	no	—
17	C₁₆×AGL(1,𝔽₅)	—	2	2	no	—
18	C₅×#364	C₄	1	1	yes	characteristic, complemented
19	#64731	C₄	1	2	yes	complemented
20	#64683	C₄	1	1	yes	characteristic, complemented
21	#12664	—	5	1	no	maximal, Sylow
22	#4130	C₂×C₄	1	2	yes	—
23	D₅⋊M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
26	C₄²×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂×#674	C₂×C₄	1	2	yes	—
28	C₂×#674	C₂×C₄	1	2	yes	—
29	#3996	C₈	1	2	yes	—
30	#3995	C₈	1	2	yes	—
31	#3994	C₈	1	2	yes	—
32	#3993	C₈	1	2	yes	—
33	#3992	—	2	2	no	—
34	C₈×AGL(1,𝔽₅)	—	2	2	no	—
35	D₅⋊C₁₆	—	2	1	no	—
36	D₅⋊C₁₆	—	2	1	no	—
37	D₅⋊C₁₆	D₄	1	1	yes	characteristic
38	D₅⋊C₁₆	D₄	1	1	yes	characteristic
39	D₅⋊C₁₆	Q₈	1	2	yes	—
40	C₅×M₃₂	C₂×C₄	1	2	yes	—
41	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
44	#3944	C₂×C₄	1	2	yes	—
45	C₄×#168	C₂×C₄	1	1	yes	characteristic
46	#3930	—	2	2	no	—
47	D₅×C₁₆	—	2	2	no	—
48	C₂×#364	—	5	1	no	—
49	#2108	—	5	2	no	—
50	C₂×C₄×C₁₆	—	5	1	no	—
51	#1772	—	5	1	no	—
52	#1379	—	5	2	no	—
53	C₂×C₄×D₅	—	2	1	no	—
54	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
55	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
56	#704	—	2	2	no	—
57	C₄⋊AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
58	C₄×AGL(1,𝔽₅)	—	2	2	no	—
59	#700	C₂×C₈	1	4	yes	—
60	D₅⋊C₈	—	2	2	no	—
61	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₄²	1	1	yes	characteristic
64	C₄×Dic₅	—	2	1	no	—
65	C₄×Dic₅	C₄²	1	1	yes	characteristic
66	C₂×#168	C₄²	1	1	yes	characteristic
67	C₂×#168	C₂²×C₄	1	1	yes	characteristic
68	C₈×D₅	C₂×C₈	1	1	yes	characteristic
69	C₈×D₅	C₂×C₈	1	1	yes	characteristic
70	C₈×D₅	C₂×C₈	1	1	yes	characteristic
71	C₈×D₅	C₂×C₈	1	1	yes	characteristic
72	C₈×D₅	C₂×D₄	1	1	yes	characteristic
73	C₈×D₅	C₂×Q₈	1	1	yes	characteristic
74	C₈×D₅	#41	1	1	yes	characteristic
75	C₈×D₅	#41	1	1	yes	characteristic
76	#674	—	2	2	no	—
77	#674	C₄⋊C₄	1	2	yes	—
78	#674	C₄⋊C₄	1	2	yes	—
79	C₅×C₁₆	—	2	2	no	—
80	#672	—	2	2	no	—
81	C₂×M₃₂	—	5	2	no	—
82	C₂²×C₁₆	—	5	1	no	—
83	C₂²×C₁₆	—	5	1	no	—
84	C₂×#108	—	5	1	no	—
85	#427	—	5	1	no	—
86	#405	—	5	2	no	—
87	C₂×C₄×C₈	—	5	1	no	—
88	#364	—	5	2	no	—
89	#364	—	5	1	no	—
90	#364	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
91	#346	—	10	2	no	—
92	C₄×C₁₆	—	5	2	no	—
93	C₄×C₁₆	—	5	2	no	—
94	C₄×C₁₆	—	10	2	no	—
95	#339	—	5	2	no	—
96	C₂²×D₅	#131	1	1	yes	characteristic
97	C₂×AGL(1,𝔽₅)	—	2	4	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₂×Dic₅	—	2	1	no	—
101	C₂×Dic₅	#131	1	1	yes	characteristic
102	C₄×D₅	—	2	4	no	—
103	C₄×D₅	C₄×D₄	1	1	yes	characteristic
104	C₄×D₅	C₄×Q₈	1	1	yes	characteristic
105	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
106	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
107	#170	—	2	4	no	—
108	C₅×C₈	C₄×C₈	1	1	yes	characteristic
109	C₅×C₈	C₄×C₈	1	1	yes	characteristic
110	C₅×C₈	C₄×D₄	1	1	yes	characteristic
111	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
112	#168	C₄×C₈	1	1	yes	characteristic
113	#168	C₄×C₈	1	1	yes	characteristic
114	#168	C₂×C₄⋊C₄	1	1	yes	characteristic
115	#168	#117	1	1	yes	characteristic
116	C₂×M₁₆	—	5	2	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	5	1	no	—
119	#117	—	5	2	no	—
120	C₂×C₄²	—	5	1	no	—
121	M₃₂	—	5	2	no	—
122	M₃₂	—	10	2	no	—
123	M₃₂	C₂×AGL(1,𝔽₅)	1	2	yes	—
124	C₂×C₁₆	—	5	2	no	—
125	C₂×C₁₆	—	5	1	no	—
126	C₂×C₁₆	—	5	1	no	—
127	C₂×C₁₆	—	5	2	no	—
128	C₂×C₁₆	—	5	2	no	—
129	C₂×C₁₆	—	10	2	no	—
130	C₂×C₁₆	—	10	1	no	—
131	C₂×C₁₆	—	10	1	no	—
132	#108	—	5	2	no	—
133	#108	—	5	2	no	—
134	#107	—	5	2	no	—
135	#106	—	5	2	no	—
136	#97	—	10	2	no	—
137	C₄×C₈	—	5	1	no	—
138	C₄×C₈	—	5	1	no	—
139	C₄×C₈	—	5	1	no	—
140	C₄×C₈	—	10	2	no	—
141	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂²×C₅	#405	1	1	yes	characteristic
143	C₂×D₅	—	2	2	no	—
144	C₂×D₅	C₈×D₄	1	1	yes	characteristic
145	C₂×D₅	C₈×Q₈	1	1	yes	characteristic
146	AGL(1,𝔽₅)	—	4	4	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
150	Dic₅	—	2	2	no	—
151	Dic₅	C₈×D₄	1	1	yes	characteristic
152	Dic₅	C₈×Q₈	1	1	yes	characteristic
153	C₂²×C₄	—	5	1	no	—
154	C₂²×C₄	—	10	1	no	—
155	M₁₆	—	5	4	no	—
156	M₁₆	—	10	2	no	—
157	C₂×C₈	—	5	1	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	1	no	—
163	C₂×C₈	—	5	1	no	—
164	C₂×C₈	—	5	1	no	—
165	C₂×C₈	—	5	1	no	—
166	C₂×C₈	—	5	1	no	—
167	C₂×C₈	—	10	2	no	—
168	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
169	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₄⋊C₄	—	5	4	no	—
171	C₂²⋊C₄	—	10	2	no	—
172	C₄²	—	5	1	no	—
173	C₄²	—	10	1	no	—
174	C₄²	—	10	2	no	—
175	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₁₆	—	2	2	no	—
177	C₁₆	—	5	2	no	—
178	C₁₆	—	5	2	no	—
179	C₁₆	—	10	2	no	—
180	C₁₆	—	10	2	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
183	D₅	—	2	1	no	—
184	D₅	#2170	1	2	yes	—
185	C₂³	—	5	1	no	—
186	C₂×C₄	—	2	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	—	5	1	no	—
192	C₂×C₄	—	10	4	no	—
193	C₂×C₄	—	10	1	no	—
194	C₂×C₄	—	10	4	no	—
195	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
196	C₈	—	5	1	no	—
197	C₈	—	5	1	no	—
198	C₈	—	5	1	no	—
199	C₈	—	5	1	no	—
200	C₈	—	10	4	no	—
201	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₈	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₈	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₅	#12664	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	5	1	no	—
207	C₂²	—	5	1	no	—
208	C₂²	—	10	2	no	—
209	C₂²	#65728	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	5	1	no	—
212	C₄	—	5	1	no	—
213	C₄	—	10	2	no	—
214	C₄	—	20	4	no	—
215	C₄	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	5	2	no	—
219	C₂	—	10	1	no	—
220	C₂	#10604033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604113

C₂

yes

maximal

#10603851

C₂

yes

maximal, characteristic

C₄×D₅⋊C₁₆

C₂

yes

maximal, characteristic

D₅×#364

C₂

yes

maximal, characteristic

#10598673

C₂

yes

maximal

D₅⋊#108

C₄

yes

characteristic

#65721

C₄

yes

characteristic

#65718

C₂²

yes

—

C₂×D₅⋊C₁₆

C₂²

yes

characteristic

C₂×D₅⋊C₁₆

C₂²

yes

characteristic

D₅×M₃₂

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64885

C₄

yes

complemented

#64862

C₄

yes

—

C₄×#674

C₄

yes

complemented

#64844

—

C₁₆×AGL(1,𝔽₅)

—

C₅×#364

C₄

yes

characteristic, complemented

#64731

C₄

yes

complemented

#64683

C₄

yes

characteristic, complemented

#12664

—

maximal, Sylow

#4130

C₂×C₄

yes

—

D₅⋊M₁₆

C₂×C₄

yes

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

C₂³

yes

characteristic

C₄²×D₅

C₂×C₄

yes

characteristic

C₂×#674

C₂×C₄

yes

—

C₂×#674

C₂×C₄

yes

—

#3996

C₈

yes

—

#3995

C₈

yes

—

#3994

C₈

yes

—

#3993

C₈

yes

—

#3992

—

C₈×AGL(1,𝔽₅)

—

D₅⋊C₁₆

—

D₅⋊C₁₆

—

D₅⋊C₁₆

D₄

yes

characteristic

D₅⋊C₁₆

D₄

yes

characteristic

D₅⋊C₁₆

Q₈

yes

—

C₅×M₃₂

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

#3944

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

#3930

—

D₅×C₁₆

—

C₂×#364

—

#2108

—

C₂×C₄×C₁₆

—

#1772

—

#1379

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂²×C₄

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

—

#704

—

C₄⋊AGL(1,𝔽₅)

C₂×C₈

yes

—

C₄×AGL(1,𝔽₅)

—

#700

C₂×C₈

yes

—

D₅⋊C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₄²

yes

characteristic

C₄×Dic₅

—

C₄×Dic₅

C₄²

yes

characteristic

C₂×#168

C₄²

yes

characteristic

C₂×#168

C₂²×C₄

yes

characteristic

C₈×D₅

C₂×C₈

yes

characteristic

C₈×D₅

C₂×C₈

yes

characteristic

C₈×D₅

C₂×C₈

yes

characteristic

C₈×D₅

C₂×C₈

yes

characteristic

C₈×D₅

C₂×D₄

yes

characteristic

C₈×D₅

C₂×Q₈

yes

C₈×D₅

yes

C₈×D₅

yes

—

C₄⋊C₄

yes

—

#674

C₄⋊C₄

yes

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

—

—

yes

characteristic, complemented

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#339

—

C₂²×D₅

#131

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

100

C₂×Dic₅

—

101

C₂×Dic₅

#131

yes

characteristic

102

C₄×D₅

—

103

C₄×D₅

C₄×D₄

yes

characteristic

104

C₄×D₅

C₄×Q₈

yes

characteristic

105

C₄×D₅

C₂²×C₈

yes

characteristic

106

C₄×D₅

C₂²×C₈

yes

characteristic

107

#170

—

108

C₅×C₈

C₄×C₈

yes

characteristic

109

C₅×C₈

C₄×C₈

yes

characteristic

110

C₅×C₈

C₄×D₄

yes

characteristic

111

C₅×C₈

C₄×Q₈

yes

characteristic

112

#168

C₄×C₈

yes

characteristic

113

#168

C₄×C₈

yes

characteristic

114

#168

C₂×C₄⋊C₄

yes

115

yes

116

C₂×M₁₆

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

#117

—

120

C₂×C₄²

—

121

M₃₂

—

122

M₃₂

—

123

M₃₂

C₂×AGL(1,𝔽₅)

yes

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

#108

—

133

#108

—

134

#107

—

135

#106

—

136

#97

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

142

C₂²×C₅

#405

yes

characteristic

143

C₂×D₅

—

144

C₂×D₅

C₈×D₄

yes

characteristic

145

C₂×D₅

C₈×Q₈

yes

characteristic

146

AGL(1,𝔽₅)

—

147

C₄×C₅

—

148

C₄×C₅

C₄×C₄⋊C₄

yes

characteristic

149

C₄×C₅

C₂×C₄×C₈

yes

characteristic

150

Dic₅

—

151

Dic₅

C₈×D₄

yes

characteristic

152

Dic₅

C₈×Q₈

yes

characteristic

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

C₄×AGL(1,𝔽₅)

yes

characteristic

169

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

170

C₄⋊C₄

—

171

C₂²⋊C₄

—

172

C₄²

—

173

C₄²

—

174

C₄²

—

175

C₄²

C₄×AGL(1,𝔽₅)

yes

characteristic

176

C₁₆

—

177

C₁₆

—

178

C₁₆

—

179

C₁₆

—

180

C₁₆

—

181

C₂×C₅

—

182

C₂×C₅

C₈×C₄⋊C₄

yes

characteristic

183

D₅

—

184

D₅

#2170

yes

—

185

C₂³

—

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

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

C₈×AGL(1,𝔽₅)

yes

characteristic

202

C₈

C₈×AGL(1,𝔽₅)

yes

characteristic

203

C₈

D₄×AGL(1,𝔽₅)

yes

characteristic

204

C₈

Q₈×AGL(1,𝔽₅)

yes

characteristic

205

C₅

#12664

yes

characteristic, complemented, Sylow

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#65728

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

C₄⋊C₄×AGL(1,𝔽₅)

yes

characteristic

216

C₄

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

#10604033

yes

characteristic

	subgroup	G/H
center	C₈	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#364	C₄
socle	C₂×C₅	C₈×C₄⋊C₄
prime radical	C₂²×D₅	#131
cosocle	C₂×C₈×D₅	C₂³
hypercenter	#364	AGL(1,𝔽₅)
hypocenter	C₅	#12664
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#364

C₄

socle

C₂×C₅

C₈×C₄⋊C₄

prime radical

C₂²×D₅

#131

cosocle

C₂×C₈×D₅

C₂³

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
13. #64885	C₄	semidirect
15. C₄×#674	C₄	semidirect
18. C₅×#364	C₄	semidirect
19. #64731	C₄	semidirect
20. #64683	C₄	semidirect
90. #364	AGL(1,𝔽₅)	semidirect
205. C₅	#12664	semidirect

H ≅ G/N

type

13. #64885

C₄

semidirect

15. C₄×#674

C₄

semidirect

18. C₅×#364

C₄

semidirect

19. #64731

C₄

semidirect

20. #64683

C₄

semidirect

90. #364

AGL(1,𝔽₅)

semidirect

205. C₅

#12664

semidirect

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

subgroup	index
#10604113	2
#10603851	2
C₄×D₅⋊C₁₆	2
D₅×#364	2
#10598673	2
#12664	5

subgroup

index

C₄×D₅⋊C₁₆

Subgroups — #13026000

Conjugacy classes of subgroups

Complements

Maximal subgroups