Group #13026005 — subgroups

Subgroups — #13026005

Order 1280.

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

normal subgroups only

This group has 326 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	#10603823	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	#64886	C₄	1	2	yes	complemented
14	#64862	C₄	1	2	yes	—
15	#64859	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	#12665	—	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	2	yes	—
38	D₅⋊C₁₆	Q₈	1	2	yes	—
39	C₅×M₃₂	C₂×C₄	1	2	yes	—
40	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
43	#3944	C₂×C₄	1	2	yes	—
44	C₄×#168	C₂×C₄	1	1	yes	characteristic
45	#3930	—	2	2	no	—
46	D₅×C₁₆	—	2	2	no	—
47	C₂×#364	—	5	1	no	—
48	#2108	—	5	2	no	—
49	C₂×#346	—	5	1	no	—
50	#1772	—	5	1	no	—
51	#1379	—	5	2	no	—
52	C₂×C₄×D₅	—	2	1	no	—
53	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
54	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
55	#704	—	2	2	no	—
56	C₄⋊AGL(1,𝔽₅)	C₂×C₈	1	4	yes	—
57	C₄×AGL(1,𝔽₅)	—	2	2	no	—
58	#700	C₂×C₈	1	4	yes	—
59	D₅⋊C₈	—	2	2	no	—
60	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₅	C₄²	1	1	yes	characteristic
63	C₄×Dic₅	—	2	1	no	—
64	C₄×Dic₅	C₄²	1	1	yes	characteristic
65	C₂×#168	C₄²	1	1	yes	characteristic
66	C₂×#168	C₂²×C₄	1	1	yes	characteristic
67	C₈×D₅	C₂×C₈	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×D₄	1	1	yes	characteristic
72	C₈×D₅	C₂×Q₈	1	1	yes	characteristic
73	C₈×D₅	#41	1	1	yes	characteristic
74	C₈×D₅	#41	1	1	yes	characteristic
75	#674	—	2	2	no	—
76	#674	C₄⋊C₄	1	4	yes	—
77	C₅×C₁₆	—	2	2	no	—
78	#672	—	2	2	no	—
79	C₂×M₃₂	—	5	2	no	—
80	C₂²×C₁₆	—	5	1	no	—
81	C₂²×C₁₆	—	5	1	no	—
82	C₂×#108	—	5	1	no	—
83	#427	—	5	1	no	—
84	#405	—	5	2	no	—
85	C₂×C₄×C₈	—	5	1	no	—
86	#364	—	5	2	no	—
87	#364	—	5	1	no	—
88	#364	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
89	#346	—	5	2	no	—
90	#346	—	5	2	no	—
91	#346	—	10	2	no	—
92	C₄×C₁₆	—	10	2	no	—
93	#339	—	5	2	no	—
94	C₂²×D₅	#131	1	1	yes	characteristic
95	C₂×AGL(1,𝔽₅)	—	2	4	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
98	C₂×Dic₅	—	2	1	no	—
99	C₂×Dic₅	#131	1	1	yes	characteristic
100	C₄×D₅	—	4	2	no	—
101	C₄×D₅	C₄×D₄	1	1	yes	characteristic
102	C₄×D₅	C₄×Q₈	1	1	yes	characteristic
103	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
104	C₄×D₅	C₂²×C₈	1	1	yes	characteristic
105	#170	—	2	4	no	—
106	C₅×C₈	C₄×C₈	1	1	yes	characteristic
107	C₅×C₈	C₄×C₈	1	1	yes	characteristic
108	C₅×C₈	C₄×D₄	1	1	yes	characteristic
109	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
110	#168	C₄×C₈	1	1	yes	characteristic
111	#168	C₄×C₈	1	1	yes	characteristic
112	#168	C₂×C₄⋊C₄	1	1	yes	characteristic
113	#168	#117	1	1	yes	characteristic
114	C₂×M₁₆	—	5	2	no	—
115	C₂²×C₈	—	5	1	no	—
116	C₂²×C₈	—	5	1	no	—
117	#117	—	5	2	no	—
118	C₂×C₄²	—	5	1	no	—
119	M₃₂	—	5	2	no	—
120	M₃₂	—	10	2	no	—
121	M₃₂	C₂×AGL(1,𝔽₅)	1	2	yes	—
122	C₂×C₁₆	—	5	2	no	—
123	C₂×C₁₆	—	5	2	no	—
124	C₂×C₁₆	—	5	2	no	—
125	C₂×C₁₆	—	5	2	no	—
126	C₂×C₁₆	—	10	2	no	—
127	C₂×C₁₆	—	10	1	no	—
128	C₂×C₁₆	—	10	1	no	—
129	#108	—	5	2	no	—
130	#108	—	5	2	no	—
131	#107	—	5	2	no	—
132	#106	—	5	2	no	—
133	#97	—	10	2	no	—
134	C₄×C₈	—	5	1	no	—
135	C₄×C₈	—	5	1	no	—
136	C₄×C₈	—	5	1	no	—
137	C₄×C₈	—	10	2	no	—
138	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₂²×C₅	#405	1	1	yes	characteristic
140	C₂×D₅	—	2	2	no	—
141	C₂×D₅	C₈×D₄	1	1	yes	characteristic
142	C₂×D₅	C₈×Q₈	1	1	yes	characteristic
143	AGL(1,𝔽₅)	—	4	4	no	—
144	C₄×C₅	—	4	1	no	—
145	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
147	Dic₅	—	4	1	no	—
148	Dic₅	C₈×D₄	1	1	yes	characteristic
149	Dic₅	C₈×Q₈	1	1	yes	characteristic
150	C₂²×C₄	—	5	1	no	—
151	C₂²×C₄	—	10	1	no	—
152	M₁₆	—	5	4	no	—
153	M₁₆	—	10	2	no	—
154	C₂×C₈	—	5	1	no	—
155	C₂×C₈	—	5	1	no	—
156	C₂×C₈	—	5	1	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	—	10	2	no	—
165	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₄⋊C₄	—	5	4	no	—
168	C₂²⋊C₄	—	10	2	no	—
169	C₄²	—	5	1	no	—
170	C₄²	—	10	1	no	—
171	C₄²	—	10	2	no	—
172	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₁₆	—	2	2	no	—
174	C₁₆	—	5	4	no	—
175	C₁₆	—	10	2	no	—
176	C₁₆	—	10	2	no	—
177	C₂×C₅	—	2	1	no	—
178	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
179	D₅	—	2	1	no	—
180	D₅	#2171	1	2	yes	—
181	C₂³	—	5	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	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	—	10	4	no	—
189	C₂×C₄	—	10	1	no	—
190	C₂×C₄	—	20	2	no	—
191	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
192	C₈	—	5	1	no	—
193	C₈	—	5	1	no	—
194	C₈	—	5	1	no	—
195	C₈	—	5	1	no	—
196	C₈	—	10	4	no	—
197	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₈	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
199	C₈	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₈	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₅	#12665	1	1	yes	characteristic, complemented, Sylow
202	C₂²	—	5	1	no	—
203	C₂²	—	5	1	no	—
204	C₂²	—	10	2	no	—
205	C₂²	#65728	1	1	yes	characteristic
206	C₄	—	4	1	no	—
207	C₄	—	5	1	no	—
208	C₄	—	5	1	no	—
209	C₄	—	20	1	no	—
210	C₄	—	20	4	no	—
211	C₄	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	5	2	no	—
215	C₂	—	10	1	no	—
216	C₂	#10604033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604113

C₂

yes

maximal

#10603851

C₂

yes

maximal, characteristic

#10603823

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

#64886

C₄

yes

complemented

#64862

C₄

yes

—

#64859

C₄

yes

complemented

#64844

—

C₁₆×AGL(1,𝔽₅)

—

C₅×#364

C₄

yes

characteristic, complemented

#64731

C₄

yes

complemented

#64683

C₄

yes

characteristic, complemented

#12665

—

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

—

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×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

—

C₅×C₁₆

—

#672

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

—

—

yes

characteristic, complemented

#346

—

#346

—

#346

—

C₄×C₁₆

—

#339

—

C₂²×D₅

#131

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

#131

yes

characteristic

100

C₄×D₅

—

101

C₄×D₅

C₄×D₄

yes

characteristic

102

C₄×D₅

C₄×Q₈

yes

characteristic

103

C₄×D₅

C₂²×C₈

yes

characteristic

104

C₄×D₅

C₂²×C₈

yes

characteristic

105

#170

—

106

C₅×C₈

C₄×C₈

yes

characteristic

107

C₅×C₈

C₄×C₈

yes

characteristic

108

C₅×C₈

C₄×D₄

yes

characteristic

109

C₅×C₈

C₄×Q₈

yes

characteristic

110

#168

C₄×C₈

yes

characteristic

111

#168

C₄×C₈

yes

characteristic

112

#168

C₂×C₄⋊C₄

yes

113

yes

114

C₂×M₁₆

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

#117

—

118

C₂×C₄²

—

119

M₃₂

—

120

M₃₂

—

121

M₃₂

C₂×AGL(1,𝔽₅)

yes

—

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

#108

—

130

#108

—

131

#107

—

132

#106

—

133

#97

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

139

C₂²×C₅

#405

yes

characteristic

140

C₂×D₅

—

141

C₂×D₅

C₈×D₄

yes

characteristic

142

C₂×D₅

C₈×Q₈

yes

characteristic

143

AGL(1,𝔽₅)

—

144

C₄×C₅

—

145

C₄×C₅

C₄×C₄⋊C₄

yes

characteristic

146

C₄×C₅

C₂×C₄×C₈

yes

characteristic

147

Dic₅

—

148

Dic₅

C₈×D₄

yes

characteristic

149

Dic₅

C₈×Q₈

yes

characteristic

150

C₂²×C₄

—

151

C₂²×C₄

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

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

C₄×AGL(1,𝔽₅)

yes

characteristic

166

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

167

C₄⋊C₄

—

168

C₂²⋊C₄

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

C₄×AGL(1,𝔽₅)

yes

characteristic

173

C₁₆

—

174

C₁₆

—

175

C₁₆

—

176

C₁₆

—

177

C₂×C₅

—

178

C₂×C₅

C₈×C₄⋊C₄

yes

characteristic

179

D₅

—

180

D₅

#2171

yes

—

181

C₂³

—

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

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

C₈×AGL(1,𝔽₅)

yes

characteristic

198

C₈

C₈×AGL(1,𝔽₅)

yes

characteristic

199

C₈

D₄×AGL(1,𝔽₅)

yes

characteristic

200

C₈

Q₈×AGL(1,𝔽₅)

yes

characteristic

201

C₅

#12665

yes

characteristic, complemented, Sylow

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

#65728

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

C₄⋊C₄×AGL(1,𝔽₅)

yes

characteristic

212

C₄

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

213

C₂

—

214

C₂

—

215

C₂

—

216

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	#12665
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. #64886	C₄	semidirect
15. #64859	C₄	semidirect
18. C₅×#364	C₄	semidirect
19. #64731	C₄	semidirect
20. #64683	C₄	semidirect
88. #364	AGL(1,𝔽₅)	semidirect
201. C₅	#12665	semidirect

H ≅ G/N

type

13. #64886

C₄

semidirect

15. #64859

C₄

semidirect

18. C₅×#364

C₄

semidirect

19. #64731

C₄

semidirect

20. #64683

C₄

semidirect

88. #364

AGL(1,𝔽₅)

semidirect

201. C₅

#12665

semidirect

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

subgroup	index
#10604113	2
#10603851	2
#10603823	2
D₅×#364	2
#10598673	2
#12665	5

subgroup

index

Subgroups — #13026005

Conjugacy classes of subgroups

Complements

Maximal subgroups