Group #13031477 — subgroups

Subgroups — #13031477

Order 1280.

702 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 270 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	#10617185	C₂	1	2	yes	maximal
2	C₈×#713	C₂	1	1	yes	maximal, characteristic
3	#10604016	C₂	1	6	yes	maximal, complemented
4	#10603819	C₂	1	3	yes	maximal, complemented
5	D₅⋊#335	C₂	1	3	yes	maximal, complemented
6	C₂×#4133	—	2	2	no	—
7	C₄×#713	C₂²	1	1	yes	characteristic
8	#65708	—	2	6	no	—
9	#65681	C₂²	1	6	yes	—
10	D₅⋊(C₄⋊C₈)	C₂²	1	6	yes	—
11	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
12	Q₈×#168	C₄	1	1	yes	characteristic
13	#65123	C₄	1	3	yes	—
14	#65122	C₂²	1	3	yes	—
15	C₈×D₂₀	C₄	1	3	yes	—
16	C₄×C₈×D₅	C₂²	1	3	yes	—
17	#64884	C₂²	1	1	yes	characteristic
18	#64884	C₂²	1	3	yes	—
19	#64884	C₂²	1	6	yes	—
20	#64882	C₂²	1	6	yes	—
21	#17828	—	5	1	no	maximal, Sylow
22	C₂×#713	D₄	1	1	yes	characteristic
23	#4133	—	4	8	no	—
24	C₂×#700	—	2	6	no	—
25	C₂×D₅⋊C₈	—	2	6	no	—
26	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
27	Q₈×Dic₅	Q₈	1	1	yes	characteristic
28	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
29	C₂×C₈×D₅	C₂×C₄	1	3	yes	—
30	#4039	Q₈	1	3	yes	—
31	#4038	D₄	1	3	yes	—
32	C₄×D₂₀	C₂×C₄	1	3	yes	—
33	C₄²×D₅	C₂³	1	3	yes	—
34	#4004	—	2	6	no	—
35	C₄⋊#170	—	2	6	no	—
36	C₄⋊#170	C₂³	1	2	yes	—
37	C₄⋊#170	C₂³	1	6	yes	—
38	C₅×C₄⋊C₈	C₂×C₄	1	3	yes	—
39	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
40	#3952	C₂×C₄	1	6	yes	—
41	C₈×Dic₅	C₂³	1	3	yes	—
42	C₈×Dic₅	C₂³	1	1	yes	characteristic
43	C₄⋊#168	C₂×C₄	1	3	yes	—
44	C₄×#168	C₂×C₄	1	3	yes	—
45	C₈×#41	—	5	1	no	—
46	#2906	—	5	2	no	—
47	#1571	—	5	6	no	—
48	#1567	—	5	3	no	—
49	C₂×#335	—	5	3	no	—
50	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
51	#713	D₈	1	4	yes	—
52	#713	Q₁₆	1	4	yes	—
53	C₂×D₂₀	C₄⋊C₄	1	3	yes	—
54	C₂×C₄×D₅	C₂²×C₄	1	3	yes	—
55	C₂×C₄×D₅	C₂×D₄	1	3	yes	—
56	C₂×C₄×D₅	C₂×Q₈	1	3	yes	—
57	C₂×#170	—	2	2	no	—
58	C₂×#170	—	2	6	no	—
59	#700	—	4	12	no	—
60	D₅⋊C₈	—	4	12	no	—
61	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₅×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
64	C₄²×C₅	C₂²×C₄	1	3	yes	—
65	#685	C₄⋊C₄	1	6	yes	—
66	C₄⋊Dic₅	C₄⋊C₄	1	3	yes	—
67	C₄×Dic₅	C₂×D₄	1	3	yes	—
68	C₄×Dic₅	C₂×Q₈	1	3	yes	—
69	C₄×Dic₅	C₂⁴	1	1	yes	characteristic
70	C₂×#168	C₂²×C₄	1	3	yes	—
71	C₂×#168	C₂²×C₄	1	1	yes	characteristic
72	C₈×D₅	—	2	6	no	—
73	C₈×D₅	—	4	3	no	—
74	C₂×#131	—	10	2	no	—
75	C₄×#41	—	5	1	no	—
76	C₈×Q₈	—	5	1	no	—
77	C₈×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₈×D₄	—	5	3	no	—
79	C₈×D₄	—	5	3	no	—
80	#431	—	5	3	no	—
81	#424	—	10	6	no	—
82	#423	—	5	6	no	—
83	C₂×C₄⋊C₈	—	5	6	no	—
84	C₂×C₄×C₈	—	5	3	no	—
85	#335	—	5	1	no	—
86	#335	—	5	3	no	—
87	#335	—	5	6	no	—
88	#335	—	5	6	no	—
89	C₂²×D₅	C₂×C₄⋊C₄	1	3	yes	—
90	C₅×Q₈	#107	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
94	C₂×Dic₅	C₂×C₄⋊C₄	1	3	yes	—
95	C₂×Dic₅	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₅	C₂²×Q₈	1	1	yes	characteristic
97	D₂₀	#107	1	12	yes	—
98	C₄×D₅	—	2	3	no	—
99	C₄×D₅	—	2	3	no	—
100	C₄×D₅	—	4	3	no	—
101	C₄×D₅	C₂×D₈	1	6	yes	—
102	C₄×D₅	C₂×Q₁₆	1	6	yes	—
103	#170	—	4	2	no	—
104	#170	—	4	6	no	—
105	C₅×C₈	—	2	3	no	—
106	C₅×C₈	—	2	1	no	—
107	#168	—	2	3	no	—
108	#168	#131	1	2	yes	complemented
109	C₂×#41	—	5	1	no	—
110	#131	—	20	8	no	—
111	C₂×M₁₆	—	10	6	no	—
112	C₂²×C₈	—	5	3	no	—
113	C₂²×C₈	—	5	3	no	—
114	C₂²×C₈	—	10	6	no	—
115	C₄×Q₈	—	5	1	no	—
116	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₄×D₄	—	5	3	no	—
118	C₄×D₄	—	5	3	no	—
119	#117	—	5	3	no	—
120	C₂×C₄²	—	5	3	no	—
121	C₄⋊C₈	—	5	2	no	—
122	C₄⋊C₈	—	5	6	no	—
123	C₄⋊C₈	—	5	3	no	—
124	C₄⋊C₈	—	10	6	no	—
125	C₄⋊C₈	—	10	6	no	—
126	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
127	C₂²⋊C₈	—	5	6	no	—
128	C₄×C₈	—	5	1	no	—
129	C₄×C₈	—	5	3	no	—
130	C₄×C₈	—	5	3	no	—
131	C₄×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	—
132	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂×D₅	—	2	3	no	—
134	C₂×D₅	—	2	3	no	—
135	C₂×D₅	C₂×#107	1	6	yes	—
136	C₄×C₅	—	2	3	no	—
137	C₄×C₅	C₂×#107	1	6	yes	—
138	C₄×C₅	C₂×#131	1	1	yes	characteristic
139	C₄×C₅	#568	1	1	yes	characteristic
140	Dic₅	—	2	3	no	—
141	Dic₅	—	2	1	no	—
142	Dic₅	C₂²×D₈	1	1	yes	characteristic
143	Dic₅	C₂²×Q₁₆	1	1	yes	characteristic
144	#41	—	5	4	no	—
145	#41	—	5	4	no	—
146	C₂×Q₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×D₄	—	5	3	no	—
148	C₂²×C₄	—	5	3	no	—
149	C₂²×C₄	—	5	3	no	—
150	C₂²×C₄	—	5	3	no	—
151	M₁₆	—	20	12	no	—
152	C₂×C₈	—	5	1	no	—
153	C₂×C₈	—	5	3	no	—
154	C₂×C₈	—	10	6	no	—
155	C₂×C₈	—	10	2	no	—
156	C₂×C₈	—	10	6	no	—
157	C₂×C₈	—	20	3	no	—
158	C₂×C₈	—	20	12	no	—
159	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
161	C₄⋊C₄	—	5	3	no	—
162	C₄⋊C₄	C₄⋊AGL(1,𝔽₅)	1	3	yes	—
163	C₂²⋊C₄	—	5	6	no	—
164	C₄²	—	5	3	no	—
165	C₄²	—	5	3	no	—
166	C₄²	—	5	1	no	—
167	C₄²	C₂²×AGL(1,𝔽₅)	1	3	yes	—
168	C₂×C₅	#2906	1	1	yes	characteristic
169	C₂×C₅	C₂²×#107	1	1	yes	characteristic
170	C₂×C₅	#2916	1	1	yes	characteristic
171	D₅	—	2	6	no	—
172	C₂³	—	5	3	no	—
173	Q₈	#3994	1	4	yes	—
174	D₄	—	5	12	no	—
175	C₂×C₄	—	5	1	no	—
176	C₂×C₄	—	5	6	no	—
177	C₂×C₄	—	5	6	no	—
178	C₂×C₄	—	5	1	no	—
179	C₂×C₄	—	5	3	no	—
180	C₂×C₄	—	10	3	no	—
181	C₂×C₄	—	10	3	no	—
182	C₂×C₄	—	20	3	no	—
183	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
184	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
185	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	3	no	—
188	C₈	—	5	2	no	—
189	C₈	—	10	3	no	—
190	C₈	—	20	2	no	—
191	C₈	—	20	6	no	—
192	C₅	#17828	1	1	yes	characteristic, complemented, Sylow
193	C₂²	—	5	6	no	—
194	C₂²	—	10	3	no	—
195	C₂²	—	10	3	no	—
196	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₄	—	2	3	no	—
198	C₄	—	5	1	no	—
199	C₄	—	5	1	no	—
200	C₄	—	10	1	no	—
201	C₄	—	10	3	no	—
202	C₄	C₂×#3994	1	6	yes	—
203	C₄	#66260	1	1	yes	characteristic
204	C₄	D₅⋊#131	1	1	yes	characteristic
205	C₂	—	10	6	no	—
206	C₂	#10617186	1	1	yes	characteristic
207	C₂	C₂²×#3994	1	1	yes	characteristic
208	C₂	#10617418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10617185

C₂

yes

maximal

C₈×#713

C₂

yes

maximal, characteristic

#10604016

C₂

yes

maximal, complemented

#10603819

C₂

yes

maximal, complemented

D₅⋊#335

C₂

yes

maximal, complemented

C₂×#4133

—

C₄×#713

C₂²

yes

characteristic

#65708

—

#65681

C₂²

yes

—

D₅⋊(C₄⋊C₈)

C₂²

yes

—

C₅×C₈×Q₈

C₄

yes

characteristic

Q₈×#168

C₄

yes

characteristic

#65123

C₄

yes

—

#65122

C₂²

yes

—

C₈×D₂₀

C₄

yes

—

C₄×C₈×D₅

C₂²

yes

—

#64884

C₂²

yes

characteristic

#64884

C₂²

yes

—

#64884

C₂²

yes

—

#64882

C₂²

yes

—

#17828

—

maximal, Sylow

C₂×#713

D₄

yes

characteristic

#4133

—

C₂×#700

—

C₂×D₅⋊C₈

—

C₄×C₅×Q₈

C₂×C₄

yes

characteristic

Q₈×Dic₅

Q₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

#4039

Q₈

yes

—

#4038

D₄

yes

—

C₄×D₂₀

C₂×C₄

yes

—

C₄²×D₅

C₂³

yes

—

#4004

—

C₄⋊#170

—

C₄⋊#170

C₂³

yes

—

C₄⋊#170

C₂³

yes

—

C₅×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

#3952

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₄⋊#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

—

C₈×#41

—

—

—

—

—

C₂×C₅×Q₈

C₄⋊C₄

yes

characteristic

#713

D₈

yes

—

#713

Q₁₆

yes

—

C₂×D₂₀

C₄⋊C₄

yes

—

C₂×C₄×D₅

C₂²×C₄

yes

—

C₂×C₄×D₅

C₂×D₄

yes

—

C₂×C₄×D₅

C₂×Q₈

yes

—

C₂×#170

—

C₂×#170

—

#700

—

D₅⋊C₈

—

C₂×C₅×C₈

C₂²×C₄

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

—

#685

C₄⋊C₄

yes

—

C₄⋊Dic₅

C₄⋊C₄

yes

—

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

—

C₄×Dic₅

C₂⁴

yes

characteristic

C₂×#168

C₂²×C₄

yes

—

C₂×#168

C₂²×C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₂×#131

—

C₄×#41

—

C₈×Q₈

—

C₈×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₈×D₄

—

C₈×D₄

—

#431

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

—

—

—

—

C₂²×D₅

C₂×C₄⋊C₄

yes

—

C₅×Q₈

#107

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂³×C₄

yes

characteristic

C₂×Dic₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

C₂²×D₄

yes

characteristic

C₂×Dic₅

C₂²×Q₈

yes

characteristic

D₂₀

#107

yes

—

C₄×D₅

—

C₄×D₅

—

100

C₄×D₅

—

101

C₄×D₅

C₂×D₈

yes

—

102

C₄×D₅

C₂×Q₁₆

yes

—

103

#170

—

104

#170

—

105

C₅×C₈

—

106

C₅×C₈

—

107

#168

—

108

#168

#131

yes

complemented

109

C₂×#41

—

110

#131

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂²×C₈

—

115

C₄×Q₈

—

116

C₄×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

117

C₄×D₄

—

118

C₄×D₄

—

119

#117

—

120

C₂×C₄²

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

127

C₂²⋊C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

132

C₂²×C₅

C₂²×C₄⋊C₄

yes

characteristic

133

C₂×D₅

—

134

C₂×D₅

—

135

C₂×D₅

C₂×#107

yes

—

136

C₄×C₅

—

137

C₄×C₅

C₂×#107

yes

—

138

C₄×C₅

C₂×#131

yes

characteristic

139

C₄×C₅

#568

yes

characteristic

140

Dic₅

—

141

Dic₅

—

142

Dic₅

C₂²×D₈

yes

characteristic

143

Dic₅

C₂²×Q₁₆

yes

characteristic

144

#41

—

145

#41

—

146

C₂×Q₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

147

C₂×D₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

160

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

—

161

C₄⋊C₄

—

162

C₄⋊C₄

C₄⋊AGL(1,𝔽₅)

yes

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

C₂²×AGL(1,𝔽₅)

yes

—

168

C₂×C₅

#2906

yes

characteristic

169

C₂×C₅

C₂²×#107

yes

characteristic

170

C₂×C₅

#2916

yes

characteristic

171

D₅

—

172

C₂³

—

173

Q₈

#3994

yes

—

174

D₄

—

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

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

184

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

185

C₂×C₄

C₂³×AGL(1,𝔽₅)

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

192

C₅

#17828

yes

characteristic, complemented, Sylow

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

C₂²×C₄⋊AGL(1,𝔽₅)

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

C₂×#3994

yes

—

203

C₄

#66260

yes

characteristic

204

C₄

D₅⋊#131

yes

characteristic

205

C₂

—

206

C₂

#10617186

yes

characteristic

207

C₂

C₂²×#3994

yes

characteristic

208

C₂

#10617418

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×C₈×Q₈	C₄
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂×#713	D₄
cosocle	C₄×Dic₅	C₂⁴
hypercenter	C₈×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#17828
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂³×C₄

Frattini

C₂×C₄

C₂³×AGL(1,𝔽₅)

Fitting

C₅×C₈×Q₈

C₄

socle

C₂²×C₅

C₂²×C₄⋊C₄

prime radical

C₂×#713

D₄

cosocle

C₄×Dic₅

C₂⁴

C₈×Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
3. #10604016	C₂	semidirect
4. #10603819	C₂	semidirect
5. D₅⋊#335	C₂	semidirect
108. #168	#131	semidirect
192. C₅	#17828	semidirect

H ≅ G/N

type

3. #10604016

C₂

semidirect

4. #10603819

C₂

semidirect

5. D₅⋊#335

C₂

semidirect

108. #168

#131

semidirect

192. C₅

#17828

semidirect

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

subgroup	index
#10617185	2
C₈×#713	2
#10604016	2
#10603819	2
D₅⋊#335	2
#17828	5

subgroup

index

Subgroups — #13031477

Conjugacy classes of subgroups

Complements

Maximal subgroups