Group #13017380 — subgroups

Subgroups — #13017380

Order 1280.

358 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 136 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	#10604009	C₂	1	2	yes	maximal
2	C₄×#3996	C₂	1	1	yes	maximal, characteristic
3	#10603807	C₂	1	1	yes	maximal, characteristic
4	#10599320	C₂	1	1	yes	maximal, characteristic
5	#10598588	C₂	1	2	yes	maximal
6	C₂×#3996	C₄	1	2	yes	—
7	C₄×#700	C₂²	1	2	yes	—
8	C₄×D₅⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×D₅⋊C₈	C₂²	1	1	yes	characteristic
10	#65122	C₂²	1	2	yes	—
11	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
12	#64896	C₄	1	2	yes	—
13	#170⋊C₈	—	2	2	no	—
14	#64879	C₄	1	2	yes	—
15	#64879	C₄	1	2	yes	—
16	C₈×#170	—	2	2	no	—
17	C₅×#334	C₄	1	1	yes	characteristic
18	#64699	C₄	1	2	yes	—
19	#64677	C₄	1	1	yes	characteristic
20	#8294	—	5	1	no	maximal, Sylow
21	C₂×#700	C₂×C₄	1	4	yes	—
22	C₂×D₅⋊C₈	—	2	1	no	—
23	C₂×D₅⋊C₈	Q₈	1	2	yes	—
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₅	C₂³	1	1	yes	characteristic
27	#4002	—	2	2	no	—
28	C₄×#170	—	2	2	no	—
29	C₄×#170	—	2	1	no	—
30	C₄×#170	C₂×C₄	1	2	yes	—
31	C₄×#170	C₂×C₄	1	2	yes	—
32	C₄×#170	D₄	1	2	yes	—
33	#3996	C₈	1	8	yes	complemented
34	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	#3952	—	2	2	no	—
37	C₈×Dic₅	—	2	2	no	—
38	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#168	C₂×C₄	1	2	yes	—
41	C₄×#168	C₂×C₄	1	1	yes	characteristic
42	C₄×#108	—	5	1	no	—
43	#1564	—	5	1	no	—
44	C₈×M₁₆	—	5	2	no	—
45	C₂×#322	—	5	1	no	—
46	#1278	—	5	2	no	—
47	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×D₅	#41	1	1	yes	characteristic
50	C₂×#170	—	2	4	no	—
51	C₂×#170	—	2	2	no	—
52	C₂×#170	C₄⋊C₄	1	4	yes	—
53	#700	—	2	4	no	—
54	#700	C₂×C₈	1	8	yes	—
55	D₅⋊C₈	—	4	2	no	—
56	D₅⋊C₈	—	4	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
59	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
60	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
62	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
63	C₄×Dic₅	#41	1	1	yes	characteristic
64	C₂×#168	—	2	2	no	—
65	C₂×#168	C₄²	1	1	yes	characteristic
66	C₂×#168	C₄²	1	1	yes	characteristic
67	C₈×D₅	—	2	2	no	—
68	C₈×D₅	C₂×C₈	1	4	yes	—
69	#431	—	5	2	no	—
70	C₂×#108	—	5	2	no	—
71	C₄×M₁₆	—	5	2	no	—
72	C₂×C₄×C₈	—	5	1	no	—
73	C₂×C₄×C₈	—	5	1	no	—
74	C₂×C₄×C₈	—	5	1	no	—
75	#341	—	5	2	no	—
76	#335	—	5	2	no	—
77	#334	—	5	1	no	—
78	#334	AGL(1,𝔽₅)	1	1	yes	characteristic
79	#322	—	5	2	no	—
80	#322	—	5	2	no	—
81	#322	—	10	2	no	—
82	C₈²	—	10	2	no	—
83	C₂²×D₅	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
87	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×Dic₅	#117	1	1	yes	characteristic
89	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
90	C₄×D₅	—	2	2	no	—
91	C₄×D₅	—	2	2	no	—
92	C₄×D₅	C₂²×C₈	1	2	yes	—
93	C₄×D₅	#131	1	2	yes	—
94	#170	—	2	4	no	—
95	#170	—	2	8	no	—
96	#170	—	4	2	no	—
97	C₅×C₈	—	2	1	no	—
98	C₅×C₈	—	4	2	no	—
99	C₅×C₈	C₄×C₈	1	2	yes	—
100	#168	—	2	1	no	—
101	#168	—	4	2	no	—
102	#168	C₄×C₈	1	2	yes	—
103	C₂×M₁₆	—	5	4	no	—
104	C₂²×C₈	—	5	1	no	—
105	C₂²×C₈	—	5	2	no	—
106	C₂²×C₈	—	5	1	no	—
107	C₂²×C₈	—	10	1	no	—
108	C₂×C₄²	—	5	1	no	—
109	#108	—	5	8	no	—
110	C₄⋊C₈	—	5	2	no	—
111	C₄⋊C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
112	C₂²⋊C₈	—	10	2	no	—
113	#97	—	10	2	no	—
114	C₄×C₈	—	5	2	no	—
115	C₄×C₈	—	5	2	no	—
116	C₄×C₈	—	5	1	no	—
117	C₄×C₈	—	5	1	no	—
118	C₄×C₈	—	5	2	no	—
119	C₄×C₈	—	5	1	no	—
120	C₄×C₈	—	10	2	no	—
121	C₄×C₈	—	10	2	no	—
122	C₄×C₈	—	10	1	no	—
123	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
125	C₂×D₅	—	2	1	no	—
126	C₂×D₅	—	2	1	no	—
127	C₂×D₅	C₈×Q₈	1	2	yes	—
128	C₄×C₅	—	2	1	no	—
129	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₅	#405	1	1	yes	characteristic
131	C₄×C₅	#444	1	2	yes	—
132	Dic₅	—	2	1	no	—
133	Dic₅	#430	1	2	yes	—
134	Dic₅	C₈×D₄	1	2	yes	—
135	C₂²×C₄	—	5	1	no	—
136	C₂²×C₄	—	5	1	no	—
137	C₂²×C₄	—	5	1	no	—
138	M₁₆	—	5	8	no	—
139	M₁₆	—	10	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	5	1	no	—
142	C₂×C₈	—	5	4	no	—
143	C₂×C₈	—	5	4	no	—
144	C₂×C₈	—	5	1	no	—
145	C₂×C₈	—	10	2	no	—
146	C₂×C₈	—	10	4	no	—
147	C₂×C₈	—	10	2	no	—
148	C₂×C₈	—	10	2	no	—
149	C₂×C₈	—	20	2	no	—
150	C₂×C₈	—	20	2	no	—
151	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₄²	—	5	1	no	—
154	C₄²	—	5	1	no	—
155	C₄²	—	5	1	no	—
156	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₅	#1773	1	1	yes	characteristic
159	C₂×C₅	#1778	1	1	yes	characteristic
160	D₅	—	2	2	no	—
161	C₂³	—	5	1	no	—
162	C₂×C₄	—	5	1	no	—
163	C₂×C₄	—	5	1	no	—
164	C₂×C₄	—	5	2	no	—
165	C₂×C₄	—	5	2	no	—
166	C₂×C₄	—	5	1	no	—
167	C₂×C₄	—	10	2	no	—
168	C₂×C₄	—	10	2	no	—
169	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₈	—	2	1	no	—
173	C₈	—	4	2	no	—
174	C₈	—	5	2	no	—
175	C₈	—	10	4	no	—
176	C₈	—	10	8	no	—
177	C₈	—	10	1	no	—
178	C₈	—	20	2	no	—
179	C₈	—	20	2	no	—
180	C₈	C₈×AGL(1,𝔽₅)	1	2	yes	—
181	C₅	#8294	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	5	2	no	—
183	C₂²	—	10	1	no	—
184	C₂²	—	10	1	no	—
185	C₂²	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	5	2	no	—
188	C₄	—	5	2	no	—
189	C₄	—	10	1	no	—
190	C₄	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
191	C₄	#65728	1	1	yes	characteristic
192	C₄	#65737	1	2	yes	—
193	C₂	—	10	2	no	—
194	C₂	#10604033	1	1	yes	characteristic
195	C₂	#10604054	1	1	yes	characteristic
196	C₂	#10604086	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604009

C₂

yes

maximal

C₄×#3996

C₂

yes

maximal, characteristic

#10603807

C₂

yes

maximal, characteristic

#10599320

C₂

yes

maximal, characteristic

#10598588

C₂

yes

maximal

C₂×#3996

C₄

yes

—

C₄×#700

C₂²

yes

—

C₄×D₅⋊C₈

C₂²

yes

characteristic

C₄×D₅⋊C₈

C₂²

yes

characteristic

#65122

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64896

C₄

yes

—

#170⋊C₈

—

#64879

C₄

yes

—

#64879

C₄

yes

—

C₈×#170

—

C₅×#334

C₄

yes

characteristic

#64699

C₄

yes

—

#64677

C₄

yes

—

C₂×C₄

yes

—

C₂×D₅⋊C₈

—

C₂×D₅⋊C₈

Q₈

yes

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₄²×D₅

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₄×#170

C₂×C₄

yes

—

C₄×#170

D₄

yes

—

#3996

C₈

yes

complemented

C₅×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3952

—

C₈×Dic₅

—

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₄⋊#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#108

—

#1564

—

C₈×M₁₆

—

C₂×#322

—

#1278

—

C₂×C₄×D₅

C₂²×C₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

C₂×C₄×D₅

yes

—

—

C₄⋊C₄

yes

—

#700

—

#700

C₂×C₈

yes

—

D₅⋊C₈

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄²

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

C₄×Dic₅

yes

—

C₄²

yes

characteristic

C₂×#168

C₄²

yes

characteristic

C₈×D₅

—

C₈×D₅

C₂×C₈

yes

—

#431

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

yes

—

—

—

C₈²

—

C₂²×D₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₄×Q₈

yes

characteristic

C₂×Dic₅

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅

#117

yes

characteristic

C₂×Dic₅

C₄×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂²×C₈

yes

—

C₄×D₅

#131

yes

—

#170

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₄×C₈

yes

—

100

#168

—

101

#168

—

102

#168

C₄×C₈

yes

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₂²×C₈

—

108

C₂×C₄²

—

109

#108

—

110

C₄⋊C₈

—

111

C₄⋊C₈

C₂×AGL(1,𝔽₅)

yes

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₄×C₈

—

123

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

124

C₂²×C₅

C₄×C₄⋊C₄

yes

characteristic

125

C₂×D₅

—

126

C₂×D₅

—

127

C₂×D₅

C₈×Q₈

yes

—

128

C₄×C₅

—

129

C₄×C₅

C₂×C₄×C₈

yes

characteristic

130

C₄×C₅

#405

yes

characteristic

131

C₄×C₅

#444

yes

—

132

Dic₅

—

133

Dic₅

#430

yes

—

134

Dic₅

C₈×D₄

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

M₁₆

—

139

M₁₆

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

152

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

157

C₂×C₅

C₈×C₄⋊C₄

yes

characteristic

158

C₂×C₅

#1773

yes

characteristic

159

C₂×C₅

#1778

yes

characteristic

160

D₅

—

161

C₂³

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

170

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

171

C₂×C₄

Q₈×AGL(1,𝔽₅)

yes

characteristic

172

C₈

—

173

C₈

—

174

C₈

—

175

C₈

—

176

C₈

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

C₈×AGL(1,𝔽₅)

yes

—

181

C₅

#8294

yes

characteristic, complemented, Sylow

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

C₄⋊C₄×AGL(1,𝔽₅)

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

C₂×C₈×AGL(1,𝔽₅)

yes

characteristic

191

C₄

#65728

yes

characteristic

192

C₄

#65737

yes

—

193

C₂

—

194

C₂

#10604033

yes

characteristic

195

C₂

#10604054

yes

characteristic

196

C₂

#10604086

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#334	C₄
socle	C₂²×C₅	C₄×C₄⋊C₄
prime radical	C₂²×D₅	C₄×Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	#334	AGL(1,𝔽₅)
hypocenter	C₅	#8294
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₂×C₄×C₈

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#334

C₄

socle

C₂²×C₅

C₄×C₄⋊C₄

prime radical

C₂²×D₅

C₄×Q₈

cosocle

C₄²×D₅

C₂³

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
33. #3996	C₈	semidirect
181. C₅	#8294	semidirect

H ≅ G/N

type

33. #3996

C₈

semidirect

181. C₅

#8294

semidirect

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

subgroup	index
#10604009	2
C₄×#3996	2
#10603807	2
#10599320	2
#10598588	2
#8294	5

subgroup

index

Subgroups — #13017380

Conjugacy classes of subgroups

Complements

Maximal subgroups