Group #13017384 — subgroups

Subgroups — #13017384

Order 1280.

334 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₄×#3995	C₂	1	1	yes	maximal, characteristic
2	#10603815	C₂	1	1	yes	maximal, characteristic
3	#10603810	C₂	1	2	yes	maximal
4	D₅×#322	C₂	1	1	yes	maximal, characteristic
5	#10598531	C₂	1	2	yes	maximal
6	C₂×#3995	C₄	1	2	yes	—
7	#65680	C₂²	1	2	yes	—
8	C₄×#700	C₂²	1	2	yes	—
9	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
12	#64896	C₄	1	2	yes	—
13	#64883	C₄	1	2	yes	—
14	#64882	C₄	1	2	yes	—
15	#170⋊C₈	—	2	2	no	—
16	#64879	—	2	2	no	—
17	C₅×#322	C₄	1	1	yes	characteristic
18	#64675	C₄	1	1	yes	characteristic
19	#64674	C₄	1	1	yes	characteristic
20	#64674	C₄	1	1	yes	characteristic
21	#8295	—	5	1	no	maximal, Sylow
22	C₂×#700	C₂×C₄	1	4	yes	—
23	C₂×C₈×D₅	—	2	1	no	—
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₅	Q₈	1	2	yes	—
27	C₄²×D₅	C₂³	1	1	yes	characteristic
28	#4003	—	2	2	no	—
29	#4002	—	2	2	no	—
30	C₄⋊#170	C₂×C₄	1	4	yes	—
31	C₄×#170	—	2	2	no	—
32	C₄×#170	—	2	2	no	—
33	#3995	—	2	4	no	—
34	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₈×Dic₅	—	2	1	no	—
38	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₅	D₄	1	2	yes	—
41	C₄×#168	C₂×C₄	1	1	yes	characteristic
42	C₄×#168	C₂×C₄	1	1	yes	characteristic
43	C₄×#168	C₂×C₄	1	1	yes	characteristic
44	C₄×#108	—	5	1	no	—
45	#1564	—	5	1	no	—
46	#1451	—	5	2	no	—
47	C₂×#322	—	5	1	no	—
48	#1278	—	5	2	no	—
49	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×D₅	#41	1	1	yes	characteristic
52	C₂×#170	—	2	4	no	—
53	C₂×#170	—	2	4	no	—
54	#700	—	2	4	no	—
55	#700	—	2	4	no	—
56	C₂×C₅×C₈	—	2	1	no	—
57	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
58	C₂×C₅×C₈	C₄²	1	1	yes	characteristic
59	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
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	1	no	—
65	C₂×#168	C₄²	1	1	yes	characteristic
66	C₂×#168	C₄²	1	1	yes	characteristic
67	C₂×#168	C₄⋊C₄	1	2	yes	—
68	C₈×D₅	—	2	2	no	—
69	C₈×D₅	—	4	2	no	—
70	C₈×D₅	—	4	2	no	—
71	C₈×D₅	M₁₆	1	4	yes	—
72	#431	—	5	2	no	—
73	C₂×#108	—	5	2	no	—
74	C₄×M₁₆	—	5	2	no	—
75	C₂×C₄×C₈	—	5	1	no	—
76	C₂×C₄×C₈	—	5	1	no	—
77	C₂×C₄×C₈	—	5	1	no	—
78	#341	—	5	2	no	—
79	#335	—	5	2	no	—
80	#334	—	5	2	no	—
81	#322	—	5	1	no	—
82	#322	—	5	1	no	—
83	#322	—	5	1	no	—
84	#322	—	10	2	no	—
85	#322	—	10	2	no	—
86	#322	AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂²×D₅	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	#117	1	1	yes	characteristic
91	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
92	C₂×Dic₅	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₅	C₄×Q₈	1	1	yes	characteristic
94	C₄×D₅	—	2	2	no	—
95	C₄×D₅	—	2	2	no	—
96	C₄×D₅	C₂×M₁₆	1	2	yes	—
97	C₄×D₅	#131	1	2	yes	—
98	#170	—	4	4	no	—
99	#170	—	4	4	no	—
100	C₅×C₈	—	2	1	no	—
101	C₅×C₈	—	2	2	no	—
102	C₅×C₈	—	4	1	no	—
103	C₅×C₈	#97	1	2	yes	—
104	#168	—	2	1	no	—
105	#168	—	2	2	no	—
106	#168	—	4	1	no	—
107	#168	#97	1	2	yes	—
108	C₂×M₁₆	—	5	4	no	—
109	C₂²×C₈	—	5	1	no	—
110	C₂²×C₈	—	5	1	no	—
111	C₂²×C₈	—	5	2	no	—
112	C₂²×C₈	—	10	1	no	—
113	C₂×C₄²	—	5	1	no	—
114	#108	—	10	4	no	—
115	C₄⋊C₈	—	5	4	no	—
116	C₂²⋊C₈	—	10	2	no	—
117	#97	—	10	2	no	—
118	C₄×C₈	—	5	1	no	—
119	C₄×C₈	—	5	1	no	—
120	C₄×C₈	—	5	2	no	—
121	C₄×C₈	—	5	1	no	—
122	C₄×C₈	—	5	1	no	—
123	C₄×C₈	—	5	1	no	—
124	C₄×C₈	—	10	1	no	—
125	C₄×C₈	—	10	2	no	—
126	C₄×C₈	—	10	2	no	—
127	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂×D₅	—	2	1	no	—
132	C₂×D₅	—	2	1	no	—
133	C₂×D₅	#446	1	2	yes	—
134	C₄×C₅	—	2	1	no	—
135	C₄×C₅	C₂×#97	1	1	yes	characteristic
136	C₄×C₅	#405	1	1	yes	characteristic
137	C₄×C₅	#430	1	2	yes	—
138	Dic₅	—	2	1	no	—
139	Dic₅	#436	1	2	yes	—
140	Dic₅	#444	1	2	yes	—
141	C₂²×C₄	—	5	1	no	—
142	C₂²×C₄	—	5	1	no	—
143	C₂²×C₄	—	5	1	no	—
144	M₁₆	—	10	4	no	—
145	M₁₆	—	10	4	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	5	1	no	—
148	C₂×C₈	—	5	4	no	—
149	C₂×C₈	—	5	1	no	—
150	C₂×C₈	—	5	2	no	—
151	C₂×C₈	—	10	2	no	—
152	C₂×C₈	—	10	4	no	—
153	C₂×C₈	—	10	4	no	—
154	C₂×C₈	—	10	1	no	—
155	C₂×C₈	—	20	2	no	—
156	C₂×C₈	—	20	2	no	—
157	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
160	C₄²	—	5	1	no	—
161	C₄²	—	5	1	no	—
162	C₄²	—	5	1	no	—
163	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂×C₅	#1770	1	1	yes	characteristic
165	C₂×C₅	#1773	1	1	yes	characteristic
166	C₂×C₅	#1778	1	1	yes	characteristic
167	D₅	—	2	2	no	—
168	C₂³	—	5	1	no	—
169	C₂×C₄	—	5	1	no	—
170	C₂×C₄	—	5	1	no	—
171	C₂×C₄	—	5	2	no	—
172	C₂×C₄	—	5	2	no	—
173	C₂×C₄	—	5	1	no	—
174	C₂×C₄	—	10	2	no	—
175	C₂×C₄	—	10	2	no	—
176	C₂×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₂×C₄	#4130	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	2	2	no	—
181	C₈	—	4	1	no	—
182	C₈	—	5	2	no	—
183	C₈	—	10	1	no	—
184	C₈	—	10	2	no	—
185	C₈	—	20	1	no	—
186	C₈	—	20	4	no	—
187	C₈	—	20	4	no	—
188	C₈	#3992	1	2	yes	—
189	C₅	#8295	1	1	yes	characteristic, complemented, Sylow
190	C₂²	—	5	2	no	—
191	C₂²	—	10	1	no	—
192	C₂²	—	10	1	no	—
193	C₂²	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	5	2	no	—
196	C₄	—	5	2	no	—
197	C₄	—	10	1	no	—
198	C₄	C₂×#3992	1	1	yes	characteristic
199	C₄	#65718	1	1	yes	characteristic
200	C₄	#65729	1	2	yes	—
201	C₂	—	10	2	no	—
202	C₂	#10603842	1	1	yes	characteristic
203	C₂	#10603885	1	1	yes	characteristic
204	C₂	#10603888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#3995

C₂

yes

maximal, characteristic

#10603815

C₂

yes

maximal, characteristic

#10603810

C₂

yes

maximal

D₅×#322

C₂

yes

maximal, characteristic

#10598531

C₂

yes

maximal

C₂×#3995

C₄

yes

—

#65680

C₂²

yes

—

C₄×#700

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

C₄×C₈×D₅

C₂²

yes

characteristic

C₄×C₈×D₅

C₂²

yes

characteristic

#64896

C₄

yes

—

#64883

C₄

yes

—

#64882

C₄

yes

—

#170⋊C₈

—

#64879

—

C₅×#322

C₄

yes

characteristic

#64675

C₄

yes

characteristic

#64674

C₄

yes

characteristic

#64674

C₄

yes

—

C₂×C₄

yes

—

C₂×C₈×D₅

—

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#170

—

C₄×#170

—

#3995

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

—

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

—

—

—

—

C₂×C₄×D₅

C₂²×C₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

C₂×C₄×D₅

yes

—

—

—

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄²

yes

characteristic

C₂×C₅×C₈

C₄⋊C₄

yes

—

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

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

M₁₆

yes

—

#431

—

C₂×#108

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

—

—

—

—

yes

C₂²×D₅

C₄×Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

#117

yes

characteristic

C₂×Dic₅

C₄×D₄

yes

characteristic

C₂×Dic₅

C₄×D₄

yes

characteristic

C₂×Dic₅

C₄×Q₈

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×M₁₆

yes

—

C₄×D₅

#131

yes

—

#170

—

#170

—

100

C₅×C₈

—

101

C₅×C₈

—

102

C₅×C₈

—

103

C₅×C₈

#97

yes

—

104

#168

—

105

#168

—

106

#168

—

107

#168

#97

yes

—

108

C₂×M₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂×C₄²

—

114

#108

—

115

C₄⋊C₈

—

116

C₂²⋊C₈

—

117

#97

—

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

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

128

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

129

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

130

C₂²×C₅

C₄×C₄⋊C₄

yes

characteristic

131

C₂×D₅

—

132

C₂×D₅

—

133

C₂×D₅

#446

yes

—

134

C₄×C₅

—

135

C₄×C₅

C₂×#97

yes

characteristic

136

C₄×C₅

#405

yes

characteristic

137

C₄×C₅

#430

yes

—

138

Dic₅

—

139

Dic₅

#436

yes

—

140

Dic₅

#444

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

M₁₆

—

145

M₁₆

—

146

C₂×C₈

—

147

C₂×C₈

—

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

C₄×AGL(1,𝔽₅)

yes

characteristic

158

C₂×C₈

C₄×AGL(1,𝔽₅)

yes

characteristic

159

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

164

C₂×C₅

#1770

yes

characteristic

165

C₂×C₅

#1773

yes

characteristic

166

C₂×C₅

#1778

yes

characteristic

167

D₅

—

168

C₂³

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

177

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

178

C₂×C₄

#4130

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

#3992

yes

—

189

C₅

#8295

yes

characteristic, complemented, Sylow

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₄×C₄⋊AGL(1,𝔽₅)

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₂×#3992

yes

characteristic

199

C₄

#65718

yes

characteristic

200

C₄

#65729

yes

—

201

C₂

—

202

C₂

#10603842

yes

characteristic

203

C₂

#10603885

yes

characteristic

204

C₂

#10603888

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂²×C₅	C₄×C₄⋊C₄
prime radical	C₂²×D₅	C₄×Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#8295
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#322

C₄

socle

C₂²×C₅

C₄×C₄⋊C₄

prime radical

C₂²×D₅

C₄×Q₈

cosocle

C₄²×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
189. C₅	#8295	semidirect

H ≅ G/N

type

189. C₅

#8295

semidirect

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

subgroup	index
C₄×#3995	2
#10603815	2
#10603810	2
D₅×#322	2
#10598531	2
#8295	5

subgroup

index

Subgroups — #13017384

Conjugacy classes of subgroups

Complements

Maximal subgroups