Group #12017536 — subgroups

Subgroups — #12017536

Order 1280.

486 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 163 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	C₂×#64988	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×#64723	C₂	1	4	yes	maximal
4	#10600559	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀	—	2	1	no	—
6	C₂²×Dic₂₀	C₂²	1	2	yes	—
7	C₂×#4015	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3950	C₂²	1	2	yes	—
11	C₂×#3949	—	2	1	no	—
12	#65225	—	2	1	no	—
13	#64988	—	2	2	no	—
14	#64988	C₄	1	4	yes	complemented
15	C₅×#346	—	2	2	no	—
16	#64723	—	2	8	no	—
17	#13207	—	5	1	no	maximal, Sylow
18	C₂²×Dic₁₀	—	2	2	no	—
19	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅	—	2	2	no	—
23	C₂×C₄⋊Dic₅	—	4	2	no	—
24	C₂×Dic₂₀	—	2	4	no	—
25	C₂×Dic₂₀	—	2	4	no	—
26	C₂×Dic₂₀	—	2	4	no	—
27	C₂×Dic₂₀	—	2	2	no	—
28	C₂×Dic₂₀	—	4	2	no	—
29	C₂×Dic₂₀	—	4	2	no	—
30	C₂×Dic₂₀	C₂×C₄	1	4	yes	—
31	#4015	—	2	4	no	—
32	#4015	—	4	2	no	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₂×C₅×C₁₆	D₄	1	4	yes	—
36	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₄×C₅×C₈	D₄	1	2	yes	—
38	#3963	—	4	2	no	—
39	#3950	—	2	4	no	—
40	#3949	—	4	2	no	—
41	C₂×#494	—	5	1	no	—
42	C₂×#358	—	5	4	no	—
43	C₂×#346	D₅	1	1	yes	characteristic
44	#1846	—	5	1	no	—
45	C₂²×C₄×C₅	D₈	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	4	2	no	—
48	C₂²×Dic₅	—	4	2	no	—
49	C₂×Dic₁₀	—	2	4	no	—
50	C₂×Dic₁₀	—	4	2	no	—
51	C₂×Dic₁₀	—	4	2	no	—
52	C₂×Dic₁₀	—	4	2	no	—
53	C₂×Dic₁₀	—	4	2	no	—
54	C₂×Dic₁₀	—	4	2	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
56	C₂×C₅×C₈	D₈	1	2	yes	—
57	C₂×C₅×C₈	SD₁₆	1	2	yes	—
58	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₅×C₈	#41	1	2	yes	—
62	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅	SD₁₆	1	2	yes	—
64	C₄⋊Dic₅	—	4	8	no	—
65	C₄⋊Dic₅	—	4	2	no	—
66	C₄⋊Dic₅	—	4	2	no	—
67	Dic₂₀	—	2	8	no	—
68	Dic₂₀	—	4	8	no	—
69	Dic₂₀	—	4	4	no	—
70	C₅×C₁₆	—	2	4	no	—
71	C₂²×Q₁₆	—	5	2	no	—
72	C₂²×Q₁₆	—	10	1	no	—
73	C₂×C₄⋊Q₈	—	10	1	no	—
74	C₂²×C₁₆	C₂×D₅	1	2	yes	—
75	#494	—	5	4	no	—
76	#494	—	10	2	no	—
77	C₂×#107	—	5	2	no	—
78	C₂×#106	—	10	1	no	—
79	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
80	#382	—	10	1	no	—
81	#358	—	10	8	no	—
82	#346	—	2	2	no	—
83	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	#102	1	4	yes	—
86	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	#124	1	1	yes	characteristic
88	C₂×C₄×C₅	#127	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
92	C₂×Dic₅	—	4	4	no	—
93	C₂×Dic₅	—	8	2	no	—
94	C₂×Dic₅	—	8	2	no	—
95	C₂×Dic₅	—	8	2	no	—
96	C₂×Dic₅	—	8	4	no	—
97	Dic₁₀	—	4	4	no	—
98	Dic₁₀	—	4	4	no	—
99	Dic₁₀	—	8	2	no	—
100	Dic₁₀	—	8	2	no	—
101	C₅×C₈	#102	1	4	yes	—
102	C₅×C₈	C₄×D₄	1	2	yes	—
103	C₅×C₈	#121	1	2	yes	—
104	C₂²×Q₈	—	10	2	no	—
105	C₂×Q₁₆	—	5	4	no	—
106	C₂×Q₁₆	—	10	4	no	—
107	C₂×Q₁₆	—	10	4	no	—
108	C₂×Q₁₆	—	10	4	no	—
109	C₂×Q₁₆	—	10	2	no	—
110	C₂×Q₁₆	—	20	2	no	—
111	C₂×Q₁₆	—	20	2	no	—
112	C₂²×C₈	D₂₀	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₄⋊Q₈	—	10	4	no	—
115	C₄⋊Q₈	—	20	2	no	—
116	C₂×C₄⋊C₄	—	10	2	no	—
117	C₂×C₄⋊C₄	—	20	2	no	—
118	C₂×C₄²	D₂₀	1	1	yes	characteristic
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	D₂₀	1	4	yes	—
122	#107	—	10	4	no	—
123	#106	—	20	2	no	—
124	C₄×C₈	C₄×D₅	1	2	yes	—
125	C₄×C₈	C₂²⋊D₅	1	2	yes	—
126	#95	—	20	2	no	—
127	C₂²×C₅	C₂×#102	1	1	yes	characteristic
128	C₂²×C₅	#486	1	1	yes	characteristic
129	C₂²×C₅	#493	1	1	yes	characteristic
130	C₂²×C₅	#510	1	4	yes	—
131	C₄×C₅	—	2	4	no	—
132	C₄×C₅	#390	1	1	yes	characteristic
133	C₄×C₅	C₂×#102	1	1	yes	characteristic
134	C₄×C₅	#486	1	1	yes	characteristic
135	C₄×C₅	#492	1	1	yes	characteristic
136	Dic₅	—	8	4	no	—
137	Dic₅	—	16	2	no	—
138	C₂×Q₈	—	10	4	no	—
139	C₂×Q₈	—	20	2	no	—
140	C₂×Q₈	—	20	2	no	—
141	C₂×Q₈	—	20	2	no	—
142	C₂×Q₈	—	20	2	no	—
143	C₂×Q₈	—	20	2	no	—
144	C₂²×C₄	—	20	2	no	—
145	C₂²×C₄	—	20	2	no	—
146	C₂²×C₄	D₄₀	1	2	yes	—
147	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
148	Q₁₆	—	10	8	no	—
149	Q₁₆	—	20	8	no	—
150	Q₁₆	—	20	4	no	—
151	C₂×C₈	#677	1	2	yes	—
152	C₂×C₈	D₄₀	1	2	yes	—
153	C₂×C₈	#685	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
155	C₂×C₈	C₂×D₂₀	1	2	yes	—
156	C₂×C₈	#709	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
158	C₄⋊C₄	—	20	8	no	—
159	C₄⋊C₄	—	20	2	no	—
160	C₄⋊C₄	—	20	2	no	—
161	C₄²	#677	1	2	yes	—
162	C₄²	#685	1	2	yes	—
163	C₁₆	—	2	4	no	—
164	C₂×C₅	#1957	1	1	yes	characteristic
165	C₂×C₅	#2141	1	2	yes	—
166	C₂×C₅	#2245	1	2	yes	—
167	C₂×C₅	#2251	1	2	yes	—
168	C₂³	C₂×D₄₀	1	1	yes	characteristic
169	Q₈	—	20	4	no	—
170	Q₈	—	20	4	no	—
171	Q₈	—	40	2	no	—
172	Q₈	—	40	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	20	4	no	—
175	C₂×C₄	—	40	2	no	—
176	C₂×C₄	—	40	2	no	—
177	C₂×C₄	—	40	4	no	—
178	C₂×C₄	—	40	2	no	—
179	C₂×C₄	#3953	1	4	yes	—
180	C₂×C₄	#4020	1	1	yes	characteristic
181	C₂×C₄	#4021	1	1	yes	characteristic
182	C₂×C₄	C₂×#677	1	1	yes	characteristic
183	C₂×C₄	C₂×#677	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
185	C₂×C₄	C₂×#685	1	1	yes	characteristic
186	C₈	#3953	1	4	yes	—
187	C₈	C₄×D₂₀	1	2	yes	—
188	C₈	#4076	1	2	yes	—
189	C₅	#13207	1	1	yes	characteristic, complemented, Sylow
190	C₂²	#64983	1	1	yes	characteristic
191	C₂²	#64984	1	1	yes	characteristic
192	C₂²	#65198	1	4	yes	—
193	C₂²	C₂×#3953	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	40	4	no	—
196	C₄	—	80	2	no	—
197	C₄	#64982	1	1	yes	characteristic
198	C₄	#64983	1	1	yes	characteristic
199	C₄	#65228	1	1	yes	characteristic
200	C₄	C₂×#3953	1	1	yes	characteristic
201	C₂	#10599876	1	2	yes	—
202	C₂	#10599877	1	2	yes	—
203	C₂	#10600565	1	1	yes	characteristic
204	C₂	#10602238	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64988

C₂

yes

maximal, characteristic

C₂×C₅×#346

C₂

yes

maximal, characteristic

C₂×#64723

C₂

yes

maximal

#10600559

C₂

yes

maximal, characteristic

C₂²×Dic₂₀

—

C₂²×Dic₂₀

C₂²

yes

—

C₂×#4015

—

C₂²×C₅×C₁₆

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3950

C₂²

yes

—

—

—

—

C₄

yes

—

—

—

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

C₂×Dic₂₀

C₂×C₄

yes

—

#4015

—

#4015

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic

#1846

—

C₂²×C₄×C₅

D₈

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

SD₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

#41

yes

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

SD₁₆

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂×D₅

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

#102

yes

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅

C₂×SD₁₆

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

100

Dic₁₀

—

101

C₅×C₈

#102

yes

—

102

C₅×C₈

C₄×D₄

yes

—

103

C₅×C₈

#121

yes

—

104

C₂²×Q₈

—

105

C₂×Q₁₆

—

106

C₂×Q₁₆

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂×Q₁₆

—

110

C₂×Q₁₆

—

111

C₂×Q₁₆

—

112

C₂²×C₈

D₂₀

yes

characteristic

113

C₂²×C₈

C₂²×D₅

yes

characteristic

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

D₂₀

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

D₂₀

yes

—

122

#107

—

123

#106

—

124

C₄×C₈

C₄×D₅

yes

—

125

C₄×C₈

C₂²⋊D₅

yes

—

126

#95

—

127

C₂²×C₅

C₂×#102

yes

characteristic

128

C₂²×C₅

#486

yes

characteristic

129

C₂²×C₅

#493

yes

characteristic

130

C₂²×C₅

#510

yes

—

131

C₄×C₅

—

132

C₄×C₅

#390

yes

characteristic

133

C₄×C₅

C₂×#102

yes

characteristic

134

C₄×C₅

#486

yes

characteristic

135

C₄×C₅

#492

yes

characteristic

136

Dic₅

—

137

Dic₅

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄₀

yes

—

147

C₂²×C₄

C₂×D₂₀

yes

characteristic

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

151

C₂×C₈

#677

yes

—

152

C₂×C₈

D₄₀

yes

—

153

C₂×C₈

#685

yes

—

154

C₂×C₈

C₂×C₄×D₅

yes

characteristic

155

C₂×C₈

C₂×D₂₀

yes

—

156

C₂×C₈

#709

yes

—

157

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄²

#677

yes

—

162

C₄²

#685

yes

—

163

C₁₆

—

164

C₂×C₅

#1957

yes

characteristic

165

C₂×C₅

#2141

yes

—

166

C₂×C₅

#2245

yes

—

167

C₂×C₅

#2251

yes

—

168

C₂³

C₂×D₄₀

yes

characteristic

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

#3953

yes

—

180

C₂×C₄

#4020

yes

characteristic

181

C₂×C₄

#4021

yes

characteristic

182

C₂×C₄

C₂×#677

yes

characteristic

183

C₂×C₄

C₂×#677

yes

characteristic

184

C₂×C₄

C₂×D₄₀

yes

characteristic

185

C₂×C₄

C₂×#685

yes

characteristic

186

C₈

#3953

yes

—

187

C₈

C₄×D₂₀

yes

—

188

C₈

#4076

yes

—

189

C₅

#13207

yes

characteristic, complemented, Sylow

190

C₂²

#64983

yes

characteristic

191

C₂²

#64984

yes

characteristic

192

C₂²

#65198

yes

—

193

C₂²

C₂×#3953

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#64982

yes

characteristic

198

C₄

#64983

yes

characteristic

199

C₄

#65228

yes

characteristic

200

C₄

C₂×#3953

yes

characteristic

201

C₂

#10599876

yes

—

202

C₂

#10599877

yes

—

203

C₂

#10600565

yes

characteristic

204

C₂

#10602238

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#346	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#13207
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂³

C₂×D₄₀

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₂×C₅×#346

C₂

socle, prime radical

C₂³×C₅

C₂×D₈

cosocle

C₂²×C₅×C₈

C₂³

D₅

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
14. #64988	C₄	semidirect
189. C₅	#13207	semidirect

H ≅ G/N

type

14. #64988

C₄

semidirect

189. C₅

#13207

semidirect

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

subgroup	index
C₂×#64988	2
C₂×C₅×#346	2
C₂×#64723	2
#10600559	2
#13207	5

subgroup

index

C₂×C₅×#346

Subgroups — #12017536

Conjugacy classes of subgroups

Complements

Maximal subgroups