Group #12017532 — subgroups

Subgroups — #12017532

Order 1280.

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

normal subgroups only

This group has 590 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×#64984	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#64729	C₂	1	4	yes	maximal, complemented
4	#10600559	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₀	—	2	1	no	—
6	C₂²×D₄₀	C₂²	1	2	yes	—
7	C₂×#4020	—	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	#64984	—	2	2	no	—
14	#64984	C₄	1	4	yes	complemented
15	C₅×#346	—	2	2	no	—
16	#64729	—	2	8	no	—
17	#13206	—	5	1	no	maximal, Sylow
18	C₂²×D₂₀	—	2	2	no	—
19	C₂²×D₂₀	—	4	2	no	—
20	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	—	2	2	no	—
24	C₂×D₄₀	—	2	4	no	—
25	C₂×D₄₀	—	2	4	no	—
26	C₂×D₄₀	—	2	4	no	—
27	C₂×D₄₀	—	2	2	no	—
28	C₂×D₄₀	—	4	2	no	—
29	C₂×D₄₀	—	4	2	no	—
30	C₂×D₄₀	C₂×C₄	1	4	yes	—
31	#4020	—	2	4	no	—
32	#4020	—	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	complemented
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×#493	—	5	1	no	—
42	C₂×#357	—	5	4	no	—
43	C₂×#346	D₅	1	1	yes	characteristic, complemented
44	#1846	—	5	1	no	—
45	C₂³×D₅	—	4	2	no	—
46	C₂²×C₄×C₅	D₈	1	2	yes	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅	—	4	2	no	—
49	C₂×D₂₀	—	2	4	no	—
50	C₂×D₂₀	—	4	2	no	—
51	C₂×D₂₀	—	4	2	no	—
52	C₂×D₂₀	—	4	2	no	—
53	C₂×D₂₀	—	4	2	no	—
54	C₂×D₂₀	—	4	8	no	—
55	C₂×D₂₀	—	4	2	no	—
56	C₂×D₂₀	—	8	8	no	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	D₈	1	2	yes	—
59	C₂×C₅×C₈	SD₁₆	1	2	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₅×C₈	#41	1	2	yes	—
64	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅	SD₁₆	1	2	yes	—
66	C₄⋊Dic₅	—	4	2	no	—
67	C₄⋊Dic₅	—	4	2	no	—
68	D₄₀	—	2	8	no	—
69	D₄₀	—	4	8	no	—
70	D₄₀	—	4	4	no	—
71	C₅×C₁₆	—	2	4	no	—
72	C₂²×D₈	—	5	2	no	—
73	C₂²×D₈	—	10	1	no	—
74	C₂×#127	—	10	1	no	—
75	C₂²×C₁₆	C₂×D₅	1	2	yes	—
76	#493	—	5	4	no	—
77	#493	—	10	2	no	—
78	C₂×#107	—	5	2	no	—
79	C₂×#106	—	10	1	no	—
80	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
81	#382	—	10	1	no	—
82	#357	—	10	8	no	—
83	#346	—	2	2	no	—
84	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
85	C₂²×D₅	—	4	4	no	—
86	C₂²×D₅	—	8	2	no	—
87	C₂²×D₅	—	8	2	no	—
88	C₂²×D₅	—	8	8	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	#102	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	#124	1	1	yes	characteristic
93	C₂×C₄×C₅	#127	1	1	yes	characteristic
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×Dic₅	—	8	2	no	—
98	C₂×Dic₅	—	8	4	no	—
99	D₂₀	—	4	4	no	—
100	D₂₀	—	4	4	no	—
101	D₂₀	—	8	16	no	—
102	D₂₀	—	8	2	no	—
103	D₂₀	—	8	2	no	—
104	C₅×C₈	#102	1	4	yes	—
105	C₅×C₈	C₄×D₄	1	2	yes	—
106	C₅×C₈	#121	1	2	yes	—
107	C₂²×D₄	—	10	2	no	—
108	C₂²×D₄	—	20	2	no	—
109	C₂×D₈	—	5	4	no	—
110	C₂×D₈	—	10	4	no	—
111	C₂×D₈	—	10	4	no	—
112	C₂×D₈	—	10	4	no	—
113	C₂×D₈	—	10	2	no	—
114	C₂×D₈	—	20	2	no	—
115	C₂×D₈	—	20	2	no	—
116	C₂²×C₈	D₂₀	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
118	#127	—	10	4	no	—
119	#127	—	20	2	no	—
120	C₂×C₄⋊C₄	—	10	2	no	—
121	C₂×C₄²	D₂₀	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	D₂₀	1	4	yes	complemented
125	#107	—	10	4	no	—
126	#106	—	20	2	no	—
127	C₄×C₈	C₄×D₅	1	2	yes	—
128	C₄×C₈	C₂²⋊D₅	1	2	yes	—
129	#95	—	20	2	no	—
130	C₂²×C₅	C₂×#102	1	1	yes	characteristic
131	C₂²×C₅	#486	1	1	yes	characteristic
132	C₂²×C₅	#493	1	1	yes	characteristic
133	C₂²×C₅	#509	1	4	yes	—
134	C₂×D₅	—	8	8	no	—
135	C₂×D₅	—	8	4	no	—
136	C₂×D₅	—	16	4	no	—
137	C₂×D₅	—	16	4	no	—
138	C₄×C₅	—	2	4	no	—
139	C₄×C₅	#390	1	1	yes	characteristic
140	C₄×C₅	C₂×#102	1	1	yes	characteristic
141	C₄×C₅	#486	1	1	yes	characteristic
142	C₄×C₅	#492	1	1	yes	characteristic
143	Dic₅	—	16	2	no	—
144	C₂⁴	—	20	2	no	—
145	C₂×D₄	—	10	4	no	—
146	C₂×D₄	—	20	8	no	—
147	C₂×D₄	—	20	2	no	—
148	C₂×D₄	—	20	2	no	—
149	C₂×D₄	—	20	2	no	—
150	C₂×D₄	—	20	2	no	—
151	C₂×D₄	—	20	2	no	—
152	C₂×D₄	—	40	8	no	—
153	C₂²×C₄	—	20	2	no	—
154	C₂²×C₄	D₄₀	1	2	yes	—
155	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
156	D₈	—	10	8	no	—
157	D₈	—	20	8	no	—
158	D₈	—	20	4	no	—
159	C₂×C₈	#677	1	2	yes	—
160	C₂×C₈	D₄₀	1	2	yes	—
161	C₂×C₈	#685	1	2	yes	—
162	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂×C₈	C₂×D₂₀	1	2	yes	—
164	C₂×C₈	#709	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
166	C₄⋊C₄	—	20	2	no	—
167	C₄⋊C₄	—	20	2	no	—
168	C₄²	#677	1	2	yes	—
169	C₄²	#685	1	2	yes	—
170	C₁₆	—	2	4	no	—
171	C₂×C₅	#1957	1	1	yes	characteristic
172	C₂×C₅	#2140	1	2	yes	—
173	C₂×C₅	#2244	1	2	yes	—
174	C₂×C₅	#2250	1	2	yes	—
175	D₅	—	16	4	no	—
176	C₂³	—	20	4	no	—
177	C₂³	—	40	8	no	—
178	C₂³	—	40	2	no	—
179	C₂³	—	40	2	no	—
180	C₂³	C₂×D₄₀	1	1	yes	characteristic
181	D₄	—	20	4	no	—
182	D₄	—	20	4	no	—
183	D₄	—	40	16	no	—
184	D₄	—	40	2	no	—
185	D₄	—	40	2	no	—
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	40	4	no	—
188	C₂×C₄	—	40	2	no	—
189	C₂×C₄	#3953	1	4	yes	—
190	C₂×C₄	#4020	1	1	yes	characteristic
191	C₂×C₄	#4021	1	1	yes	characteristic
192	C₂×C₄	C₂×#677	1	1	yes	characteristic
193	C₂×C₄	C₂×#677	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
195	C₂×C₄	C₂×#685	1	1	yes	characteristic
196	C₈	#3953	1	4	yes	—
197	C₈	C₄×D₂₀	1	2	yes	—
198	C₈	#4076	1	2	yes	—
199	C₅	#13206	1	1	yes	characteristic, complemented, Sylow
200	C₂²	—	40	4	no	—
201	C₂²	—	40	8	no	—
202	C₂²	—	80	4	no	—
203	C₂²	—	80	4	no	—
204	C₂²	#64983	1	1	yes	characteristic
205	C₂²	#64984	1	1	yes	characteristic
206	C₂²	#65197	1	4	yes	—
207	C₂²	C₂×#3953	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	80	2	no	—
210	C₄	#64982	1	1	yes	characteristic
211	C₄	#64983	1	1	yes	characteristic
212	C₄	#65228	1	1	yes	characteristic
213	C₄	C₂×#3953	1	1	yes	characteristic
214	C₂	—	80	4	no	—
215	C₂	#10599873	1	2	yes	—
216	C₂	#10599874	1	2	yes	—
217	C₂	#10600565	1	1	yes	characteristic
218	C₂	#10602233	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64984

C₂

yes

maximal, characteristic

C₂×C₅×#346

C₂

yes

maximal, characteristic, complemented

C₂×#64729

C₂

yes

maximal, complemented

#10600559

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀

—

C₂²×D₄₀

C₂²

yes

—

C₂×#4020

—

C₂²×C₅×C₁₆

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3950

C₂²

yes

—

—

—

—

C₄

yes

—

—

—

C₂²×D₂₀

—

C₂²×D₂₀

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

—

C₂×D₄₀

C₂×C₄

yes

—

#4020

—

#4020

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

D₄

yes

complemented

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

D₅

yes

characteristic, complemented

#1846

—

C₂³×D₅

—

C₂²×C₄×C₅

D₈

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

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

—

D₄₀

—

D₄₀

—

D₄₀

—

C₅×C₁₆

—

C₂²×D₈

—

C₂²×D₈

—

C₂×#127

—

C₂²×C₁₆

C₂×D₅

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

C₂×D₈

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

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

—

D₂₀

—

100

D₂₀

—

101

D₂₀

—

102

D₂₀

—

103

D₂₀

—

104

C₅×C₈

#102

yes

—

105

C₅×C₈

C₄×D₄

yes

—

106

C₅×C₈

#121

yes

—

107

C₂²×D₄

—

108

C₂²×D₄

—

109

C₂×D₈

—

110

C₂×D₈

—

111

C₂×D₈

—

112

C₂×D₈

—

113

C₂×D₈

—

114

C₂×D₈

—

115

C₂×D₈

—

116

C₂²×C₈

D₂₀

yes

characteristic

117

C₂²×C₈

C₂²×D₅

yes

characteristic

118

#127

—

119

#127

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

D₂₀

yes

characteristic

122

C₂×C₁₆

—

123

C₂×C₁₆

—

124

C₂×C₁₆

D₂₀

yes

complemented

125

#107

—

126

#106

—

127

C₄×C₈

C₄×D₅

yes

—

128

C₄×C₈

C₂²⋊D₅

yes

—

129

#95

—

130

C₂²×C₅

C₂×#102

yes

characteristic

131

C₂²×C₅

#486

yes

characteristic

132

C₂²×C₅

#493

yes

characteristic

133

C₂²×C₅

#509

yes

—

134

C₂×D₅

—

135

C₂×D₅

—

136

C₂×D₅

—

137

C₂×D₅

—

138

C₄×C₅

—

139

C₄×C₅

#390

yes

characteristic

140

C₄×C₅

C₂×#102

yes

characteristic

141

C₄×C₅

#486

yes

characteristic

142

C₄×C₅

#492

yes

characteristic

143

Dic₅

—

144

C₂⁴

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂²×C₄

—

154

C₂²×C₄

D₄₀

yes

—

155

C₂²×C₄

C₂×D₂₀

yes

characteristic

156

D₈

—

157

D₈

—

158

D₈

—

159

C₂×C₈

#677

yes

—

160

C₂×C₈

D₄₀

yes

—

161

C₂×C₈

#685

yes

—

162

C₂×C₈

C₂×C₄×D₅

yes

characteristic

163

C₂×C₈

C₂×D₂₀

yes

—

164

C₂×C₈

#709

yes

—

165

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

#677

yes

—

169

C₄²

#685

yes

—

170

C₁₆

—

171

C₂×C₅

#1957

yes

characteristic

172

C₂×C₅

#2140

yes

—

173

C₂×C₅

#2244

yes

—

174

C₂×C₅

#2250

yes

—

175

D₅

—

176

C₂³

—

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

C₂×D₄₀

yes

characteristic

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

#3953

yes

—

190

C₂×C₄

#4020

yes

characteristic

191

C₂×C₄

#4021

yes

characteristic

192

C₂×C₄

C₂×#677

yes

characteristic

193

C₂×C₄

C₂×#677

yes

characteristic

194

C₂×C₄

C₂×D₄₀

yes

characteristic

195

C₂×C₄

C₂×#685

yes

characteristic

196

C₈

#3953

yes

—

197

C₈

C₄×D₂₀

yes

—

198

C₈

#4076

yes

—

199

C₅

#13206

yes

characteristic, complemented, Sylow

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

#64983

yes

characteristic

205

C₂²

#64984

yes

characteristic

206

C₂²

#65197

yes

—

207

C₂²

C₂×#3953

yes

characteristic

208

C₄

—

209

C₄

—

210

C₄

#64982

yes

characteristic

211

C₄

#64983

yes

characteristic

212

C₄

#65228

yes

characteristic

213

C₄

C₂×#3953

yes

characteristic

214

C₂

—

215

C₂

#10599873

yes

—

216

C₂

#10599874

yes

—

217

C₂

#10600565

yes

characteristic

218

C₂

#10602233

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	C₂³×C₅	C₂×D₈
prime radical	C₂×#64984	C₂
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#13206
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

C₂³×C₅

C₂×D₈

prime radical

C₂×#64984

C₂

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
2. C₂×C₅×#346	C₂	semidirect
3. C₂×#64729	C₂	semidirect
4. #10600559	C₂	semidirect
14. #64984	C₄	semidirect
35. C₂×C₅×C₁₆	D₄	semidirect
43. C₂×#346	D₅	semidirect
124. C₂×C₁₆	D₂₀	semidirect
199. C₅	#13206	semidirect

H ≅ G/N

type

2. C₂×C₅×#346

C₂

semidirect

3. C₂×#64729

C₂

semidirect

4. #10600559

C₂

semidirect

14. #64984

C₄

semidirect

35. C₂×C₅×C₁₆

D₄

semidirect

43. C₂×#346

D₅

semidirect

124. C₂×C₁₆

D₂₀

semidirect

199. C₅

#13206

semidirect

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

subgroup	index
C₂×#64984	2
C₂×C₅×#346	2
C₂×#64729	2
#10600559	2
#13206	5

subgroup

index

C₂×C₅×#346

Subgroups — #12017532

Conjugacy classes of subgroups

Complements

Maximal subgroups