Group #12017537 — subgroups

Subgroups — #12017537

Order 1280.

486 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 167 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×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#64784	C₂	1	4	yes	maximal
3	#10600559	C₂	1	1	yes	maximal, characteristic
4	C₂×#64681	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅×Q₁₆	—	2	1	no	—
6	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₅×#494	—	2	2	no	—
9	C₅×#494	C₄	1	4	yes	complemented
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×#3950	C₂²	1	2	yes	—
12	C₂×#3949	—	2	1	no	—
13	C₂²×#672	C₂²	1	2	yes	—
14	#65225	—	2	1	no	—
15	#64784	—	2	8	no	—
16	#64681	—	2	2	no	—
17	#13207	—	5	1	no	maximal, Sylow
18	C₂²×C₅×Q₈	—	2	2	no	—
19	C₂×C₅×Q₁₆	—	2	4	no	—
20	C₂×C₅×Q₁₆	—	2	4	no	—
21	C₂×C₅×Q₁₆	—	2	2	no	—
22	C₂×C₅×Q₁₆	—	2	4	no	—
23	C₂×C₅×Q₁₆	—	4	2	no	—
24	C₂×C₅×Q₁₆	—	4	2	no	—
25	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
28	C₅×C₄⋊Q₈	—	2	4	no	—
29	C₅×C₄⋊Q₈	—	4	2	no	—
30	C₂×C₅×C₄⋊C₄	—	4	2	no	—
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₅	—	2	2	no	—
33	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₄×C₅×C₈	D₄	1	2	yes	—
35	#3963	—	4	2	no	—
36	#3950	—	2	4	no	—
37	#3949	—	4	2	no	—
38	C₂×#672	—	2	2	no	—
39	C₂×#672	—	2	2	no	—
40	C₂×#672	D₄	1	4	yes	—
41	C₂×#494	D₅	1	1	yes	characteristic
42	C₂×#358	—	5	4	no	—
43	C₂×#346	—	5	1	no	—
44	#1846	—	5	1	no	—
45	C₂×C₅×Q₈	—	2	4	no	—
46	C₂×C₅×Q₈	—	4	2	no	—
47	C₂×C₅×Q₈	—	4	2	no	—
48	C₂×C₅×Q₈	—	4	2	no	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	—	4	2	no	—
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	D₈	1	2	yes	—
53	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₅	—	4	2	no	—
55	C₅×Q₁₆	—	2	8	no	—
56	C₅×Q₁₆	—	4	8	no	—
57	C₅×Q₁₆	—	4	4	no	—
58	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	D₈	1	2	yes	—
60	C₂×C₅×C₈	SD₁₆	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	#41	1	2	yes	—
65	C₅×C₄⋊C₄	—	4	8	no	—
66	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅	SD₁₆	1	2	yes	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₄⋊Dic₅	—	4	2	no	—
70	#672	—	2	4	no	—
71	C₂²×Q₁₆	—	2	1	no	—
72	C₂²×Q₁₆	C₂×D₅	1	2	yes	—
73	C₂×C₄⋊Q₈	—	2	1	no	—
74	C₂²×C₁₆	—	5	2	no	—
75	#494	—	2	2	no	—
76	#494	Dic₅	1	4	yes	complemented
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	—	10	2	no	—
83	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
84	C₅×Q₈	—	4	4	no	—
85	C₅×Q₈	—	4	4	no	—
86	C₅×Q₈	—	8	2	no	—
87	C₅×Q₈	—	8	2	no	—
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	8	2	no	—
91	C₂×C₄×C₅	—	8	2	no	—
92	C₂×C₄×C₅	#102	1	4	yes	—
93	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅	#124	1	1	yes	characteristic
95	C₂×C₄×C₅	#127	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×Dic₅	—	8	2	no	—
100	C₂×Dic₅	—	8	4	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	—	2	2	no	—
105	C₂×Q₁₆	—	2	4	no	—
106	C₂×Q₁₆	—	2	4	no	—
107	C₂×Q₁₆	—	2	4	no	—
108	C₂×Q₁₆	—	2	2	no	—
109	C₂×Q₁₆	—	4	2	no	—
110	C₂×Q₁₆	—	4	2	no	—
111	C₂×Q₁₆	C₂×Dic₅	1	4	yes	—
112	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	4	no	—
115	C₄⋊Q₈	—	4	2	no	—
116	C₂×C₄⋊C₄	—	4	2	no	—
117	C₂×C₄⋊C₄	—	10	2	no	—
118	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
119	C₂×C₁₆	—	5	4	no	—
120	C₂×C₁₆	—	10	2	no	—
121	C₂×C₁₆	—	10	2	no	—
122	#107	—	10	4	no	—
123	#106	—	20	2	no	—
124	C₄×C₈	C₂×Dic₅	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	—	8	4	no	—
133	C₄×C₅	#390	1	1	yes	characteristic
134	C₄×C₅	C₂×#102	1	1	yes	characteristic
135	C₄×C₅	#486	1	1	yes	characteristic
136	C₄×C₅	#492	1	1	yes	characteristic
137	Dic₅	—	16	2	no	—
138	C₂×Q₈	—	2	4	no	—
139	C₂×Q₈	—	4	2	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	20	2	no	—
146	C₂²×C₄	#686	1	2	yes	—
147	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
148	Q₁₆	—	2	8	no	—
149	Q₁₆	—	4	8	no	—
150	Q₁₆	—	4	4	no	—
151	C₂×C₈	#686	1	2	yes	—
152	C₂×C₈	#687	1	2	yes	—
153	C₂×C₈	C₂²⋊Dic₅	1	2	yes	—
154	C₂×C₈	D₄×D₅	1	2	yes	—
155	C₂×C₈	#711	1	2	yes	—
156	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
158	C₄⋊C₄	—	4	8	no	—
159	C₄⋊C₄	—	20	2	no	—
160	C₄⋊C₄	—	20	2	no	—
161	C₄²	#687	1	2	yes	—
162	C₄²	C₂²⋊Dic₅	1	2	yes	—
163	C₁₆	—	10	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×#686	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	4	4	no	—
171	Q₈	—	8	2	no	—
172	Q₈	—	8	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	40	2	no	—
178	C₂×C₄	—	40	4	no	—
179	C₂×C₄	#3964	1	4	yes	—
180	C₂×C₄	C₂×#686	1	1	yes	characteristic
181	C₂×C₄	C₂×#687	1	1	yes	characteristic
182	C₂×C₄	C₂×#687	1	1	yes	characteristic
183	C₂×C₄	#4082	1	1	yes	characteristic
184	C₂×C₄	#4086	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
186	C₈	#3964	1	4	yes	—
187	C₈	D₄×Dic₅	1	2	yes	—
188	C₈	#4084	1	2	yes	—
189	C₅	#13207	1	1	yes	characteristic, complemented, Sylow
190	C₂²	#65359	1	1	yes	characteristic
191	C₂²	#65362	1	1	yes	characteristic
192	C₂²	#65468	1	4	yes	—
193	C₂²	C₂×#3964	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	80	2	no	—
197	C₄	#65359	1	1	yes	characteristic
198	C₄	#65365	1	1	yes	characteristic
199	C₄	C₂×#3964	1	1	yes	characteristic
200	C₄	#65510	1	1	yes	characteristic
201	C₂	#10602104	1	2	yes	—
202	C₂	#10602107	1	2	yes	—
203	C₂	#10602557	1	1	yes	characteristic
204	C₂	#10602854	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#494

C₂

yes

maximal, characteristic

C₂×#64784

C₂

yes

maximal

#10600559

C₂

yes

maximal, characteristic

C₂×#64681

C₂

yes

maximal, characteristic

C₂²×C₅×Q₁₆

—

C₂²×C₅×Q₁₆

C₂²

yes

—

C₂×C₅×C₄⋊Q₈

—

C₅×#494

—

C₅×#494

C₄

yes

complemented

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₂×#3950

C₂²

yes

—

C₂×#3949

—

C₂²×#672

C₂²

yes

—

—

—

—

—

C₂²×C₅×Q₈

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

—

C₂×C₅×Q₁₆

C₂×C₄

yes

—

C₂²×C₅×C₈

D₄

yes

characteristic

C₂²×C₅×C₈

C₂³

yes

characteristic

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

D₄

yes

characteristic

C₂×C₄⋊Dic₅

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂×#494

D₅

yes

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

D₈

yes

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

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

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²⋊C₄

yes

—

C₄²×C₅

SD₁₆

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#672

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×D₅

yes

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

#494

—

#494

Dic₅

yes

complemented

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₅

yes

—

—

—

C₂³×C₅

C₂×D₈

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

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

—

100

C₂×Dic₅

—

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

C₂×Dic₅

yes

—

112

C₂²×C₈

C₂²⋊D₅

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

C₂²⋊D₅

yes

characteristic

119

C₂×C₁₆

—

120

C₂×C₁₆

—

121

C₂×C₁₆

—

122

#107

—

123

#106

—

124

C₄×C₈

C₂×Dic₅

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

—

133

C₄×C₅

#390

yes

characteristic

134

C₄×C₅

C₂×#102

yes

characteristic

135

C₄×C₅

#486

yes

characteristic

136

C₄×C₅

#492

yes

characteristic

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

#686

yes

—

147

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

148

Q₁₆

—

149

Q₁₆

—

150

Q₁₆

—

151

C₂×C₈

#686

yes

—

152

C₂×C₈

#687

yes

—

153

C₂×C₈

C₂²⋊Dic₅

yes

—

154

C₂×C₈

D₄×D₅

yes

—

155

C₂×C₈

#711

yes

—

156

C₂×C₈

C₂²×Dic₅

yes

characteristic

157

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄²

#687

yes

—

162

C₄²

C₂²⋊Dic₅

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×#686

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

#3964

yes

—

180

C₂×C₄

C₂×#686

yes

characteristic

181

C₂×C₄

C₂×#687

yes

characteristic

182

C₂×C₄

C₂×#687

yes

characteristic

183

C₂×C₄

#4082

yes

characteristic

184

C₂×C₄

#4086

yes

characteristic

185

C₂×C₄

C₂×C₂²⋊Dic₅

yes

characteristic

186

C₈

#3964

yes

—

187

C₈

D₄×Dic₅

yes

—

188

C₈

#4084

yes

—

189

C₅

#13207

yes

characteristic, complemented, Sylow

190

C₂²

#65359

yes

characteristic

191

C₂²

#65362

yes

characteristic

192

C₂²

#65468

yes

—

193

C₂²

C₂×#3964

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#65359

yes

characteristic

198

C₄

#65365

yes

characteristic

199

C₄

C₂×#3964

yes

characteristic

200

C₄

#65510

yes

characteristic

201

C₂

#10602104

yes

—

202

C₂

#10602107

yes

—

203

C₂

#10602557

yes

characteristic

204

C₂

#10602854

yes

—

	subgroup	G/H
center	C₂³	C₂×#686
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#494	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#494	D₅
hypocenter	C₅	#13207
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#686

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅

Fitting

C₂×C₅×#494

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×#494

N	H ≅ G/N	type
9. C₅×#494	C₄	semidirect
76. #494	Dic₅	semidirect
189. C₅	#13207	semidirect

H ≅ G/N

type

9. C₅×#494

C₄

semidirect

76. #494

Dic₅

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×C₅×#494	2
C₂×#64784	2
#10600559	2
C₂×#64681	2
#13207	5

subgroup

index

C₂×C₅×#494

Subgroups — #12017537

Conjugacy classes of subgroups

Complements

Maximal subgroups