Group #11990315 — subgroups

Subgroups — #11990315

Order 1280.

436 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 133 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	#10615006	C₂	1	1	yes	maximal, characteristic
2	C₅×#1525	C₂	1	1	yes	maximal, characteristic
3	#10602598	C₂	1	1	yes	maximal, characteristic
4	#10600640	C₂	1	2	yes	maximal, complemented
5	#10598160	C₂	1	2	yes	maximal, complemented
6	C₅×#532	C₂²	1	1	yes	characteristic
7	C₂×#4094	C₄	1	2	yes	—
8	C₄×C₅×M₁₆	C₂²	1	2	yes	—
9	#65523	C₄	1	2	yes	—
10	#65301	—	2	2	no	—
11	D₄×#168	—	2	2	no	—
12	#65214	C₂²	1	2	yes	—
13	#65213	C₂²	1	1	yes	characteristic
14	C₂×C₄×#168	C₂²	1	1	yes	characteristic
15	C₅×#329	C₄	1	4	yes	—
16	#64766	C₄	1	2	yes	—
17	#64764	C₄	1	2	yes	—
18	#64674	—	2	2	no	—
19	#64674	—	2	2	no	—
20	#8243	—	5	1	no	maximal, Sylow
21	C₂×C₅×#41	C₂×C₄	1	2	yes	—
22	C₂×C₅×M₁₆	—	2	2	no	—
23	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
24	C₅×#127	C₂×C₄	1	1	yes	characteristic
25	C₅×#124	C₂×C₄	1	2	yes	—
26	C₅×#121	—	2	2	no	—
27	C₄×C₅×D₄	—	2	2	no	—
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	#4094	—	4	4	no	—
30	C₂×#681	—	2	2	no	—
31	C₂×#681	C₂×C₄	1	2	yes	—
32	C₂²×#168	—	2	2	no	—
33	C₂²×#168	C₂×C₄	1	1	yes	characteristic
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₅×#97	C₂×C₄	1	4	yes	—
36	C₄×C₅×C₈	—	2	2	no	—
37	C₂²⋊#168	—	2	2	no	—
38	C₂²⋊#168	—	2	4	no	—
39	C₄⋊#168	—	2	2	no	—
40	C₄⋊#168	C₂×C₄	1	4	yes	—
41	C₄×#168	—	2	2	no	—
42	C₄×#168	D₄	1	2	yes	—
43	C₄×#168	D₄	1	2	yes	—
44	#2931	—	5	1	no	—
45	#1543	—	5	1	no	—
46	#1525	D₅	1	1	yes	characteristic
47	#1451	—	5	2	no	—
48	#1299	—	5	2	no	—
49	C₅×#41	—	4	2	no	—
50	C₅×#41	—	4	2	no	—
51	C₂×C₅×Q₈	C₄²	1	2	yes	—
52	C₂×C₅×D₄	—	2	2	no	—
53	C₂×C₅×D₄	—	2	2	no	—
54	C₂×C₅×D₄	C₄²	1	2	yes	—
55	C₂²×C₄×C₅	—	2	2	no	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	#41	1	2	yes	—
58	C₅×M₁₆	—	4	4	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₅×C₄⋊C₄	—	2	2	no	—
61	C₅×C₂²⋊C₄	—	2	4	no	—
62	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₅	C₂×D₄	1	2	yes	—
65	#681	—	4	4	no	—
66	#681	—	4	2	no	—
67	C₂×#168	—	2	2	no	—
68	C₂×#168	—	2	2	no	—
69	C₂×#168	—	2	4	no	—
70	C₂×#168	—	4	1	no	—
71	C₂×#168	—	4	4	no	—
72	C₂×#168	C₂²⋊C₄	1	2	yes	—
73	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×#168	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#131	—	5	2	no	—
76	#532	C₂×D₅	1	1	yes	characteristic
77	#436	—	10	2	no	—
78	C₈×D₄	—	10	2	no	—
79	#431	—	5	2	no	—
80	#423	—	5	1	no	—
81	#408	—	5	2	no	—
82	C₄×M₁₆	C₂×D₅	1	2	yes	—
83	C₂×C₄×C₈	—	5	1	no	—
84	#333	—	5	2	no	—
85	#331	—	5	2	no	—
86	#329	Dic₅	1	4	yes	—
87	#322	—	10	2	no	—
88	#322	—	10	2	no	—
89	C₂³×C₅	—	2	2	no	—
90	C₂³×C₅	#117	1	1	yes	characteristic
91	C₅×Q₈	—	4	2	no	—
92	C₅×D₄	—	4	2	no	—
93	C₅×D₄	—	4	4	no	—
94	C₅×D₄	—	4	4	no	—
95	C₂×C₄×C₅	—	2	1	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	—	4	2	no	—
101	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
105	C₅×C₈	—	4	4	no	—
106	#168	—	2	1	no	—
107	#168	—	2	2	no	—
108	#168	—	4	2	no	—
109	#168	—	4	4	no	—
110	#168	#100	1	2	yes	—
111	C₂×#41	C₄×D₅	1	2	yes	—
112	#131	—	20	4	no	—
113	C₂×M₁₆	—	2	2	no	—
114	C₂×M₁₆	—	5	2	no	—
115	C₂×M₁₆	—	10	2	no	—
116	C₂²×C₈	—	5	1	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	10	2	no	—
119	C₄⋊Q₈	C₄×D₅	1	1	yes	characteristic
120	#127	C₄×D₅	1	1	yes	characteristic
121	#124	C₂×Dic₅	1	2	yes	—
122	#121	—	2	2	no	—
123	C₄×D₄	—	2	2	no	—
124	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
125	C₄⋊C₈	—	5	4	no	—
126	C₄⋊C₈	—	10	2	no	—
127	C₂²⋊C₈	—	10	4	no	—
128	C₂²⋊C₈	—	10	2	no	—
129	#97	C₂×Dic₅	1	4	yes	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	5	2	no	—
132	C₄×C₈	—	5	2	no	—
133	C₄×C₈	—	10	2	no	—
134	C₂²×C₅	—	2	1	no	—
135	C₂²×C₅	—	2	2	no	—
136	C₂²×C₅	—	4	2	no	—
137	C₂²×C₅	—	4	4	no	—
138	C₂²×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	2	2	no	—
141	C₄×C₅	—	4	2	no	—
142	C₄×C₅	C₂×#100	1	1	yes	characteristic
143	C₄×C₅	#413	1	1	yes	characteristic
144	C₄×C₅	#444	1	4	yes	—
145	#41	—	4	2	no	—
146	#41	—	4	2	no	—
147	C₂×Q₈	C₄×Dic₅	1	2	yes	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	C₄×Dic₅	1	2	yes	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
153	C₂²×C₄	#711	1	2	yes	—
154	M₁₆	—	4	4	no	—
155	M₁₆	—	20	4	no	—
156	M₁₆	—	20	2	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	2	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	10	2	no	—
162	C₂×C₈	—	10	2	no	—
163	C₂×C₈	—	10	4	no	—
164	C₂×C₈	—	20	4	no	—
165	C₂×C₈	—	20	1	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
169	C₄²	D₄×D₅	1	2	yes	—
170	C₄²	C₂²×Dic₅	1	1	yes	characteristic
171	C₂×C₅	—	4	1	no	—
172	C₂×C₅	—	8	2	no	—
173	C₂×C₅	C₄×#100	1	1	yes	characteristic
174	C₂×C₅	#1765	1	2	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	#4023	1	1	yes	characteristic
177	Q₈	—	4	2	no	—
178	D₄	—	4	2	no	—
179	D₄	—	4	4	no	—
180	D₄	—	4	4	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₄	#4027	1	2	yes	—
189	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
190	C₂×C₄	D₄×Dic₅	1	2	yes	—
191	C₈	—	4	4	no	—
192	C₈	—	5	2	no	—
193	C₈	—	10	1	no	—
194	C₈	—	10	2	no	—
195	C₈	—	20	2	no	—
196	C₈	—	20	4	no	—
197	C₅	#8243	1	1	yes	characteristic, complemented, Sylow
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	D₅×#100	1	1	yes	characteristic
207	C₄	#65034	1	1	yes	characteristic
208	C₄	#65112	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	Dic₅×#100	1	1	yes	characteristic
212	C₂	#10601163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10615006

C₂

yes

maximal, characteristic

C₅×#1525

C₂

yes

maximal, characteristic

#10602598

C₂

yes

maximal, characteristic

#10600640

C₂

yes

maximal, complemented

#10598160

C₂

yes

maximal, complemented

C₅×#532

C₂²

yes

characteristic

C₂×#4094

C₄

yes

—

C₄×C₅×M₁₆

C₂²

yes

—

#65523

C₄

yes

—

#65301

—

D₄×#168

—

#65214

C₂²

yes

—

#65213

C₂²

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

C₅×#329

C₄

yes

—

#64766

C₄

yes

—

#64764

C₄

yes

—

—

—

—

C₂×C₅×#41

C₂×C₄

yes

—

C₂×C₅×M₁₆

—

C₅×C₄⋊Q₈

C₂×C₄

yes

characteristic

C₅×#127

C₂×C₄

yes

characteristic

C₅×#124

C₂×C₄

yes

—

C₅×#121

—

C₄×C₅×D₄

—

C₂×C₄²×C₅

C₂³

yes

—

—

C₂×C₄

yes

—

C₂²×#168

—

C₂²×#168

C₂×C₄

yes

characteristic

C₂²×#168

C₂×C₄

yes

characteristic

C₅×#97

C₂×C₄

yes

—

C₄×C₅×C₈

—

C₂²⋊#168

—

C₂²⋊#168

—

C₄⋊#168

—

C₄⋊#168

C₂×C₄

yes

—

C₄×#168

—

C₄×#168

D₄

yes

—

C₄×#168

D₄

yes

—

#2931

—

#1543

—

#1525

D₅

yes

characteristic

#1451

—

#1299

—

C₅×#41

—

C₅×#41

—

C₂×C₅×Q₈

C₄²

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₄²

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

#41

yes

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

—

—

—

—

—

—

—

—

C₂²⋊C₄

yes

—

C₂×#168

C₂²⋊C₄

yes

characteristic

C₂×#168

C₂²⋊C₄

yes

characteristic

C₂×#131

—

#532

C₂×D₅

yes

characteristic

#436

—

C₈×D₄

—

#431

—

#423

—

#408

—

C₄×M₁₆

C₂×D₅

yes

—

C₂×C₄×C₈

—

#333

—

#331

—

#329

Dic₅

yes

—

#322

—

#322

—

C₂³×C₅

—

C₂³×C₅

#117

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

100

C₂×C₄×C₅

—

101

C₂×C₄×C₅

C₂×C₄²

yes

characteristic

102

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₄×C₅

C₄×D₄

yes

—

104

C₂×C₄×C₅

C₄×D₄

yes

—

105

C₅×C₈

—

106

#168

—

107

#168

—

108

#168

—

109

#168

—

110

#168

#100

yes

—

111

C₂×#41

C₄×D₅

yes

—

112

#131

—

113

C₂×M₁₆

—

114

C₂×M₁₆

—

115

C₂×M₁₆

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

—

119

C₄⋊Q₈

C₄×D₅

yes

characteristic

120

#127

C₄×D₅

yes

characteristic

121

#124

C₂×Dic₅

yes

—

122

#121

—

123

C₄×D₄

—

124

C₂×C₄²

C₂²×D₅

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₂²⋊C₈

—

128

C₂²⋊C₈

—

129

#97

C₂×Dic₅

yes

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₂²×C₅

—

135

C₂²×C₅

—

136

C₂²×C₅

—

137

C₂²×C₅

—

138

C₂²×C₅

C₄×C₂²⋊C₄

yes

characteristic

139

C₄×C₅

—

140

C₄×C₅

—

141

C₄×C₅

—

142

C₄×C₅

C₂×#100

yes

characteristic

143

C₄×C₅

#413

yes

characteristic

144

C₄×C₅

#444

yes

—

145

#41

—

146

#41

—

147

C₂×Q₈

C₄×Dic₅

yes

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

C₄×Dic₅

yes

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

153

C₂²×C₄

#711

yes

—

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₄⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

C₂×C₄×D₅

yes

characteristic

169

C₄²

D₄×D₅

yes

—

170

C₄²

C₂²×Dic₅

yes

characteristic

171

C₂×C₅

—

172

C₂×C₅

—

173

C₂×C₅

C₄×#100

yes

characteristic

174

C₂×C₅

#1765

yes

—

175

C₂³

—

176

C₂³

#4023

yes

characteristic

177

Q₈

—

178

D₄

—

179

D₄

—

180

D₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

D₅×C₂²⋊C₄

yes

characteristic

188

C₂×C₄

#4027

yes

—

189

C₂×C₄

C₂×C₄×Dic₅

yes

characteristic

190

C₂×C₄

D₄×Dic₅

yes

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₅

#8243

yes

characteristic, complemented, Sylow

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

—

202

C₂²

C₂²⋊C₄×Dic₅

yes

characteristic

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

D₅×#100

yes

characteristic

207

C₄

#65034

yes

characteristic

208

C₄

#65112

yes

—

209

C₂

—

210

C₂

—

211

C₂

Dic₅×#100

yes

characteristic

212

C₂

#10601163

yes

—

	subgroup	G/H
center	C₂×C₄	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1525	C₂
socle	C₂²×C₅	C₄×C₂²⋊C₄
prime radical	C₅×#532	C₂²
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1525	D₅
hypocenter	C₅	#8243
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1525

subgroup

G/H

center

C₂×C₄

D₅×C₂²⋊C₄

commutator

C₂×C₄×C₅

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₅×#1525

C₂

socle

C₂²×C₅

C₄×C₂²⋊C₄

prime radical

C₅×#532

C₂²

cosocle

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1525

N	H ≅ G/N	type
4. #10600640	C₂	semidirect
5. #10598160	C₂	semidirect
197. C₅	#8243	semidirect

H ≅ G/N

type

4. #10600640

C₂

semidirect

5. #10598160

C₂

semidirect

197. C₅

#8243

semidirect

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

subgroup	index
#10615006	2
C₅×#1525	2
#10602598	2
#10600640	2
#10598160	2
#8243	5

subgroup

index

Subgroups — #11990315

Conjugacy classes of subgroups

Complements

Maximal subgroups