Group #11999331 — subgroups

Subgroups — #11999331

Order 1280.

486 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 159 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×#4016	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
3	#10600187	C₂	1	1	yes	maximal, characteristic
4	#10598146	C₂	1	4	yes	maximal
5	C₂×#4016	C₄	1	1	yes	characteristic
6	C₂×C₅×#97	—	2	2	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	#65225	C₄	1	1	yes	characteristic
10	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
11	#65222	—	2	2	no	—
12	C₄×#683	—	2	2	no	—
13	#65217	C₄	1	2	yes	—
14	C₂×C₄⋊#168	—	2	2	no	—
15	C₂×C₄⋊#168	C₂²	1	2	yes	—
16	#9876	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅	D₄	1	2	yes	—
21	C₂×C₄²×C₅	D₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
25	C₂×#683	—	2	2	no	—
26	C₂×#683	—	2	2	no	—
27	C₂×C₄×Dic₅	—	2	2	no	—
28	C₂²×#168	—	2	2	no	—
29	#4016	—	2	4	no	—
30	C₅×#97	—	2	8	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	C₄×C₅×C₈	Q₈	1	4	yes	—
33	#3963	—	2	2	no	—
34	#3963	—	2	2	no	—
35	#3963	—	2	2	no	—
36	C₄⋊#168	—	2	4	no	—
37	C₄⋊#168	—	4	4	no	—
38	C₄×#125	—	5	1	no	—
39	#1842	—	5	1	no	—
40	C₄×#97	D₅	1	1	yes	characteristic
41	#1279	—	5	4	no	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	2	2	no	—
48	C₂²×Dic₅	—	2	2	no	—
49	C₂×C₅×C₈	—	2	8	no	—
50	C₂×C₅×C₈	—	2	4	no	—
51	C₄²×C₅	—	2	4	no	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	—	2	2	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	C₂×Q₈	1	2	yes	—
57	C₄²×C₅	#41	1	2	yes	—
58	C₄⋊Dic₅	—	2	2	no	—
59	C₄⋊Dic₅	—	2	4	no	—
60	C₄⋊Dic₅	M₁₆	1	4	yes	—
61	#683	—	2	8	no	—
62	#683	—	4	4	no	—
63	C₄×Dic₅	—	4	2	no	—
64	C₄×Dic₅	—	4	2	no	—
65	C₂×#168	—	4	2	no	—
66	C₂×#168	—	4	4	no	—
67	C₂×#125	—	5	1	no	—
68	C₂×C₄⋊C₈	—	5	2	no	—
69	C₂×C₄⋊C₈	—	10	2	no	—
70	C₂×#97	—	2	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
72	#389	—	5	2	no	—
73	#387	—	10	2	no	—
74	#382	—	5	1	no	—
75	C₄×C₄⋊C₄	—	5	2	no	—
76	C₄×C₄⋊C₄	—	10	2	no	—
77	C₄³	C₂×D₅	1	1	yes	characteristic
78	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	#104	1	4	yes	—
86	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
88	C₂×C₄×C₅	#124	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅	#131	1	2	yes	—
92	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₅	#137	1	2	yes	—
94	C₂×Dic₅	—	2	4	no	—
95	C₂×Dic₅	—	4	2	no	—
96	C₂×Dic₅	—	4	2	no	—
97	C₂×Dic₅	—	4	2	no	—
98	C₂×Dic₅	—	4	2	no	—
99	C₂×Dic₅	—	4	2	no	—
100	C₅×C₈	—	2	8	no	—
101	#168	—	8	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	—	10	2	no	—
104	#125	—	10	4	no	—
105	C₂×C₄⋊C₄	—	5	2	no	—
106	C₂×C₄⋊C₄	—	5	2	no	—
107	C₂×C₄⋊C₄	—	10	2	no	—
108	C₂×C₄⋊C₄	—	10	2	no	—
109	C₂×C₄²	—	10	2	no	—
110	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
111	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
112	C₂×C₄²	D₂₀	1	2	yes	—
113	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
114	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
115	C₄⋊C₈	—	10	4	no	—
116	C₄⋊C₈	—	20	4	no	—
117	#97	—	2	8	no	—
118	C₄×C₈	—	2	2	no	—
119	C₄×C₈	Dic₁₀	1	4	yes	—
120	#95	—	10	2	no	—
121	#95	—	10	2	no	—
122	#95	—	10	2	no	—
123	C₂²×C₅	#391	1	1	yes	characteristic
124	C₂²×C₅	#407	1	1	yes	characteristic
125	C₂²×C₅	#408	1	1	yes	characteristic
126	C₂²×C₅	#417	1	2	yes	—
127	C₂²×C₅	C₂×#104	1	2	yes	—
128	C₄×C₅	—	2	2	no	—
129	C₄×C₅	—	2	4	no	—
130	C₄×C₅	—	4	2	no	—
131	C₄×C₅	#446	1	4	yes	—
132	C₄×C₅	#490	1	1	yes	characteristic
133	C₄×C₅	#491	1	1	yes	characteristic
134	C₄×C₅	#501	1	2	yes	—
135	Dic₅	—	4	4	no	—
136	Dic₅	—	8	2	no	—
137	C₂²×C₄	—	10	2	no	—
138	C₂²×C₄	—	10	2	no	—
139	C₂²×C₄	#685	1	2	yes	—
140	C₂²×C₄	#685	1	2	yes	—
141	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
143	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	20	2	no	—
147	C₂×C₈	—	20	4	no	—
148	C₄⋊C₄	—	5	4	no	—
149	C₄⋊C₄	—	10	8	no	—
150	C₄⋊C₄	—	10	2	no	—
151	C₄⋊C₄	—	10	4	no	—
152	C₄⋊C₄	—	20	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	20	2	no	—
159	C₄²	—	20	2	no	—
160	C₄²	C₂×Dic₁₀	1	2	yes	—
161	C₄²	#709	1	2	yes	—
162	C₂×C₅	#1483	1	2	yes	—
163	C₂×C₅	#1489	1	2	yes	—
164	C₂×C₅	#1985	1	1	yes	characteristic
165	C₂×C₅	#1992	1	1	yes	characteristic
166	C₂×C₅	#1994	1	1	yes	characteristic
167	C₂³	C₂×#685	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	10	4	no	—
175	C₂×C₄	—	20	2	no	—
176	C₂×C₄	—	20	2	no	—
177	C₂×C₄	—	20	2	no	—
178	C₂×C₄	—	20	2	no	—
179	C₂×C₄	—	20	2	no	—
180	C₂×C₄	#3937	1	4	yes	—
181	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
182	C₂×C₄	#4015	1	1	yes	characteristic
183	C₂×C₄	#4021	1	1	yes	characteristic
184	C₂×C₄	D₅×M₁₆	1	2	yes	—
185	C₂×C₄	#4053	1	2	yes	—
186	C₂×C₄	#4054	1	2	yes	—
187	C₂×C₄	#4055	1	2	yes	—
188	C₂×C₄	#4070	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₈	—	40	2	no	—
191	C₅	#9876	1	1	yes	characteristic, complemented, Sylow
192	C₂²	C₂×#3937	1	2	yes	—
193	C₂²	#65218	1	1	yes	characteristic
194	C₂²	#65415	1	1	yes	characteristic
195	C₂²	#65418	1	1	yes	characteristic
196	C₂²	#65422	1	2	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	20	4	no	—
201	C₄	—	40	2	no	—
202	C₄	#64990	1	4	yes	—
203	C₄	#64991	1	2	yes	—
204	C₄	#64992	1	1	yes	characteristic
205	C₄	#65002	1	1	yes	characteristic
206	C₂	#10600189	1	1	yes	characteristic
207	C₂	#10600620	1	1	yes	characteristic
208	C₂	#10600631	1	1	yes	characteristic
209	C₂	#10600645	1	2	yes	—
210	C₂	#10600647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#4016

C₂

yes

maximal, characteristic

C₄×C₅×#97

C₂

yes

maximal, characteristic

#10600187

C₂

yes

maximal, characteristic

#10598146

C₂

yes

maximal

C₂×#4016

C₄

yes

characteristic

C₂×C₅×#97

—

C₂×C₄×C₅×C₈

C₂²

yes

—

C₄³×C₅

C₂²

yes

characteristic

#65225

C₄

yes

characteristic

C₄×C₄⋊Dic₅

C₂²

yes

—

#65222

—

C₄×#683

—

#65217

C₄

yes

—

C₂×C₄⋊#168

—

C₂×C₄⋊#168

C₂²

yes

—

#9876

—

maximal, Sylow

C₂²×C₅×C₈

—

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

C₂×C₄

yes

characteristic

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

C₂×C₄

yes

—

C₂×C₄⋊Dic₅

C₂×C₄

yes

—

C₂×#683

—

C₂×#683

—

C₂×C₄×Dic₅

—

C₂²×#168

—

#4016

—

C₅×#97

—

C₄×C₅×C₈

—

C₄×C₅×C₈

Q₈

yes

—

—

—

—

—

—

—

—

C₄×#97

D₅

yes

characteristic

#1279

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

M₁₆

yes

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₅

yes

—

#389

—

#387

—

#382

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂×D₅

yes

characteristic

C₂³×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#104

yes

—

C₂×C₄×C₅

C₄×Q₈

yes

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

#124

yes

characteristic

C₂×C₄×C₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×M₁₆

yes

—

C₂×C₄×C₅

#131

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

100

C₅×C₈

—

101

#168

—

102

C₂²×C₈

—

103

C₂²×C₈

—

104

#125

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄²

—

110

C₂×C₄²

C₄×D₅

yes

characteristic

111

C₂×C₄²

C₄×D₅

yes

characteristic

112

C₂×C₄²

D₂₀

yes

—

113

C₂×C₄²

C₂²⋊D₅

yes

—

114

C₂×C₄²

C₂²×D₅

yes

characteristic

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

Dic₁₀

yes

—

120

#95

—

121

#95

—

122

#95

—

123

C₂²×C₅

#391

yes

characteristic

124

C₂²×C₅

#407

yes

characteristic

125

C₂²×C₅

#408

yes

characteristic

126

C₂²×C₅

#417

yes

—

127

C₂²×C₅

C₂×#104

yes

—

128

C₄×C₅

—

129

C₄×C₅

—

130

C₄×C₅

—

131

C₄×C₅

#446

yes

—

132

C₄×C₅

#490

yes

characteristic

133

C₄×C₅

#491

yes

characteristic

134

C₄×C₅

#501

yes

—

135

Dic₅

—

136

Dic₅

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

#685

yes

—

140

C₂²×C₄

#685

yes

—

141

C₂²×C₄

C₂×C₄×D₅

yes

characteristic

142

C₂²×C₄

C₂×D₂₀

yes

characteristic

143

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×Dic₁₀

yes

—

161

C₄²

#709

yes

—

162

C₂×C₅

#1483

yes

—

163

C₂×C₅

#1489

yes

—

164

C₂×C₅

#1985

yes

characteristic

165

C₂×C₅

#1992

yes

characteristic

166

C₂×C₅

#1994

yes

characteristic

167

C₂³

C₂×#685

yes

characteristic

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#3937

yes

—

181

C₂×C₄

C₄×Dic₁₀

yes

—

182

C₂×C₄

#4015

yes

characteristic

183

C₂×C₄

#4021

yes

characteristic

184

C₂×C₄

D₅×M₁₆

yes

—

185

C₂×C₄

#4053

yes

—

186

C₂×C₄

#4054

yes

—

187

C₂×C₄

#4055

yes

—

188

C₂×C₄

#4070

yes

—

189

C₈

—

190

C₈

—

191

C₅

#9876

yes

characteristic, complemented, Sylow

192

C₂²

C₂×#3937

yes

—

193

C₂²

#65218

yes

characteristic

194

C₂²

#65415

yes

characteristic

195

C₂²

#65418

yes

characteristic

196

C₂²

#65422

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

#64990

yes

—

203

C₄

#64991

yes

—

204

C₄

#64992

yes

characteristic

205

C₄

#65002

yes

characteristic

206

C₂

#10600189

yes

characteristic

207

C₂

#10600620

yes

characteristic

208

C₂

#10600631

yes

characteristic

209

C₂

#10600645

yes

—

210

C₂

#10600647

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₄×C₅×#97	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	C₄×#97	D₅
hypocenter	C₅	#9876
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

subgroup

G/H

center

C₂²×C₄

C₂×D₂₀

commutator

C₂²×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

Fitting

C₄×C₅×#97

C₂

socle, prime radical

C₂³×C₅

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₅

C₂³

hypercenter

C₄×#97

D₅

hypocenter

C₅

#9876

LCS

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

N	H ≅ G/N	type
191. C₅	#9876	semidirect

H ≅ G/N

type

191. C₅

#9876

semidirect

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

subgroup	index
C₄×#4016	2
C₄×C₅×#97	2
#10600187	2
#10598146	2
#9876	5

subgroup

index

C₄×C₅×#97

Subgroups — #11999331

Conjugacy classes of subgroups

Complements

Maximal subgroups