Group #11999127 — subgroups

Subgroups — #11999127

Order 1280.

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

normal subgroups only

This group has 526 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	C₅×#1834	C₂	1	1	yes	maximal, characteristic
3	C₂×#64674	C₂	1	1	yes	maximal, characteristic
4	#10600375	C₂	1	2	yes	maximal
5	#10600375	C₂	1	2	yes	maximal
6	C₂²×Dic₂₀	—	2	2	no	—
7	C₂×#4015	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#384	C₂²	1	2	yes	—
10	C₂×C₄×#168	C₂²	1	1	yes	characteristic
11	C₂×C₄×#168	C₂²	1	1	yes	characteristic
12	#64988	—	2	2	no	—
13	#64988	C₄	1	4	yes	complemented
14	C₅×#340	—	2	2	no	—
15	#64749	—	2	4	no	—
16	#64749	—	2	4	no	—
17	#64674	—	2	2	no	—
18	#9835	—	5	1	no	maximal, Sylow
19	C₂²×Dic₁₀	—	2	2	no	—
20	C₂²×C₅×C₈	D₄	1	2	yes	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	2	2	no	—
27	C₂²×#168	D₄	1	2	yes	—
28	C₂²×#168	D₄	1	2	yes	—
29	C₂×Dic₂₀	—	2	8	no	—
30	C₂×Dic₂₀	—	4	8	no	—
31	#4015	—	2	2	no	—
32	#4015	—	2	2	no	—
33	#4015	—	2	2	no	—
34	#4015	C₂×C₄	1	4	yes	—
35	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₅×C₈	D₄	1	2	yes	—
37	C₄×#168	—	2	1	no	—
38	C₄×#168	—	2	1	no	—
39	C₄×#168	—	2	1	no	—
40	C₄×#168	—	2	1	no	—
41	C₂×#494	—	5	1	no	—
42	#1990	—	5	2	no	—
43	#1990	—	5	2	no	—
44	#1834	D₅	1	1	yes	characteristic
45	C₂×#322	—	5	1	no	—
46	C₂²×C₄×C₅	—	4	2	no	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₂×Dic₁₀	—	2	4	no	—
52	C₂×Dic₁₀	—	4	8	no	—
53	C₂×Dic₁₀	—	4	2	no	—
54	C₂×C₅×C₈	—	2	4	no	—
55	C₂×C₅×C₈	C₂²⋊C₄	1	4	yes	—
56	C₅×C₄⋊C₄	—	4	4	no	—
57	C₅×C₄⋊C₄	—	4	4	no	—
58	C₅×C₄⋊C₄	—	4	4	no	—
59	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅	#41	1	1	yes	characteristic
62	C₄²×C₅	#41	1	1	yes	characteristic
63	C₄⋊Dic₅	—	2	4	no	—
64	C₄⋊Dic₅	—	2	4	no	—
65	C₄⋊Dic₅	—	4	2	no	—
66	C₄⋊Dic₅	—	4	2	no	—
67	C₂×#168	—	2	2	no	—
68	C₂×#168	—	2	2	no	—
69	C₂×#168	—	2	2	no	—
70	C₂×#168	—	2	4	no	—
71	C₂×#168	Q₁₆	1	4	yes	complemented
72	Dic₂₀	—	4	16	no	—
73	C₂²×Q₁₆	—	10	2	no	—
74	C₂×C₄⋊Q₈	—	5	2	no	—
75	#494	—	5	4	no	—
76	#494	—	10	2	no	—
77	C₂×C₄×C₈	—	5	1	no	—
78	C₂×C₄×C₈	—	5	1	no	—
79	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
80	#384	C₂×D₅	1	2	yes	—
81	#340	—	2	2	no	—
82	#340	—	10	4	no	—
83	#340	—	10	4	no	—
84	#322	—	10	2	no	—
85	C₂³×C₅	#127	1	1	yes	characteristic
86	C₂×C₄×C₅	—	8	2	no	—
87	C₂×C₄×C₅	—	8	2	no	—
88	C₂×C₄×C₅	—	8	2	no	—
89	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	#121	1	1	yes	characteristic
93	C₂×C₄×C₅	#121	1	1	yes	characteristic
94	C₂×C₄×C₅	#124	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₅	#135	1	2	yes	—
97	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
98	C₂×C₄×C₅	#137	1	4	yes	—
99	C₂×Dic₅	—	4	4	no	—
100	C₂×Dic₅	—	8	4	no	—
101	Dic₁₀	—	4	8	no	—
102	Dic₁₀	—	8	4	no	—
103	C₅×C₈	—	2	4	no	—
104	#168	—	2	4	no	—
105	#168	—	4	2	no	—
106	C₂²×Q₈	—	10	2	no	—
107	C₂×Q₁₆	—	10	8	no	—
108	C₂×Q₁₆	—	20	8	no	—
109	C₂²×C₈	—	5	2	no	—
110	C₂²×C₈	—	5	2	no	—
111	C₂²×C₈	C₂²⋊D₅	1	2	yes	—
112	C₄⋊Q₈	—	5	4	no	—
113	C₄⋊Q₈	—	10	2	no	—
114	C₄⋊Q₈	—	10	2	no	—
115	C₄⋊Q₈	—	10	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	10	2	no	—
120	C₂×C₄⋊C₄	—	10	2	no	—
121	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
122	C₄×C₈	—	10	1	no	—
123	C₄×C₈	—	10	1	no	—
124	C₄×C₈	—	10	1	no	—
125	C₄×C₈	—	10	1	no	—
126	C₄×C₈	C₄×D₅	1	2	yes	—
127	C₄×C₈	D₂₀	1	2	yes	—
128	C₂²×C₅	#390	1	1	yes	characteristic
129	C₂²×C₅	#494	1	1	yes	characteristic
130	C₂²×C₅	#495	1	1	yes	characteristic
131	C₂²×C₅	#496	1	2	yes	—
132	C₂²×C₅	#497	1	2	yes	—
133	C₄×C₅	—	16	2	no	—
134	C₄×C₅	#418	1	2	yes	—
135	C₄×C₅	C₄×Q₁₆	1	2	yes	—
136	C₄×C₅	#441	1	2	yes	—
137	C₄×C₅	#467	1	2	yes	—
138	C₄×C₅	#470	1	2	yes	—
139	C₄×C₅	#490	1	2	yes	—
140	Dic₅	—	8	4	no	—
141	C₂×Q₈	—	10	4	no	—
142	C₂×Q₈	—	20	2	no	—
143	C₂×Q₈	—	20	8	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	20	2	no	—
146	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
149	Q₁₆	—	20	16	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	5	4	no	—
152	C₂×C₈	—	10	2	no	—
153	C₂×C₈	—	10	2	no	—
154	C₂×C₈	—	10	2	no	—
155	C₂×C₈	—	10	4	no	—
156	C₂×C₈	#685	1	4	yes	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	10	4	no	—
161	C₄⋊C₄	—	10	4	no	—
162	C₄⋊C₄	—	20	2	no	—
163	C₄⋊C₄	—	20	2	no	—
164	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
165	C₄²	C₂×D₂₀	1	1	yes	characteristic
166	C₄²	#713	1	1	yes	characteristic
167	C₄²	#713	1	1	yes	characteristic
168	C₂×C₅	#1720	1	2	yes	—
169	C₂×C₅	#1723	1	2	yes	—
170	C₂×C₅	#1961	1	1	yes	characteristic
171	C₂×C₅	#1969	1	1	yes	characteristic
172	C₂×C₅	#1971	1	1	yes	characteristic
173	C₂³	#4086	1	1	yes	characteristic
174	Q₈	—	20	8	no	—
175	Q₈	—	40	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	20	4	no	—
180	C₂×C₄	—	40	4	no	—
181	C₂×C₄	#4039	1	1	yes	characteristic
182	C₂×C₄	#4039	1	1	yes	characteristic
183	C₂×C₄	#4041	1	1	yes	characteristic
184	C₂×C₄	#4041	1	1	yes	characteristic
185	C₂×C₄	#4058	1	2	yes	—
186	C₂×C₄	#4061	1	4	yes	—
187	C₂×C₄	D₅×Q₁₆	1	2	yes	—
188	C₂×C₄	C₂×#685	1	1	yes	characteristic
189	C₂×C₄	#4093	1	1	yes	characteristic
190	C₂×C₄	#4095	1	4	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	10	4	no	—
193	C₈	—	20	2	no	—
194	C₅	#9835	1	1	yes	characteristic, complemented, Sylow
195	C₂²	#65277	1	1	yes	characteristic
196	C₂²	#65347	1	2	yes	—
197	C₂²	#65444	1	1	yes	characteristic
198	C₂²	#65456	1	2	yes	—
199	C₂²	#65470	1	1	yes	characteristic
200	C₄	—	16	2	no	—
201	C₄	—	40	4	no	—
202	C₄	#65142	1	2	yes	—
203	C₄	#65157	1	2	yes	—
204	C₄	#65162	1	2	yes	—
205	C₄	#65176	1	2	yes	—
206	C₄	#65290	1	2	yes	—
207	C₄	#65356	1	2	yes	—
208	C₂	#10600346	1	1	yes	characteristic
209	C₂	#10601368	1	1	yes	characteristic
210	C₂	#10601420	1	1	yes	characteristic
211	C₂	#10601986	1	2	yes	—
212	C₂	#10602019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#64988

C₂

yes

maximal, characteristic

C₅×#1834

C₂

yes

maximal, characteristic

C₂×#64674

C₂

yes

maximal, characteristic

#10600375

C₂

yes

maximal

#10600375

C₂

yes

maximal

C₂²×Dic₂₀

—

C₂×#4015

C₂²

yes

—

C₂×C₄×C₅×C₈

C₂²

yes

characteristic

C₅×#384

C₂²

yes

—

C₂×C₄×#168

C₂²

yes

characteristic

C₂×C₄×#168

C₂²

yes

characteristic

#64988

—

#64988

C₄

yes

—

—

—

—

—

C₂²×Dic₁₀

—

C₂²×C₅×C₈

D₄

yes

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂²×#168

D₄

yes

—

C₂²×#168

D₄

yes

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

—

—

—

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

D₄

yes

—

—

—

—

—

—

—

—

D₅

yes

characteristic

C₂×#322

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×C₄×C₅

C₂×D₄

yes

characteristic

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

C₄²×C₅

yes

C₄²×C₅

yes

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

Q₁₆

yes

complemented

Dic₂₀

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅

yes

characteristic

#384

C₂×D₅

yes

—

—

—

—

—

C₂³×C₅

#127

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×C₄×C₅

C₄×D₄

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

yes

C₂×C₄×C₅

C₂×Q₁₆

yes

—

C₂×C₄×C₅

#135

yes

—

C₂×C₄×C₅

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅

#137

yes

—

C₂×Dic₅

—

100

C₂×Dic₅

—

101

Dic₁₀

—

102

Dic₁₀

—

103

C₅×C₈

—

104

#168

—

105

#168

—

106

C₂²×Q₈

—

107

C₂×Q₁₆

—

108

C₂×Q₁₆

—

109

C₂²×C₈

—

110

C₂²×C₈

—

111

C₂²×C₈

C₂²⋊D₅

yes

—

112

C₄⋊Q₈

—

113

C₄⋊Q₈

—

114

C₄⋊Q₈

—

115

C₄⋊Q₈

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄²

C₂²×D₅

yes

characteristic

122

C₄×C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₄×D₅

yes

—

127

C₄×C₈

D₂₀

yes

—

128

C₂²×C₅

#390

yes

characteristic

129

C₂²×C₅

#494

yes

characteristic

130

C₂²×C₅

#495

yes

characteristic

131

C₂²×C₅

#496

yes

—

132

C₂²×C₅

#497

yes

—

133

C₄×C₅

—

134

C₄×C₅

#418

yes

—

135

C₄×C₅

C₄×Q₁₆

yes

—

136

C₄×C₅

#441

yes

—

137

C₄×C₅

#467

yes

—

138

C₄×C₅

#470

yes

—

139

C₄×C₅

#490

yes

—

140

Dic₅

—

141

C₂×Q₈

—

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

D₄×D₅

yes

characteristic

147

C₂²×C₄

D₄×D₅

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₅

yes

characteristic

149

Q₁₆

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#685

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄²

C₂×C₄×D₅

yes

characteristic

165

C₄²

C₂×D₂₀

yes

characteristic

166

C₄²

#713

yes

characteristic

167

C₄²

#713

yes

characteristic

168

C₂×C₅

#1720

yes

—

169

C₂×C₅

#1723

yes

—

170

C₂×C₅

#1961

yes

characteristic

171

C₂×C₅

#1969

yes

characteristic

172

C₂×C₅

#1971

yes

characteristic

173

C₂³

#4086

yes

characteristic

174

Q₈

—

175

Q₈

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#4039

yes

characteristic

182

C₂×C₄

#4039

yes

characteristic

183

C₂×C₄

#4041

yes

characteristic

184

C₂×C₄

#4041

yes

characteristic

185

C₂×C₄

#4058

yes

—

186

C₂×C₄

#4061

yes

—

187

C₂×C₄

D₅×Q₁₆

yes

—

188

C₂×C₄

C₂×#685

yes

characteristic

189

C₂×C₄

#4093

yes

characteristic

190

C₂×C₄

#4095

yes

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₅

#9835

yes

characteristic, complemented, Sylow

195

C₂²

#65277

yes

characteristic

196

C₂²

#65347

yes

—

197

C₂²

#65444

yes

characteristic

198

C₂²

#65456

yes

—

199

C₂²

#65470

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

#65142

yes

—

203

C₄

#65157

yes

—

204

C₄

#65162

yes

—

205

C₄

#65176

yes

—

206

C₄

#65290

yes

—

207

C₄

#65356

yes

—

208

C₂

#10600346

yes

characteristic

209

C₂

#10601368

yes

characteristic

210

C₂

#10601420

yes

characteristic

211

C₂

#10601986

yes

—

212

C₂

#10602019

yes

—

	subgroup	G/H
center	C₂³	#4086
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1834	C₂
socle, prime radical	C₂³×C₅	#127
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1834	D₅
hypocenter	C₅	#9835
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

subgroup

G/H

center

C₂³

#4086

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

C₂×C₄²×C₅

C₂³

D₅

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

N	H ≅ G/N	type
13. #64988	C₄	semidirect
71. C₂×#168	Q₁₆	semidirect
194. C₅	#9835	semidirect

H ≅ G/N

type

13. #64988

C₄

semidirect

71. C₂×#168

Q₁₆

semidirect

194. C₅

#9835

semidirect

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

subgroup	index
C₂×#64988	2
C₅×#1834	2
C₂×#64674	2
#10600375	2
#9835	5

subgroup

index

Subgroups — #11999127

Conjugacy classes of subgroups

Complements

Maximal subgroups