Group #20244619 — subgroups

Subgroups — #20244619

Order 6016.

402 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 95 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×#13093945	C₂	1	2	yes	maximal, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal, complemented
3	#15689021	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₇⋊C₄	—	2	1	no	—
10	#13093945	—	2	4	no	—
11	#13093945	—	2	4	no	—
12	C₄₇⋊#95	—	2	1	no	—
13	C₂²×D₄×C₄₇	—	2	1	no	—
14	C₂²×D₄×C₄₇	—	2	1	no	—
15	C₂²×D₄×C₄₇	C₂²	1	2	yes	—
16	#127×C₄₇	—	2	4	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	4	no	—
20	C₂×D₄×C₄₇	—	2	4	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	2	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	4	2	no	—
27	C₂×D₄×C₄₇	—	4	2	no	—
28	C₂×D₄×C₄₇	—	4	2	no	—
29	C₂×D₄×C₄₇	—	4	2	no	—
30	C₂×D₄×C₄₇	D₄	1	4	yes	—
31	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₇	—	2	1	no	—
35	C₂²×Dic₄₇	—	2	1	no	—
36	C₄²×C₄₇	—	2	2	no	—
37	C₄⋊Dic₄₇	—	2	1	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₄⋊Dic₄₇	D₄	1	2	yes	complemented
40	Dic₄₇⋊C₄	—	4	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂³×C₄₇	—	2	4	no	—
45	C₂³×C₄₇	—	4	2	no	—
46	C₂³×C₄₇	—	4	2	no	—
47	C₂³×C₄₇	—	4	2	no	—
48	C₂³×C₄₇	—	4	2	no	—
49	C₂³×C₄₇	—	4	2	no	—
50	C₂³×C₄₇	—	4	2	no	—
51	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
52	D₄×C₄₇	—	2	4	no	—
53	D₄×C₄₇	—	2	4	no	—
54	D₄×C₄₇	—	4	8	no	—
55	D₄×C₄₇	—	4	8	no	—
56	D₄×C₄₇	—	4	2	no	—
57	D₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	D₈	1	2	yes	—
63	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₄₇	#41	1	2	yes	—
67	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
68	C₂×Dic₄₇	—	2	2	no	—
69	C₂×Dic₄₇	—	4	1	no	—
70	C₂×Dic₄₇	—	4	1	no	—
71	C₂×Dic₄₇	—	4	1	no	—
72	C₂×Dic₄₇	—	4	1	no	—
73	C₂×Dic₄₇	—	4	1	no	—
74	C₄₇⋊C₈	—	4	2	no	—
75	C₂²×C₄₇	—	4	4	no	—
76	C₂²×C₄₇	—	4	4	no	—
77	C₂²×C₄₇	—	4	4	no	—
78	C₂²×C₄₇	—	8	2	no	—
79	C₂²×C₄₇	—	8	2	no	—
80	C₂²×C₄₇	—	8	2	no	—
81	C₂²×C₄₇	—	8	2	no	—
82	C₂²×C₄₇	#120	1	1	yes	characteristic
83	C₂²×C₄₇	#121	1	1	yes	characteristic
84	C₂²×C₄₇	#124	1	1	yes	characteristic
85	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
86	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₄×C₄₇	—	2	2	no	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	#121	1	2	yes	—
92	C₄×C₄₇	#123	1	2	yes	—
93	Dic₄₇	—	4	2	no	—
94	Dic₄₇	—	8	1	no	—
95	#2041	—	47	1	no	maximal, Sylow
96	C₂×C₄₇	—	8	4	no	—
97	C₂×C₄₇	#394	1	1	yes	characteristic
98	C₂×C₄₇	#447	1	1	yes	characteristic
99	C₂×C₄₇	#450	1	1	yes	characteristic
100	C₂×C₄₇	#459	1	1	yes	characteristic
101	C₂×C₄₇	#460	1	1	yes	characteristic
102	C₂×C₄₇	#486	1	1	yes	characteristic
103	C₂×C₄₇	#490	1	1	yes	characteristic
104	C₂×#127	D₄₇	1	1	yes	characteristic
105	C₂×#102	—	47	2	no	—
106	C₂×#102	—	47	2	no	—
107	#389	—	47	1	no	—
108	#336	—	47	1	no	—
109	C₄₇	#2041	1	1	yes	characteristic, complemented, Sylow
110	C₂²×D₄	—	2	1	no	—
111	C₂²×D₄	—	2	1	no	—
112	C₂²×D₄	C₂×D₄₇	1	2	yes	—
113	C₂²×C₈	—	47	2	no	—
114	#127	—	2	4	no	—
115	C₂×C₄⋊C₄	—	47	1	no	—
116	C₂×C₄⋊C₄	—	47	1	no	—
117	C₂×C₄⋊C₄	—	94	1	no	—
118	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
119	#102	—	94	4	no	—
120	#102	—	94	4	no	—
121	#95	—	94	1	no	—
122	C₂⁴	—	2	2	no	—
123	C₂×D₄	—	2	4	no	—
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	—	2	2	no	—
130	C₂×D₄	—	4	2	no	—
131	C₂×D₄	—	4	2	no	—
132	C₂×D₄	—	4	2	no	—
133	C₂×D₄	—	4	2	no	—
134	C₂×D₄	C₂²⋊D₄₇	1	4	yes	—
135	C₂²×C₄	—	94	1	no	—
136	C₂²×C₄	—	94	1	no	—
137	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
139	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₄⋊C₄	—	47	2	no	—
144	C₄⋊C₄	—	94	1	no	—
145	C₄⋊C₄	—	94	2	no	—
146	C₄⋊C₄	—	188	2	no	—
147	C₄²	—	2	2	no	—
148	C₂³	—	2	4	no	—
149	C₂³	—	4	2	no	—
150	C₂³	—	4	2	no	—
151	C₂³	—	4	2	no	—
152	C₂³	—	4	2	no	—
153	C₂³	—	4	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
156	D₄	—	2	4	no	—
157	D₄	—	2	4	no	—
158	D₄	—	4	8	no	—
159	D₄	—	4	8	no	—
160	D₄	—	4	2	no	—
161	D₄	—	4	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	94	2	no	—
167	C₂×C₄	—	188	1	no	—
168	C₂×C₄	—	188	1	no	—
169	C₂×C₄	—	188	1	no	—
170	C₂×C₄	—	188	1	no	—
171	C₂×C₄	—	188	1	no	—
172	C₂×C₄	#10625334	1	2	yes	—
173	C₂×C₄	#10625335	1	2	yes	—
174	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
175	C₂×C₄	#10625352	1	2	yes	—
176	C₂×C₄	#10625352	1	1	yes	characteristic
177	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
178	C₈	—	188	2	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	4	no	—
182	C₂²	—	8	2	no	—
183	C₂²	—	8	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	2	no	—
186	C₂²	#13093993	1	1	yes	characteristic
187	C₂²	#13093995	1	1	yes	characteristic
188	C₂²	#13094011	1	1	yes	characteristic
189	C₂²	C₂×#10625334	1	1	yes	characteristic
190	C₂²	C₂×#10625335	1	1	yes	characteristic
191	C₂²	#13094050	1	1	yes	characteristic
192	C₂²	#13094050	1	1	yes	characteristic
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	376	1	no	—
196	C₄	—	188	2	no	—
197	C₄	#13093996	1	2	yes	—
198	C₄	#13094005	1	2	yes	—
199	C₂	—	8	4	no	—
200	C₂	#15689037	1	1	yes	characteristic
201	C₂	#15689257	1	1	yes	characteristic
202	C₂	#15689274	1	1	yes	characteristic
203	C₂	#15689315	1	1	yes	characteristic
204	C₂	#15689320	1	1	yes	characteristic
205	C₂	#15689469	1	1	yes	characteristic
206	C₂	#15689487	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093945

C₂

yes

maximal, complemented

C₂×#13093945

C₂

yes

maximal, complemented

#15689021

C₂

yes

maximal, characteristic, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×Dic₄₇⋊C₄

—

#13093945

—

#13093945

—

C₄₇⋊#95

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂²

yes

—

#127×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

complemented

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂²×C₄₇

C₂×SD₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

#2041

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#459

yes

characteristic

101

C₂×C₄₇

#460

yes

characteristic

102

C₂×C₄₇

#486

yes

characteristic

103

C₂×C₄₇

#490

yes

characteristic

104

C₂×#127

D₄₇

yes

characteristic

105

C₂×#102

—

106

C₂×#102

—

107

#389

—

108

#336

—

109

C₄₇

#2041

yes

characteristic, complemented, Sylow

110

C₂²×D₄

—

111

C₂²×D₄

—

112

C₂²×D₄

C₂×D₄₇

yes

—

113

C₂²×C₈

—

114

#127

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₂×D₄₇

yes

characteristic

119

#102

—

120

#102

—

121

#95

—

122

C₂⁴

—

123

C₂×D₄

—

124

C₂×D₄

—

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂×D₄

—

132

C₂×D₄

—

133

C₂×D₄

—

134

C₂×D₄

C₂²⋊D₄₇

yes

—

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

138

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

139

C₂²×C₄

C₂²×D₄₇

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

188

—

147

C₄²

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

156

D₄

—

157

D₄

—

158

D₄

—

159

D₄

—

160

D₄

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

188

—

168

C₂×C₄

—

188

—

169

C₂×C₄

—

188

—

170

C₂×C₄

—

188

—

171

C₂×C₄

—

188

—

172

C₂×C₄

#10625334

yes

—

173

C₂×C₄

#10625335

yes

—

174

C₂×C₄

D₄×D₄₇

yes

characteristic

175

C₂×C₄

#10625352

yes

—

176

C₂×C₄

#10625352

yes

characteristic

177

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

178

C₈

—

188

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

#13093993

yes

characteristic

187

C₂²

#13093995

yes

characteristic

188

C₂²

#13094011

yes

characteristic

189

C₂²

C₂×#10625334

yes

characteristic

190

C₂²

C₂×#10625335

yes

characteristic

191

C₂²

#13094050

yes

characteristic

192

C₂²

#13094050

yes

characteristic

193

C₄

—

194

C₄

—

195

C₄

—

376

—

196

C₄

—

188

—

197

C₄

#13093996

yes

—

198

C₄

#13094005

yes

—

199

C₂

—

200

C₂

#15689037

yes

characteristic

201

C₂

#15689257

yes

characteristic

202

C₂

#15689274

yes

characteristic

203

C₂

#15689315

yes

characteristic

204

C₂

#15689320

yes

characteristic

205

C₂

#15689469

yes

characteristic

206

C₂

#15689487

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#127×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#127	D₄₇
hypocenter	C₄₇	#2041
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#127×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#13093945	C₂	semidirect
2. C₂×#13093945	C₂	semidirect
3. #15689021	C₂	semidirect
4. #15688878	C₂	semidirect
39. C₄⋊Dic₄₇	D₄	semidirect
109. C₄₇	#2041	semidirect

H ≅ G/N

type

1. C₂×#13093945

C₂

semidirect

2. C₂×#13093945

C₂

semidirect

3. #15689021

C₂

semidirect

4. #15688878

C₂

semidirect

39. C₄⋊Dic₄₇

D₄

semidirect

109. C₄₇

#2041

semidirect

Subgroups — #20244619

Conjugacy classes of subgroups

Complements