Group #20254783 — subgroups

Subgroups — #20254783

Order 6016.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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	#15689654	C₂	1	2	yes	maximal, complemented
2	#15689433	C₂	1	2	yes	maximal, complemented
3	#15689425	C₂	1	2	yes	maximal, complemented
4	#15689381	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689367	C₂	1	2	yes	maximal, complemented
6	#15689344	C₂	1	2	yes	maximal, complemented
7	#15689336	C₂	1	2	yes	maximal, complemented
8	#15689146	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#469	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#10625335	C₂²	1	2	yes	—
12	C₂×#10625329	C₂²	1	1	yes	characteristic
13	#13094017	—	2	2	no	—
14	#13094015	C₂²	1	2	yes	—
15	#13094011	C₂²	1	2	yes	—
16	#13094005	—	2	2	no	—
17	#13094003	C₂²	1	2	yes	—
18	#13093989	C₂²	1	2	yes	—
19	D₄×Dic₄₇	—	2	2	no	—
20	D₄×Dic₄₇	C₂²	1	2	yes	—
21	#13093980	—	2	2	no	—
22	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
23	#13093957	C₂²	1	2	yes	—
24	#13093956	C₂²	1	1	yes	characteristic
25	Dic₄₇⋊C₈	C₂²	1	2	yes	—
26	Dic₄₇⋊C₈	C₂²	1	2	yes	—
27	#13093949	C₂²	1	2	yes	—
28	#13093947	C₂²	1	2	yes	—
29	#13093945	C₂²	1	2	yes	—
30	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
32	#121×C₄₇	C₂²	1	2	yes	—
33	#106×C₄₇	C₂²	1	1	yes	characteristic
34	#102×C₄₇	C₂²	1	2	yes	—
35	C₂×D₄×C₄₇	—	2	2	no	—
36	C₂×D₄×C₄₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	2	no	—
39	C₂²×Dic₄₇	D₄	1	2	yes	—
40	C₂×Dic₉₄	C₂³	1	2	yes	—
41	D₈×C₄₇	—	4	1	no	—
42	M₁₆×C₄₇	—	4	1	no	—
43	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
46	C₂²⋊C₄×C₄₇	—	2	2	no	—
47	C₂²⋊Dic₄₇	—	2	2	no	—
48	C₂²⋊Dic₄₇	—	2	2	no	—
49	C₂²⋊Dic₄₇	—	2	2	no	—
50	C₂²⋊Dic₄₇	—	2	2	no	—
51	#10625335	—	4	2	no	—
52	C₄⋊Dic₄₇	—	2	2	no	—
53	C₄⋊Dic₄₇	C₂³	1	2	yes	—
54	Dic₄₇⋊C₄	—	2	2	no	—
55	Dic₄₇⋊C₄	—	2	2	no	—
56	Dic₄₇⋊C₄	—	2	2	no	—
57	C₄×Dic₄₇	—	2	2	no	—
58	C₄×Dic₄₇	D₄	1	1	yes	characteristic
59	C₄×Dic₄₇	D₄	1	1	yes	characteristic
60	C₄×Dic₄₇	C₂³	1	2	yes	—
61	#10625329	—	4	1	no	—
62	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
63	Dic₁₈₈	—	4	1	no	—
64	C₂³×C₄₇	—	2	2	no	—
65	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
66	D₄×C₄₇	—	2	2	no	—
67	D₄×C₄₇	—	2	2	no	—
68	D₄×C₄₇	—	4	4	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	2	2	no	—
76	C₂×Dic₄₇	—	4	2	no	—
77	C₂×Dic₄₇	—	4	4	no	—
78	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
79	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
80	Dic₉₄	—	2	2	no	—
81	Dic₉₄	—	2	2	no	—
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₄₇⋊C₈	—	2	2	no	—
85	C₂²×C₄₇	—	2	1	no	—
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	4	2	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	4	2	no	—
93	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
94	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
95	Dic₄₇	—	2	2	no	—
96	Dic₄₇	—	2	2	no	—
97	Dic₄₇	—	4	2	no	—
98	Dic₄₇	—	4	2	no	—
99	#3256	—	47	1	no	maximal, Sylow
100	C₂×C₄₇	—	4	1	no	—
101	C₂×C₄₇	—	8	2	no	—
102	C₂×C₄₇	#537	1	1	yes	characteristic
103	C₂×C₄₇	#575	1	2	yes	—
104	#529	—	47	2	no	—
105	#479	—	47	2	no	—
106	#478	—	47	2	no	—
107	#470	—	47	1	no	—
108	#469	D₄₇	1	1	yes	characteristic
109	#468	—	47	2	no	—
110	#464	—	47	2	no	—
111	#463	—	47	2	no	—
112	#423	—	47	1	no	—
113	C₄₇	#3256	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₁₆	—	47	1	no	—
115	C₂×SD₁₆	—	47	2	no	—
116	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
117	C₂×M₁₆	—	47	1	no	—
118	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
119	#126	—	94	2	no	—
120	#125	—	47	2	no	—
121	#124	—	47	2	no	—
122	#123	—	94	2	no	—
123	C₂²⋊Q₈	—	47	2	no	—
124	#121	C₂×D₄₇	1	2	yes	—
125	C₄×Q₈	—	47	2	no	—
126	C₄×D₄	—	47	2	no	—
127	C₄×D₄	—	94	2	no	—
128	#117	—	94	2	no	—
129	C₂×C₄²	—	47	1	no	—
130	#107	—	47	2	no	—
131	#106	—	47	1	no	—
132	#106	C₂×D₄₇	1	1	yes	characteristic
133	C₄⋊C₈	—	47	2	no	—
134	C₄⋊C₈	—	47	2	no	—
135	#103	—	47	2	no	—
136	#103	—	47	2	no	—
137	#102	—	47	2	no	—
138	#102	C₂×D₄₇	1	2	yes	—
139	C₂×Q₈	—	47	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₂²×D₄₇	1	2	yes	—
142	C₂²×C₄	—	47	2	no	—
143	C₂²×C₄	—	94	2	no	—
144	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
145	Q₁₆	—	188	1	no	—
146	SD₁₆	—	188	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	188	1	no	—
150	C₂×C₈	—	47	2	no	—
151	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
152	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
153	C₄⋊C₄	—	47	2	no	—
154	C₄⋊C₄	—	94	2	no	—
155	C₄⋊C₄	—	94	2	no	—
156	C₄⋊C₄	—	94	2	no	—
157	C₄⋊C₄	—	94	2	no	—
158	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	94	2	no	—
161	C₂²⋊C₄	—	94	2	no	—
162	C₂²⋊C₄	—	94	2	no	—
163	C₂²⋊C₄	—	94	2	no	—
164	C₄²	—	47	1	no	—
165	C₄²	—	47	1	no	—
166	C₄²	—	47	2	no	—
167	C₄²	—	94	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	D₄×D₄₇	1	1	yes	characteristic
170	Q₈	—	94	2	no	—
171	Q₈	—	94	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	47	2	no	—
179	C₂×C₄	—	47	2	no	—
180	C₂×C₄	—	94	2	no	—
181	C₂×C₄	—	94	2	no	—
182	C₂×C₄	—	188	2	no	—
183	C₂×C₄	—	188	4	no	—
184	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
185	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	1	no	—
188	C₈	—	94	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	94	2	no	—
198	C₄	—	94	2	no	—
199	C₄	—	188	2	no	—
200	C₄	—	188	2	no	—
201	C₄	#13094069	1	1	yes	characteristic
202	C₄	#13094071	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	#15689700	1	1	yes	characteristic
206	C₂	#15689919	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689654

C₂

yes

maximal, complemented

#15689433

C₂

yes

maximal, complemented

#15689425

C₂

yes

maximal, complemented

#15689381

C₂

yes

maximal, characteristic, complemented

#15689367

C₂

yes

maximal, complemented

#15689344

C₂

yes

maximal, complemented

#15689336

C₂

yes

maximal, complemented

#15689146

C₂

yes

maximal, characteristic, complemented

C₄₇×#469

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×#10625335

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

#13094017

—

#13094015

C₂²

yes

—

#13094011

C₂²

yes

—

#13094005

—

#13094003

C₂²

yes

—

#13093989

C₂²

yes

—

D₄×Dic₄₇

—

D₄×Dic₄₇

C₂²

yes

—

#13093980

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093957

C₂²

yes

—

#13093956

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

#13093949

C₂²

yes

—

#13093947

C₂²

yes

—

#13093945

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#121×C₄₇

C₂²

yes

—

#106×C₄₇

C₂²

yes

characteristic

#102×C₄₇

C₂²

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

—

C₂×Dic₉₄

C₂³

yes

—

D₈×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#3256

—

maximal, Sylow

100

C₂×C₄₇

—

101

C₂×C₄₇

—

102

C₂×C₄₇

#537

yes

characteristic

103

C₂×C₄₇

#575

yes

—

104

#529

—

105

#479

—

106

#478

—

107

#470

—

108

#469

D₄₇

yes

characteristic

109

#468

—

110

#464

—

111

#463

—

112

#423

—

113

C₄₇

#3256

yes

characteristic, complemented, Sylow

114

C₂×Q₁₆

—

115

C₂×SD₁₆

—

116

C₂×D₈

C₂×D₄₇

yes

characteristic

117

C₂×M₁₆

—

118

C₂×M₁₆

C₂×D₄₇

yes

characteristic

119

#126

—

120

#125

—

121

#124

—

122

#123

—

123

C₂²⋊Q₈

—

124

#121

C₂×D₄₇

yes

—

125

C₄×Q₈

—

126

C₄×D₄

—

127

C₄×D₄

—

128

#117

—

129

C₂×C₄²

—

130

#107

—

131

#106

—

132

#106

C₂×D₄₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

#103

—

136

#103

—

137

#102

—

138

#102

C₂×D₄₇

yes

—

139

C₂×Q₈

—

140

C₂×D₄

—

141

C₂×D₄

C₂²×D₄₇

yes

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

C₂²×D₄₇

yes

characteristic

145

Q₁₆

—

188

—

146

SD₁₆

—

188

—

147

D₈

—

148

M₁₆

—

149

M₁₆

—

188

—

150

C₂×C₈

—

151

C₂×C₈

C₂²×D₄₇

yes

characteristic

152

C₂×C₈

C₂²×D₄₇

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

C₂²×D₄₇

yes

—

159

C₂²⋊C₄

—

160

C₂²⋊C₄

—

161

C₂²⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

—

169

C₂³

D₄×D₄₇

yes

characteristic

170

Q₈

—

171

Q₈

—

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

188

—

183

C₂×C₄

—

188

—

184

C₂×C₄

D₄×D₄₇

yes

characteristic

185

C₂×C₄

C₂³×D₄₇

yes

characteristic

186

C₈

—

187

C₈

—

188

C₈

—

189

C₂²

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

C₂×D₄×D₄₇

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

188

—

200

C₄

—

188

—

201

C₄

#13094069

yes

characteristic

202

C₄

#13094071

yes

characteristic

203

C₂

—

204

C₂

—

205

C₂

#15689700

yes

characteristic

206

C₂

#15689919

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#469	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#469	D₄₇
hypocenter	C₄₇	#3256
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#469

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #469

N	H ≅ G/N	type
1. #15689654	C₂	semidirect
2. #15689433	C₂	semidirect
3. #15689425	C₂	semidirect
4. #15689381	C₂	semidirect
5. #15689367	C₂	semidirect
6. #15689344	C₂	semidirect
7. #15689336	C₂	semidirect
8. #15689146	C₂	semidirect
113. C₄₇	#3256	semidirect

H ≅ G/N

type

1. #15689654

C₂

semidirect

2. #15689433

C₂

semidirect

3. #15689425

C₂

semidirect

4. #15689381

C₂

semidirect

5. #15689367

C₂

semidirect

6. #15689344

C₂

semidirect

7. #15689336

C₂

semidirect

8. #15689146

C₂

semidirect

113. C₄₇

#3256

semidirect

Subgroups — #20254783

Conjugacy classes of subgroups

Complements