Group #20256479 — subgroups

Subgroups — #20256479

Order 6016.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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	#15689855	C₂	1	2	yes	maximal, complemented
2	#15689509	C₂	1	2	yes	maximal, complemented
3	#15689347	C₂	1	2	yes	maximal, complemented
4	#15689335	C₂	1	2	yes	maximal, complemented
5	#15689316	C₂	1	2	yes	maximal, complemented
6	#15689253	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689224	C₂	1	2	yes	maximal, complemented
8	C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#495	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625336	C₂²	1	2	yes	—
11	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
12	C₂×#10625334	C₂²	1	2	yes	—
13	#13094020	C₂²	1	2	yes	—
14	#13094013	C₂²	1	2	yes	—
15	#13094010	—	2	2	no	—
16	#13094008	—	2	2	no	—
17	#13094006	—	2	2	no	—
18	Q₈×Dic₄₇	C₂²	1	2	yes	—
19	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
20	C₄×D₁₈₈	C₂²	1	2	yes	—
21	D₄×Dic₄₇	C₂²	1	2	yes	—
22	(C₂²⋊D₄₇)⋊C₄	—	2	2	no	—
23	#13093958	C₂²	1	1	yes	characteristic
24	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
25	Dic₄₇⋊C₈	C₂²	1	2	yes	—
26	#13093948	C₂²	1	2	yes	—
27	#13093946	C₂²	1	2	yes	—
28	#13093945	C₂²	1	2	yes	—
29	#13093934	C₂²	1	2	yes	—
30	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
31	C₂×SD₁₆×C₄₇	C₂²	1	2	yes	—
32	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
33	#124×C₄₇	C₂²	1	2	yes	—
34	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
36	C₂×D₄×C₄₇	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₄₇	—	2	2	no	—
38	C₂×C₂²⋊D₄₇	—	2	2	no	—
39	C₂²×Dic₄₇	—	2	2	no	—
40	C₂×D₁₈₈	C₂³	1	2	yes	—
41	C₂×C₄×D₄₇	—	2	2	no	—
42	C₂×Dic₉₄	D₄	1	1	yes	characteristic
43	Q₁₆×C₄₇	—	4	1	no	—
44	SD₁₆×C₄₇	—	2	4	no	—
45	D₈×C₄₇	—	4	1	no	—
46	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
48	C₂²⋊C₄×C₄₇	—	2	2	no	—
49	C₂²⋊C₄×C₄₇	—	2	2	no	—
50	C₄²×C₄₇	C₂³	1	1	yes	characteristic
51	C₂²⋊Dic₄₇	—	2	2	no	—
52	#10625336	—	4	2	no	—
53	#10625334	—	4	2	no	—
54	(C₂×D₄₇)⋊C₄	—	2	2	no	—
55	(C₂×D₄₇)⋊C₄	—	2	2	no	—
56	(C₂×D₄₇)⋊C₄	—	2	2	no	—
57	C₄⋊Dic₄₇	—	2	2	no	—
58	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
59	C₄⋊Dic₄₇	C₂³	1	2	yes	—
60	Dic₄₇⋊C₄	D₄	1	2	yes	—
61	C₄×Dic₄₇	—	2	2	no	—
62	C₄×Dic₄₇	C₂³	1	2	yes	—
63	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
64	D₃₇₆	—	2	2	no	—
65	C₂³×C₄₇	—	2	2	no	—
66	C₂²×D₄₇	—	2	2	no	—
67	Q₈×C₄₇	—	2	2	no	—
68	Q₈×C₄₇	—	2	2	no	—
69	D₄×C₄₇	—	2	2	no	—
70	D₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
75	C₂²⋊D₄₇	—	4	4	no	—
76	C₂²⋊D₄₇	—	8	2	no	—
77	C₂×Dic₄₇	—	2	2	no	—
78	C₂×Dic₄₇	—	8	2	no	—
79	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
80	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
81	D₁₈₈	—	2	2	no	—
82	D₁₈₈	—	2	2	no	—
83	C₄×D₄₇	—	8	2	no	—
84	Dic₉₄	—	4	1	no	—
85	C₈×C₄₇	—	2	1	no	—
86	C₈×C₄₇	#41	1	2	yes	—
87	C₄₇⋊C₈	—	2	2	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	—	4	2	no	—
91	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
92	C₂×D₄₇	—	4	2	no	—
93	C₂×D₄₇	—	4	2	no	—
94	C₂×D₄₇	—	4	2	no	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	4	2	no	—
98	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
99	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
100	Dic₄₇	—	2	2	no	—
101	Dic₄₇	—	2	2	no	—
102	Dic₄₇	—	4	2	no	—
103	#3403	—	47	1	no	maximal, Sylow
104	C₂×C₄₇	—	8	2	no	—
105	C₂×C₄₇	#550	1	1	yes	characteristic
106	C₂×C₄₇	#575	1	1	yes	characteristic
107	C₂×C₄₇	#576	1	1	yes	characteristic
108	D₄₇	—	8	2	no	—
109	#559	—	47	2	no	—
110	#496	—	47	2	no	—
111	#495	D₄₇	1	1	yes	characteristic, complemented
112	#464	—	47	2	no	—
113	#463	—	47	2	no	—
114	#459	—	47	2	no	—
115	#446	—	47	1	no	—
116	#440	—	47	2	no	—
117	C₄×D₈	—	47	1	no	—
118	C₄₇	#3403	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
120	C₂×SD₁₆	—	47	2	no	—
121	C₂×SD₁₆	C₂×D₄₇	1	2	yes	—
122	C₂×D₈	—	47	2	no	—
123	C₂×D₈	—	47	1	no	—
124	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
125	#127	—	47	2	no	—
126	#124	—	47	2	no	—
127	#124	—	94	2	no	—
128	#124	C₂×D₄₇	1	2	yes	—
129	#123	—	94	2	no	—
130	#123	—	94	2	no	—
131	C₄×Q₈	—	47	1	no	—
132	C₄×Q₈	—	47	2	no	—
133	C₄×D₄	—	47	2	no	—
134	C₄×D₄	—	47	2	no	—
135	C₄×D₄	—	94	2	no	—
136	#107	—	47	1	no	—
137	C₄⋊C₈	—	47	2	no	—
138	C₄⋊C₈	—	47	1	no	—
139	#103	—	47	2	no	—
140	#102	—	47	2	no	—
141	#102	—	47	2	no	—
142	#97	—	47	2	no	—
143	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×Q₈	—	47	1	no	—
145	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
146	C₂×D₄	—	47	2	no	—
147	C₂×D₄	—	94	2	no	—
148	C₂×D₄	—	94	2	no	—
149	C₂×D₄	C₂²×D₄₇	1	2	yes	—
150	C₂²×C₄	—	94	2	no	—
151	C₂²×C₄	—	94	2	no	—
152	Q₁₆	—	4	1	no	—
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	188	2	no	—
155	D₈	—	4	1	no	—
156	D₈	—	94	2	no	—
157	D₈	—	188	2	no	—
158	C₂×C₈	—	47	2	no	—
159	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
160	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	C₄⋊C₄	—	47	2	no	—
162	C₄⋊C₄	—	47	1	no	—
163	C₄⋊C₄	—	47	2	no	—
164	C₄⋊C₄	—	94	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	94	2	no	—
168	C₂²⋊C₄	—	94	2	no	—
169	C₂²⋊C₄	—	94	2	no	—
170	C₂²⋊C₄	—	94	2	no	—
171	C₄²	—	47	2	no	—
172	C₄²	—	94	2	no	—
173	C₄²	C₂²×D₄₇	1	1	yes	characteristic
174	C₂³	—	2	2	no	—
175	C₂³	—	94	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	188	1	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	94	2	no	—
182	D₄	—	94	2	no	—
183	D₄	—	376	2	no	—
184	D₄	—	188	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	47	2	no	—
187	C₂×C₄	—	47	2	no	—
188	C₂×C₄	—	94	2	no	—
189	C₂×C₄	—	376	2	no	—
190	C₂×C₄	—	376	2	no	—
191	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	94	2	no	—
196	C₈	#10625352	1	2	yes	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	188	2	no	—
201	C₂²	—	188	2	no	—
202	C₂²	—	188	2	no	—
203	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	4	2	no	—
207	C₄	—	94	2	no	—
208	C₄	—	94	2	no	—
209	C₄	—	188	2	no	—
210	C₄	C₂×#10625352	1	1	yes	characteristic
211	C₄	#13094069	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	—	376	2	no	—
214	C₂	#15689793	1	1	yes	characteristic
215	C₂	#15689920	1	1	yes	characteristic
216	C₂	#15689925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689855

C₂

yes

maximal, complemented

#15689509

C₂

yes

maximal, complemented

#15689347

C₂

yes

maximal, complemented

#15689335

C₂

yes

maximal, complemented

#15689316

C₂

yes

maximal, complemented

#15689253

C₂

yes

maximal, characteristic, complemented

#15689224

C₂

yes

maximal, complemented

C₄×D₃₇₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#495

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

characteristic

C₂×#10625334

C₂²

yes

—

#13094020

C₂²

yes

—

#13094013

C₂²

yes

—

#13094010

—

#13094008

—

#13094006

—

Q₈×Dic₄₇

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

—

D₄×Dic₄₇

C₂²

yes

—

(C₂²⋊D₄₇)⋊C₄

—

#13093958

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

#13093948

C₂²

yes

—

#13093946

C₂²

yes

—

#13093945

C₂²

yes

—

#13093934

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

#124×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×D₄×C₄₇

C₂³

yes

—

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

characteristic

Q₁₆×C₄₇

—

SD₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

#10625336

—

#10625334

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

D₃₇₆

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

100

Dic₄₇

—

101

Dic₄₇

—

102

Dic₄₇

—

103

#3403

—

maximal, Sylow

104

C₂×C₄₇

—

105

C₂×C₄₇

#550

yes

characteristic

106

C₂×C₄₇

#575

yes

characteristic

107

C₂×C₄₇

#576

yes

characteristic

108

D₄₇

—

109

#559

—

110

#496

—

111

#495

D₄₇

yes

characteristic, complemented

112

#464

—

113

#463

—

114

#459

—

115

#446

—

116

#440

—

117

C₄×D₈

—

118

C₄₇

#3403

yes

characteristic, complemented, Sylow

119

C₂×Q₁₆

C₂×D₄₇

yes

characteristic

120

C₂×SD₁₆

—

121

C₂×SD₁₆

C₂×D₄₇

yes

—

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

C₂×D₄₇

yes

characteristic

125

#127

—

126

#124

—

127

#124

—

128

#124

C₂×D₄₇

yes

—

129

#123

—

130

#123

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₄×D₄

—

134

C₄×D₄

—

135

C₄×D₄

—

136

#107

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

#103

—

140

#102

—

141

#102

—

142

#97

—

143

C₄×C₈

C₂×D₄₇

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

C₂²×D₄₇

yes

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

C₂²×D₄₇

yes

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

153

SD₁₆

—

154

SD₁₆

—

188

—

155

D₈

—

156

D₈

—

157

D₈

—

188

—

158

C₂×C₈

—

159

C₂×C₈

C₂²×D₄₇

yes

characteristic

160

C₂×C₈

C₂²×D₄₇

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

—

173

C₄²

C₂²×D₄₇

yes

characteristic

174

C₂³

—

175

C₂³

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

188

—

179

D₄

—

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

376

—

184

D₄

—

188

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

376

—

190

C₂×C₄

—

376

—

191

C₂×C₄

D₄×D₄₇

yes

characteristic

192

C₂×C₄

D₄×D₄₇

yes

characteristic

193

C₂×C₄

C₂³×D₄₇

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

#10625352

yes

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

188

—

201

C₂²

—

188

—

202

C₂²

—

188

—

203

C₂²

C₂×D₄×D₄₇

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

188

—

210

C₄

C₂×#10625352

yes

characteristic

211

C₄

#13094069

yes

characteristic

212

C₂

—

213

C₂

—

376

—

214

C₂

#15689793

yes

characteristic

215

C₂

#15689920

yes

characteristic

216

C₂

#15689925

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#495	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#495	D₄₇
hypocenter	C₄₇	#3403
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#495

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #495

N	H ≅ G/N	type
1. #15689855	C₂	semidirect
2. #15689509	C₂	semidirect
3. #15689347	C₂	semidirect
4. #15689335	C₂	semidirect
5. #15689316	C₂	semidirect
6. #15689253	C₂	semidirect
7. #15689224	C₂	semidirect
8. C₄×D₃₇₆	C₂	semidirect
9. C₄₇×#495	C₂	semidirect
111. #495	D₄₇	semidirect
118. C₄₇	#3403	semidirect

H ≅ G/N

type

1. #15689855

C₂

semidirect

2. #15689509

C₂

semidirect

3. #15689347

C₂

semidirect

4. #15689335

C₂

semidirect

5. #15689316

C₂

semidirect

6. #15689253

C₂

semidirect

7. #15689224

C₂

semidirect

8. C₄×D₃₇₆

C₂

semidirect

9. C₄₇×#495

C₂

semidirect

111. #495

D₄₇

semidirect

118. C₄₇

#3403

semidirect

Subgroups — #20256479

Conjugacy classes of subgroups

Complements