Group #20256478 — subgroups

Subgroups — #20256478

Order 6016.

380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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	#15689858	C₂	1	2	yes	maximal, complemented
2	#15689512	C₂	1	2	yes	maximal, complemented
3	#15689507	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689346	C₂	1	2	yes	maximal, complemented
5	#15689339	C₂	1	2	yes	maximal, complemented
6	#15689319	C₂	1	2	yes	maximal, complemented
7	#15689225	C₂	1	2	yes	maximal, complemented
8	C₄×D₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#10625325	C₂²	1	2	yes	—
12	C₂×#10625325	C₂²	1	2	yes	—
13	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
14	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
15	C₂×D₃₇₆	C₂²	1	2	yes	—
16	#13094021	C₂²	1	2	yes	—
17	#13094013	—	2	2	no	—
18	#13094012	C₂²	1	2	yes	—
19	#13094012	C₂²	1	2	yes	—
20	#13094009	—	2	2	no	—
21	#13094009	—	2	2	no	—
22	C₄×Dic₉₄	C₂²	1	2	yes	—
23	C₄×D₁₈₈	C₂²	1	2	yes	—
24	C₄×D₁₈₈	C₂²	1	2	yes	—
25	#13093986	—	2	2	no	—
26	#13093958	C₂²	1	1	yes	characteristic
27	#13093949	C₂²	1	2	yes	—
28	#13093946	C₂²	1	2	yes	—
29	#13093946	C₂²	1	2	yes	—
30	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
32	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
33	#97×C₄₇	C₂²	1	2	yes	—
34	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
36	C₂×D₁₈₈	—	2	2	no	—
37	C₂×D₁₈₈	—	2	2	no	—
38	C₂×D₁₈₈	C₂³	1	2	yes	—
39	C₂×D₁₈₈	C₂³	1	2	yes	—
40	C₂×C₄×D₄₇	—	2	2	no	—
41	C₂×C₄×D₄₇	—	2	2	no	—
42	C₂×Dic₉₄	C₂³	1	2	yes	—
43	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂³	1	2	yes	—
45	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
47	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂³	1	1	yes	characteristic
49	C₄²×C₄₇	C₂³	1	2	yes	—
50	(C₂×D₄₇)⋊C₄	—	2	2	no	—
51	(C₂×D₄₇)⋊C₄	—	2	2	no	—
52	(C₂×D₄₇)⋊C₄	—	2	2	no	—
53	(C₂×D₄₇)⋊C₄	—	2	2	no	—
54	(C₂×D₄₇)⋊C₄	—	2	2	no	—
55	(C₂×D₄₇)⋊C₄	—	2	2	no	—
56	C₄⋊Dic₄₇	C₂³	1	2	yes	—
57	Dic₄₇⋊C₄	—	2	2	no	—
58	C₄×Dic₄₇	—	2	2	no	—
59	Dic₁₈₈	—	4	1	no	—
60	D₃₇₆	—	2	2	no	—
61	D₃₇₆	—	4	2	no	—
62	D₃₇₆	—	4	1	no	—
63	#10625325	—	2	4	no	—
64	#10625325	—	4	2	no	—
65	C₂²×D₄₇	—	2	2	no	—
66	C₂²×D₄₇	—	2	2	no	—
67	Q₈×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
73	C₂×Dic₄₇	—	2	2	no	—
74	C₂×Dic₄₇	—	2	2	no	—
75	D₁₈₈	—	2	2	no	—
76	D₁₈₈	—	2	2	no	—
77	D₁₈₈	—	2	2	no	—
78	D₁₈₈	—	2	2	no	—
79	D₁₈₈	—	4	4	no	—
80	D₁₈₈	—	8	2	no	—
81	C₄×D₄₇	—	8	2	no	—
82	C₄×D₄₇	—	8	2	no	—
83	Dic₉₄	—	2	2	no	—
84	Dic₉₄	—	2	2	no	—
85	C₈×C₄₇	—	2	2	no	—
86	C₈×C₄₇	—	2	1	no	—
87	C₈×C₄₇	#41	1	2	yes	—
88	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
89	C₂×D₄₇	—	4	2	no	—
90	C₂×D₄₇	—	4	2	no	—
91	C₂×D₄₇	—	4	2	no	—
92	C₂×D₄₇	—	4	2	no	—
93	C₂×D₄₇	—	4	2	no	—
94	C₂×D₄₇	—	4	2	no	—
95	C₄×C₄₇	—	2	2	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	2	2	no	—
99	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
100	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
101	Dic₄₇	—	4	2	no	—
102	Dic₄₇	—	4	2	no	—
103	#3403	—	47	1	no	maximal, Sylow
104	C₂×C₄₇	#550	1	1	yes	characteristic
105	C₂×C₄₇	#575	1	1	yes	characteristic
106	C₂×C₄₇	#576	1	1	yes	characteristic
107	D₄₇	—	8	2	no	—
108	D₄₇	—	8	2	no	—
109	#559	—	47	2	no	—
110	#496	—	47	2	no	—
111	#495	—	47	1	no	—
112	#464	—	47	2	no	—
113	#463	—	47	2	no	—
114	#459	—	47	2	no	—
115	#446	D₄₇	1	1	yes	characteristic, complemented
116	#440	—	47	2	no	—
117	C₄×D₈	—	47	1	no	—
118	C₄₇	#3403	1	1	yes	characteristic, complemented, Sylow
119	C₂×Q₁₆	—	47	1	no	—
120	C₂×SD₁₆	—	47	2	no	—
121	C₂×SD₁₆	—	47	2	no	—
122	C₂×D₈	—	47	1	no	—
123	C₂×D₈	—	47	1	no	—
124	C₂×D₈	—	47	2	no	—
125	#127	—	47	2	no	—
126	#124	—	47	2	no	—
127	#124	—	47	2	no	—
128	#124	—	94	2	no	—
129	#123	—	94	2	no	—
130	#123	—	94	2	no	—
131	C₄×Q₈	—	47	2	no	—
132	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
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	C₂×D₄₇	1	2	yes	—
138	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
139	#103	—	47	2	no	—
140	#102	—	47	2	no	—
141	#102	—	47	2	no	—
142	#97	C₂×D₄₇	1	2	yes	—
143	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
144	C₂×Q₈	—	47	2	no	—
145	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
146	C₂×D₄	—	47	2	no	—
147	C₂×D₄	—	47	2	no	—
148	C₂×D₄	—	94	2	no	—
149	C₂×D₄	—	94	2	no	—
150	C₂²×C₄	—	94	2	no	—
151	C₂²×C₄	—	94	2	no	—
152	Q₁₆	—	188	1	no	—
153	SD₁₆	—	94	4	no	—
154	SD₁₆	—	188	2	no	—
155	D₈	—	94	2	no	—
156	D₈	—	188	2	no	—
157	D₈	—	188	1	no	—
158	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
159	C₂×C₈	C₂²×D₄₇	1	2	yes	—
160	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
161	C₄⋊C₄	—	47	2	no	—
162	C₄⋊C₄	—	94	2	no	—
163	C₄⋊C₄	D₁₈₈	1	2	yes	—
164	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
165	C₂²⋊C₄	—	94	2	no	—
166	C₂²⋊C₄	—	94	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	—	94	2	no	—
172	C₄²	C₂²×D₄₇	1	1	yes	characteristic
173	C₄²	C₂²×D₄₇	1	2	yes	—
174	C₂³	—	94	2	no	—
175	C₂³	—	94	2	no	—
176	Q₈	—	4	1	no	—
177	Q₈	—	94	2	no	—
178	Q₈	—	94	2	no	—
179	D₄	—	94	2	no	—
180	D₄	—	94	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	—	94	2	no	—
186	C₂×C₄	—	94	2	no	—
187	C₂×C₄	—	376	2	no	—
188	C₂×C₄	—	376	2	no	—
189	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
190	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
191	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
192	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
193	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	#10625354	1	2	yes	—
197	C₂²	—	188	2	no	—
198	C₂²	—	188	2	no	—
199	C₂²	—	188	2	no	—
200	C₂²	—	188	2	no	—
201	C₂²	—	188	2	no	—
202	C₂²	—	188	2	no	—
203	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	188	2	no	—
209	C₄	—	188	2	no	—
210	C₄	C₂×#10625354	1	1	yes	characteristic
211	C₄	#13094070	1	1	yes	characteristic
212	C₂	—	376	2	no	—
213	C₂	—	376	2	no	—
214	C₂	#15689795	1	1	yes	characteristic
215	C₂	#15689923	1	1	yes	characteristic
216	C₂	#15689928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689858

C₂

yes

maximal, complemented

#15689512

C₂

yes

maximal, complemented

#15689507

C₂

yes

maximal, characteristic, complemented

#15689346

C₂

yes

maximal, complemented

#15689339

C₂

yes

maximal, complemented

#15689319

C₂

yes

maximal, complemented

#15689225

C₂

yes

maximal, complemented

C₄×D₃₇₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#446

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×#10625325

C₂²

yes

—

C₂×#10625325

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

characteristic

C₂×D₃₇₆

C₂²

yes

characteristic

C₂×D₃₇₆

C₂²

yes

—

#13094021

C₂²

yes

—

#13094013

—

#13094012

C₂²

yes

—

#13094012

C₂²

yes

—

#13094009

—

#13094009

—

C₄×Dic₉₄

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

#13093986

—

#13093958

C₂²

yes

characteristic

#13093949

C₂²

yes

—

#13093946

C₂²

yes

—

#13093946

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

#10625325

—

#10625325

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#41

yes

characteristic

100

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

101

Dic₄₇

—

102

Dic₄₇

—

103

#3403

—

maximal, Sylow

104

C₂×C₄₇

#550

yes

characteristic

105

C₂×C₄₇

#575

yes

characteristic

106

C₂×C₄₇

#576

yes

characteristic

107

D₄₇

—

108

D₄₇

—

109

#559

—

110

#496

—

111

#495

—

112

#464

—

113

#463

—

114

#459

—

115

#446

D₄₇

yes

characteristic, complemented

116

#440

—

117

C₄×D₈

—

118

C₄₇

#3403

yes

characteristic, complemented, Sylow

119

C₂×Q₁₆

—

120

C₂×SD₁₆

—

121

C₂×SD₁₆

—

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

—

125

#127

—

126

#124

—

127

#124

—

128

#124

—

129

#123

—

130

#123

—

131

C₄×Q₈

—

132

C₄×Q₈

C₂×D₄₇

yes

characteristic

133

C₄×D₄

—

134

C₄×D₄

—

135

C₄×D₄

—

136

#107

—

137

C₄⋊C₈

C₂×D₄₇

yes

—

138

C₄⋊C₈

C₂×D₄₇

yes

characteristic

139

#103

—

140

#102

—

141

#102

—

142

#97

C₂×D₄₇

yes

—

143

C₄×C₈

C₂×D₄₇

yes

characteristic

144

C₂×Q₈

—

145

C₂×Q₈

D₁₈₈

yes

characteristic

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

Q₁₆

—

188

—

153

SD₁₆

—

154

SD₁₆

—

188

—

155

D₈

—

156

D₈

—

188

—

157

D₈

—

188

—

158

C₂×C₈

C₂²×D₄₇

yes

characteristic

159

C₂×C₈

C₂²×D₄₇

yes

—

160

C₂×C₈

C₂²×D₄₇

yes

characteristic

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

D₁₈₈

yes

—

164

C₄⋊C₄

D₁₈₈

yes

characteristic

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₂²⋊C₄

—

169

C₂²⋊C₄

—

170

C₂²⋊C₄

—

171

C₄²

—

172

C₄²

C₂²×D₄₇

yes

characteristic

173

C₄²

C₂²×D₄₇

yes

—

174

C₂³

—

175

C₂³

—

176

Q₈

—

177

Q₈

—

178

Q₈

—

179

D₄

—

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

376

—

184

D₄

—

188

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

376

—

188

C₂×C₄

—

376

—

189

C₂×C₄

C₂×D₁₈₈

yes

—

190

C₂×C₄

C₂×D₁₈₈

yes

characteristic

191

C₂×C₄

C₂×D₁₈₈

yes

characteristic

192

C₂×C₄

C₂×D₁₈₈

yes

—

193

C₂×C₄

C₂³×D₄₇

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

#10625354

yes

—

197

C₂²

—

188

—

198

C₂²

—

188

—

199

C₂²

—

188

—

200

C₂²

—

188

—

201

C₂²

—

188

—

202

C₂²

—

188

—

203

C₂²

C₂²×D₁₈₈

yes

characteristic

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

—

188

—

209

C₄

—

188

—

210

C₄

C₂×#10625354

yes

characteristic

211

C₄

#13094070

yes

characteristic

212

C₂

—

376

—

213

C₂

—

376

—

214

C₂

#15689795

yes

characteristic

215

C₂

#15689923

yes

characteristic

216

C₂

#15689928

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#446	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#446	D₄₇
hypocenter	C₄₇	#3403
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#446

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

N	H ≅ G/N	type
1. #15689858	C₂	semidirect
2. #15689512	C₂	semidirect
3. #15689507	C₂	semidirect
4. #15689346	C₂	semidirect
5. #15689339	C₂	semidirect
6. #15689319	C₂	semidirect
7. #15689225	C₂	semidirect
8. C₄×D₃₇₆	C₂	semidirect
9. C₄₇×#446	C₂	semidirect
115. #446	D₄₇	semidirect
118. C₄₇	#3403	semidirect

H ≅ G/N

type

1. #15689858

C₂

semidirect

2. #15689512

C₂

semidirect

3. #15689507

C₂

semidirect

4. #15689346

C₂

semidirect

5. #15689339

C₂

semidirect

6. #15689319

C₂

semidirect

7. #15689225

C₂

semidirect

8. C₄×D₃₇₆

C₂

semidirect

9. C₄₇×#446

C₂

semidirect

115. #446

D₄₇

semidirect

118. C₄₇

#3403

semidirect

Subgroups — #20256478

Conjugacy classes of subgroups

Complements