Group #20254734 — subgroups

Subgroups — #20254734

Order 6016.

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

normal subgroups only

This group has 360 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	#15689654	C₂	1	2	yes	maximal, complemented
2	#15689429	C₂	1	2	yes	maximal, complemented
3	#15689421	C₂	1	2	yes	maximal, complemented
4	#15689364	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689350	C₂	1	2	yes	maximal, complemented
6	#15689344	C₂	1	2	yes	maximal, complemented
7	#15689336	C₂	1	2	yes	maximal, complemented
8	C₂×Dic₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₇×#466	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
11	C₂×#10625335	C₂²	1	2	yes	—
12	C₂²×C₄₇⋊C₈	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	#13093958	C₂²	1	1	yes	characteristic
24	#13093955	C₂²	1	2	yes	—
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	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
32	#121×C₄₇	C₂²	1	2	yes	—
33	#107×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	1	yes	characteristic
40	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
41	C₂×Dic₉₄	C₂³	1	2	yes	—
42	D₈×C₄₇	—	4	1	no	—
43	C₂×C₈×C₄₇	—	4	1	no	—
44	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₄₇	—	2	2	no	—
48	C₂²⋊Dic₄₇	—	2	2	no	—
49	C₂²⋊Dic₄₇	—	2	2	no	—
50	C₂²⋊Dic₄₇	—	2	2	no	—
51	C₂²⋊Dic₄₇	—	2	2	no	—
52	#10625335	—	4	2	no	—
53	C₄⋊Dic₄₇	—	2	2	no	—
54	C₄⋊Dic₄₇	C₂³	1	2	yes	—
55	Dic₄₇⋊C₄	—	2	2	no	—
56	Dic₄₇⋊C₄	—	2	2	no	—
57	Dic₄₇⋊C₄	—	2	2	no	—
58	C₄×Dic₄₇	—	2	2	no	—
59	C₄×Dic₄₇	D₄	1	2	yes	—
60	C₄×Dic₄₇	C₂³	1	2	yes	—
61	C₂×C₄₇⋊C₈	—	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	—	2	2	no	—
77	C₂×Dic₄₇	—	4	1	no	—
78	C₂×Dic₄₇	—	4	4	no	—
79	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
80	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
81	Dic₉₄	—	2	2	no	—
82	Dic₉₄	—	2	2	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	2	1	no	—
85	C₄₇⋊C₈	—	2	2	no	—
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	—	2	1	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	—	4	2	no	—
91	C₂²×C₄₇	#135	1	2	yes	—
92	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₄₇	—	2	1	no	—
94	C₄×C₄₇	—	4	2	no	—
95	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
96	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
97	Dic₄₇	—	2	2	no	—
98	Dic₄₇	—	2	2	no	—
99	Dic₄₇	—	4	2	no	—
100	Dic₄₇	—	4	2	no	—
101	#3250	—	47	1	no	maximal, Sylow
102	C₂×C₄₇	—	2	2	no	—
103	C₂×C₄₇	—	8	2	no	—
104	C₂×C₄₇	#537	1	1	yes	characteristic
105	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
106	C₂×C₄₇	#575	1	1	yes	characteristic
107	#529	—	47	2	no	—
108	#479	—	47	2	no	—
109	#478	—	47	2	no	—
110	#467	—	47	1	no	—
111	#466	D₄₇	1	1	yes	characteristic
112	#465	—	47	2	no	—
113	#464	—	47	2	no	—
114	#463	—	47	2	no	—
115	C₂×C₄⋊C₈	—	47	1	no	—
116	C₄₇	#3250	1	1	yes	characteristic, complemented, Sylow
117	C₂×Q₁₆	—	47	1	no	—
118	C₂×SD₁₆	—	47	2	no	—
119	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
120	C₂²×C₈	—	47	1	no	—
121	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
122	#126	—	94	2	no	—
123	#125	—	47	2	no	—
124	#124	—	47	2	no	—
125	#123	—	94	2	no	—
126	C₂²⋊Q₈	—	47	2	no	—
127	#121	C₂×D₄₇	1	2	yes	—
128	C₄×Q₈	—	47	2	no	—
129	C₄×D₄	—	47	2	no	—
130	C₄×D₄	—	94	2	no	—
131	#117	—	94	2	no	—
132	C₂×C₄²	—	47	1	no	—
133	#107	—	47	1	no	—
134	#107	C₂×D₄₇	1	1	yes	characteristic
135	#106	—	47	2	no	—
136	C₄⋊C₈	—	47	2	no	—
137	C₄⋊C₈	—	47	2	no	—
138	#103	—	47	2	no	—
139	#103	—	47	2	no	—
140	#102	—	47	2	no	—
141	#102	C₂×D₄₇	1	2	yes	—
142	C₂×Q₈	—	47	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂²×D₄₇	1	2	yes	—
145	C₂²×C₄	—	47	1	no	—
146	C₂²×C₄	—	47	1	no	—
147	C₂²×C₄	—	94	2	no	—
148	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
149	Q₁₆	—	188	1	no	—
150	SD₁₆	—	188	2	no	—
151	D₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	47	2	no	—
154	C₂×C₈	—	188	1	no	—
155	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
157	C₄⋊C₄	—	47	2	no	—
158	C₄⋊C₄	—	94	2	no	—
159	C₄⋊C₄	—	94	2	no	—
160	C₄⋊C₄	—	94	2	no	—
161	C₄⋊C₄	—	94	2	no	—
162	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	94	2	no	—
165	C₂²⋊C₄	—	94	2	no	—
166	C₂²⋊C₄	—	94	2	no	—
167	C₂²⋊C₄	—	94	2	no	—
168	C₄²	—	47	2	no	—
169	C₄²	—	47	2	no	—
170	C₄²	—	94	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	D₄×D₄₇	1	1	yes	characteristic
173	Q₈	—	94	2	no	—
174	Q₈	—	94	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	4	4	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	47	2	no	—
182	C₂×C₄	—	47	2	no	—
183	C₂×C₄	—	94	2	no	—
184	C₂×C₄	—	94	2	no	—
185	C₂×C₄	—	94	2	no	—
186	C₂×C₄	—	188	1	no	—
187	C₂×C₄	—	188	4	no	—
188	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	94	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	#13094043	1	2	yes	—
199	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	94	2	no	—
203	C₄	—	94	2	no	—
204	C₄	—	188	2	no	—
205	C₄	—	188	2	no	—
206	C₄	#13094069	1	1	yes	characteristic
207	C₄	#13094071	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	#15689700	1	1	yes	characteristic
211	C₂	C₂×#13094043	1	1	yes	characteristic
212	C₂	#15689920	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689654

C₂

yes

maximal, complemented

#15689429

C₂

yes

maximal, complemented

#15689421

C₂

yes

maximal, complemented

#15689364

C₂

yes

maximal, characteristic, complemented

#15689350

C₂

yes

maximal, complemented

#15689344

C₂

yes

maximal, complemented

#15689336

C₂

yes

maximal, complemented

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#466

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×#10625335

C₂²

yes

—

C₂²×C₄₇⋊C₈

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

#13093958

C₂²

yes

characteristic

#13093955

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

#13093949

C₂²

yes

—

#13093947

C₂²

yes

—

#13093945

C₂²

yes

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#121×C₄₇

C₂²

yes

—

#107×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

characteristic

C₂²×Dic₄₇

D₄

yes

characteristic

C₂×Dic₉₄

C₂³

yes

—

D₈×C₄₇

—

C₂×C₈×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

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

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×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

#135

yes

—

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

—

100

Dic₄₇

—

101

#3250

—

maximal, Sylow

102

C₂×C₄₇

—

103

C₂×C₄₇

—

104

C₂×C₄₇

#537

yes

characteristic

105

C₂×C₄₇

C₂×#135

yes

characteristic

106

C₂×C₄₇

#575

yes

characteristic

107

#529

—

108

#479

—

109

#478

—

110

#467

—

111

#466

D₄₇

yes

characteristic

112

#465

—

113

#464

—

114

#463

—

115

C₂×C₄⋊C₈

—

116

C₄₇

#3250

yes

characteristic, complemented, Sylow

117

C₂×Q₁₆

—

118

C₂×SD₁₆

—

119

C₂×D₈

C₂×D₄₇

yes

characteristic

120

C₂²×C₈

—

121

C₂²×C₈

C₂×D₄₇

yes

characteristic

122

#126

—

123

#125

—

124

#124

—

125

#123

—

126

C₂²⋊Q₈

—

127

#121

C₂×D₄₇

yes

—

128

C₄×Q₈

—

129

C₄×D₄

—

130

C₄×D₄

—

131

#117

—

132

C₂×C₄²

—

133

#107

—

134

#107

C₂×D₄₇

yes

characteristic

135

#106

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#103

—

140

#102

—

141

#102

C₂×D₄₇

yes

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

C₂×D₄

C₂²×D₄₇

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂²×D₄₇

yes

characteristic

149

Q₁₆

—

188

—

150

SD₁₆

—

188

—

151

D₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

188

—

155

C₂×C₈

C₂²×D₄₇

yes

characteristic

156

C₂×C₈

C₂²×D₄₇

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

C₂²×D₄₇

yes

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₂²⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₂³

—

172

C₂³

D₄×D₄₇

yes

characteristic

173

Q₈

—

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

188

—

187

C₂×C₄

—

188

—

188

C₂×C₄

D₄×D₄₇

yes

characteristic

189

C₂×C₄

C₂³×D₄₇

yes

characteristic

190

C₈

—

191

C₈

—

192

C₈

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

#13094043

yes

—

199

C₂²

C₂×D₄×D₄₇

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

188

—

205

C₄

—

188

—

206

C₄

#13094069

yes

characteristic

207

C₄

#13094071

yes

characteristic

208

C₂

—

209

C₂

—

210

C₂

#15689700

yes

characteristic

211

C₂

C₂×#13094043

yes

characteristic

212

C₂

#15689920

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#466	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#466	D₄₇
hypocenter	C₄₇	#3250
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#466

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #466

N	H ≅ G/N	type
1. #15689654	C₂	semidirect
2. #15689429	C₂	semidirect
3. #15689421	C₂	semidirect
4. #15689364	C₂	semidirect
5. #15689350	C₂	semidirect
6. #15689344	C₂	semidirect
7. #15689336	C₂	semidirect
8. C₂×Dic₄₇⋊C₈	C₂	semidirect
116. C₄₇	#3250	semidirect

H ≅ G/N

type

1. #15689654

C₂

semidirect

2. #15689429

C₂

semidirect

3. #15689421

C₂

semidirect

4. #15689364

C₂

semidirect

5. #15689350

C₂

semidirect

6. #15689344

C₂

semidirect

7. #15689336

C₂

semidirect

8. C₂×Dic₄₇⋊C₈

C₂

semidirect

116. C₄₇

#3250

semidirect

Subgroups — #20254734

Conjugacy classes of subgroups

Complements