Group #20244297 — subgroups

Subgroups — #20244297

Order 6016.

316 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×#10625334	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093959	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13093941	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13093940	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×#10625334	—	2	1	no	—
10	C₂×#10625334	—	2	1	no	—
11	C₂×#10625334	—	2	2	no	—
12	C₂×#10625334	C₄	1	4	yes	complemented
13	C₂×#10625329	C₂²	1	1	yes	characteristic
14	C₂×#10625329	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	#13093959	—	2	2	no	—
17	#13093944	—	2	2	no	—
18	#13093941	—	2	2	no	—
19	#13093940	—	2	2	no	—
20	#13093935	—	2	1	no	—
21	C₄×C₄₇⋊C₈	—	2	1	no	—
22	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	#117×C₄₇	C₂²	1	1	yes	characteristic
25	#102×C₄₇	—	2	2	no	—
26	C₂⁴×C₄₇	—	2	1	no	—
27	C₂³×D₄₇	—	2	1	no	—
28	C₂×D₄×C₄₇	—	2	1	no	—
29	C₂×D₄×C₄₇	—	2	1	no	—
30	C₂×D₄×C₄₇	—	2	1	no	—
31	C₂×D₄×C₄₇	—	2	1	no	—
32	C₂×D₄×C₄₇	—	2	1	no	—
33	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×D₁₈₈	—	2	1	no	—
36	C₂×D₁₈₈	—	2	1	no	—
37	C₂×D₁₈₈	—	2	1	no	—
38	C₂×D₁₈₈	—	2	1	no	—
39	C₂×D₁₈₈	—	2	1	no	—
40	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
41	M₁₆×C₄₇	—	2	1	no	—
42	M₁₆×C₄₇	—	2	1	no	—
43	C₂×C₈×C₄₇	—	2	1	no	—
44	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
45	C₂²⋊C₄×C₄₇	—	2	1	no	—
46	C₄²×C₄₇	—	2	1	no	—
47	#10625334	—	2	4	no	—
48	#10625334	—	2	4	no	—
49	#10625329	—	2	1	no	—
50	#10625329	—	2	1	no	—
51	#10625329	—	2	1	no	—
52	#10625329	D₄	1	2	yes	—
53	C₂×C₄₇⋊C₈	—	2	1	no	—
54	C₂×C₄₇⋊C₈	—	2	1	no	—
55	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
56	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
58	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
59	C₂³×C₄₇	—	2	2	no	—
60	C₂³×C₄₇	—	4	1	no	—
61	C₂³×C₄₇	—	4	1	no	—
62	C₂³×C₄₇	—	8	2	no	—
63	C₂³×C₄₇	#41	1	1	yes	characteristic
64	C₂²×D₄₇	—	2	2	no	—
65	C₂²×D₄₇	—	4	1	no	—
66	C₂²×D₄₇	—	4	1	no	—
67	C₂²×D₄₇	—	8	2	no	—
68	D₄×C₄₇	—	2	2	no	—
69	D₄×C₄₇	—	2	2	no	—
70	D₄×C₄₇	—	4	1	no	—
71	D₄×C₄₇	—	4	1	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
79	D₁₈₈	—	2	2	no	—
80	D₁₈₈	—	2	2	no	—
81	D₁₈₈	—	4	1	no	—
82	D₁₈₈	—	4	1	no	—
83	C₈×C₄₇	—	4	1	no	—
84	C₄₇⋊C₈	—	2	2	no	—
85	C₄₇⋊C₈	—	4	1	no	—
86	C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	—	4	2	no	—
90	C₂²×C₄₇	—	8	2	no	—
91	C₂²×C₄₇	—	8	2	no	—
92	C₂²×C₄₇	—	8	2	no	—
93	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
94	C₂²×C₄₇	#121	1	1	yes	characteristic
95	C₂²×C₄₇	#124	1	1	yes	characteristic
96	C₂×D₄₇	—	4	2	no	—
97	C₂×D₄₇	—	8	2	no	—
98	C₂×D₄₇	—	8	2	no	—
99	C₂×D₄₇	—	8	2	no	—
100	C₄×C₄₇	—	4	2	no	—
101	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
103	C₄×C₄₇	#121	1	1	yes	characteristic
104	C₄×C₄₇	#127	1	1	yes	characteristic
105	#1978	—	47	1	no	maximal, Sylow
106	C₂×C₄₇	—	2	1	no	—
107	C₂×C₄₇	—	2	1	no	—
108	C₂×C₄₇	—	8	2	no	—
109	C₂×C₄₇	#390	1	1	yes	characteristic
110	C₂×C₄₇	#472	1	2	yes	—
111	D₄₇	—	8	2	no	—
112	C₂²×D₈	—	47	1	no	—
113	C₂×#108	—	47	1	no	—
114	#418	—	47	1	no	—
115	#418	D₄₇	1	1	yes	characteristic, complemented
116	C₂×#100	—	47	1	no	—
117	C₂×#100	—	47	1	no	—
118	#405	—	47	1	no	—
119	C₄₇	#1978	1	1	yes	characteristic, complemented, Sylow
120	C₂²×D₄	—	47	1	no	—
121	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
122	C₂×D₈	—	47	4	no	—
123	C₂×D₈	—	94	1	no	—
124	C₂×D₈	—	94	1	no	—
125	C₂×D₈	—	94	2	no	—
126	C₂×M₁₆	—	47	1	no	—
127	C₂×M₁₆	—	47	1	no	—
128	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
129	C₂²×C₈	—	47	1	no	—
130	#117	C₂×D₄₇	1	1	yes	characteristic
131	#108	—	94	2	no	—
132	#102	—	2	2	no	—
133	#102	—	94	2	no	—
134	#100	—	94	2	no	—
135	#100	—	94	2	no	—
136	#97	—	94	1	no	—
137	C₄×C₈	—	94	1	no	—
138	C₂⁴	—	2	1	no	—
139	C₂⁴	—	94	1	no	—
140	C₂×D₄	—	2	1	no	—
141	C₂×D₄	—	2	1	no	—
142	C₂×D₄	—	2	1	no	—
143	C₂×D₄	—	2	1	no	—
144	C₂×D₄	—	2	1	no	—
145	C₂×D₄	—	47	2	no	—
146	C₂×D₄	—	94	1	no	—
147	C₂×D₄	—	94	1	no	—
148	C₂×D₄	—	94	1	no	—
149	C₂×D₄	—	94	1	no	—
150	C₂×D₄	—	94	1	no	—
151	C₂×D₄	C₄×D₄₇	1	2	yes	—
152	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
153	D₈	—	94	4	no	—
154	D₈	—	94	4	no	—
155	M₁₆	—	2	1	no	—
156	M₁₆	—	2	1	no	—
157	M₁₆	—	47	2	no	—
158	M₁₆	—	94	1	no	—
159	M₁₆	—	94	1	no	—
160	M₁₆	—	94	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	47	2	no	—
163	C₂×C₈	—	47	1	no	—
164	C₂×C₈	—	47	1	no	—
165	C₂×C₈	—	47	2	no	—
166	C₂×C₈	—	94	1	no	—
167	C₂×C₈	—	94	1	no	—
168	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	1	no	—
173	C₂³	—	4	1	no	—
174	C₂³	—	8	2	no	—
175	C₂³	—	94	2	no	—
176	C₂³	—	376	2	no	—
177	C₂³	—	188	1	no	—
178	C₂³	—	188	1	no	—
179	C₂³	#10625352	1	1	yes	characteristic
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	4	1	no	—
183	D₄	—	4	1	no	—
184	D₄	—	94	2	no	—
185	D₄	—	94	2	no	—
186	D₄	—	188	1	no	—
187	D₄	—	188	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄	#10625350	1	1	yes	characteristic
191	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
192	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
193	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
195	C₈	—	4	1	no	—
196	C₈	—	47	2	no	—
197	C₈	—	47	2	no	—
198	C₈	—	94	2	no	—
199	C₈	—	188	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	8	2	no	—
203	C₂²	—	8	2	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	376	2	no	—
206	C₂²	—	376	2	no	—
207	C₂²	—	376	2	no	—
208	C₂²	—	188	2	no	—
209	C₂²	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
210	C₂²	#13093996	1	1	yes	characteristic
211	C₂²	#13094010	1	1	yes	characteristic
212	C₄	—	4	2	no	—
213	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
214	C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
215	C₄	#13093997	1	1	yes	characteristic
216	C₄	#13094020	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	8	2	no	—
220	C₂	—	376	2	no	—
221	C₂	#15689023	1	1	yes	characteristic
222	C₂	#15689391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625334

C₂

yes

maximal, characteristic

C₂×#13093959

C₂

yes

maximal, characteristic, complemented

#15689127

C₂

yes

maximal, characteristic, complemented

C₂×#13093941

C₂

yes

maximal, characteristic, complemented

C₂×#13093940

C₂

yes

maximal, characteristic, complemented

#15689079

C₂

yes

maximal, characteristic, complemented

C₄₇×#418

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

—

—

C₄

yes

complemented

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

—

—

—

—

C₄×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

#102×C₄₇

—

C₂⁴×C₄₇

—

C₂³×D₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂×C₄

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

—

—

—

—

—

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#41

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

100

C₄×C₄₇

—

101

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

102

C₄×C₄₇

C₄×D₄

yes

characteristic

103

C₄×C₄₇

#121

yes

characteristic

104

C₄×C₄₇

yes

105

—

106

C₂×C₄₇

—

107

C₂×C₄₇

—

108

C₂×C₄₇

—

109

C₂×C₄₇

#390

yes

characteristic

110

C₂×C₄₇

#472

yes

—

111

D₄₇

—

112

C₂²×D₈

—

113

C₂×#108

—

114

#418

—

115

#418

D₄₇

yes

characteristic, complemented

116

C₂×#100

—

117

C₂×#100

—

118

#405

—

119

C₄₇

#1978

yes

characteristic, complemented, Sylow

120

C₂²×D₄

—

121

C₂²×D₄

C₂×D₄₇

yes

characteristic

122

C₂×D₈

—

123

C₂×D₈

—

124

C₂×D₈

—

125

C₂×D₈

—

126

C₂×M₁₆

—

127

C₂×M₁₆

—

128

C₂×M₁₆

C₂×D₄₇

yes

characteristic

129

C₂²×C₈

—

130

#117

C₂×D₄₇

yes

characteristic

131

#108

—

132

#102

—

133

#102

—

134

#100

—

135

#100

—

136

#97

—

137

C₄×C₈

—

138

C₂⁴

—

139

C₂⁴

—

140

C₂×D₄

—

141

C₂×D₄

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂×D₄

C₄×D₄₇

yes

—

152

C₂²×C₄

C₂²×D₄₇

yes

characteristic

153

D₈

—

154

D₈

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

M₁₆

—

159

M₁₆

—

160

M₁₆

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₄⋊C₄

C₂²⋊D₄₇

yes

—

169

C₂²⋊C₄

—

170

C₄²

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

—

176

C₂³

—

376

—

177

C₂³

—

188

—

178

C₂³

—

188

—

179

C₂³

#10625352

yes

characteristic

180

D₄

—

181

D₄

—

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

188

—

187

D₄

—

188

—

188

C₂×C₄

—

189

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

190

C₂×C₄

#10625350

yes

characteristic

191

C₂×C₄

D₄×D₄₇

yes

characteristic

192

C₂×C₄

D₄×D₄₇

yes

characteristic

193

C₂×C₄

D₄×D₄₇

yes

characteristic

194

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

188

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

376

—

206

C₂²

—

376

—

207

C₂²

—

376

—

208

C₂²

—

188

—

209

C₂²

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

210

C₂²

#13093996

yes

characteristic

211

C₂²

#13094010

yes

characteristic

212

C₄

—

213

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

214

C₄

C₄×C₂²⋊D₄₇

yes

characteristic

215

C₄

#13093997

yes

characteristic

216

C₄

#13094020

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

—

376

—

221

C₂

#15689023

yes

characteristic

222

C₂

#15689391

yes

—

	subgroup	G/H
center	C₂²	#13093996
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#418	C₂
socle	C₂²×C₄₇	#121
hypercenter	#418	D₄₇
hypocenter	C₄₇	#1978
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

subgroup

G/H

center

C₂²

#13093996

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#418

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #418

N	H ≅ G/N	type
2. C₂×#13093959	C₂	semidirect
3. #15689127	C₂	semidirect
4. C₂×#13093941	C₂	semidirect
5. C₂×#13093940	C₂	semidirect
6. #15689079	C₂	semidirect
7. C₄₇×#418	C₂	semidirect
12. C₂×#10625334	C₄	semidirect
115. #418	D₄₇	semidirect
119. C₄₇	#1978	semidirect

H ≅ G/N

type

2. C₂×#13093959

C₂

semidirect

3. #15689127

C₂

semidirect

4. C₂×#13093941

C₂

semidirect

5. C₂×#13093940

C₂

semidirect

6. #15689079

C₂

semidirect

7. C₄₇×#418

C₂

semidirect

12. C₂×#10625334

C₄

semidirect

115. #418

D₄₇

semidirect

119. C₄₇

#1978

semidirect

Subgroups — #20244297

Conjugacy classes of subgroups

Complements