Group #20244236 — subgroups

Subgroups — #20244236

Order 6016.

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

normal subgroups only

This group has 312 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	C₂×#13094013	C₂	1	1	yes	maximal, characteristic
2	#15689146	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093943	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13093940	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093940	C₂	1	1	yes	maximal, characteristic
6	M₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#101×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	1	yes	characteristic
11	#13094013	—	2	1	no	—
12	#13094013	—	2	1	no	—
13	#13094013	—	2	2	no	—
14	C₂×(C₂×D₄₇)⋊C₄	C₄	1	4	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
16	Dic₄₇⋊C₈	—	2	2	no	—
17	#13093943	—	2	2	no	—
18	#13093940	—	2	2	no	—
19	#13093940	—	2	2	no	—
20	#13093934	—	2	1	no	—
21	C₈×Dic₄₇	—	2	1	no	—
22	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
25	#101×C₄₇	—	2	2	no	—
26	C₂³×D₄₇	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₄₇	—	2	1	no	—
28	C₂×Q₈×C₄₇	—	2	1	no	—
29	C₂×Q₈×C₄₇	—	2	1	no	—
30	C₂×Q₈×C₄₇	—	2	1	no	—
31	C₂×Q₈×C₄₇	—	2	1	no	—
32	C₂×Q₈×C₄₇	—	2	1	no	—
33	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
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	—	2	1	no	—
41	C₂×D₁₈₈	—	2	1	no	—
42	M₁₆×C₄₇	—	2	1	no	—
43	M₁₆×C₄₇	—	2	1	no	—
44	M₁₆×C₄₇	—	2	1	no	—
45	M₁₆×C₄₇	D₄	1	2	yes	—
46	C₂×C₈×C₄₇	—	2	1	no	—
47	C₂×C₈×C₄₇	—	2	1	no	—
48	(C₂×D₄₇)⋊C₄	—	2	4	no	—
49	(C₂×D₄₇)⋊C₄	—	2	4	no	—
50	C₄×Dic₄₇	D₄	1	1	yes	characteristic
51	C₄×Dic₄₇	D₄	1	1	yes	characteristic
52	C₄×Dic₄₇	D₄	1	2	yes	—
53	#10625329	—	2	1	no	—
54	#10625329	—	2	1	no	—
55	#10625329	—	2	1	no	—
56	#10625329	D₄	1	2	yes	—
57	C₂×C₄₇⋊C₈	—	2	1	no	—
58	C₂×C₄₇⋊C₈	—	2	1	no	—
59	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×D₄₇	—	2	2	no	—
61	C₂²×D₄₇	—	2	2	no	—
62	C₂²×D₄₇	—	2	2	no	—
63	C₂²×D₄₇	—	4	4	no	—
64	Q₈×C₄₇	—	4	1	no	—
65	Q₈×C₄₇	—	4	1	no	—
66	Q₈×C₄₇	—	4	1	no	—
67	Q₈×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	—	2	2	no	—
69	C₂×C₄×C₄₇	—	2	2	no	—
70	C₂×C₄×C₄₇	—	2	2	no	—
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
77	C₂×Dic₄₇	—	4	2	no	—
78	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
79	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
80	D₁₈₈	—	4	1	no	—
81	D₁₈₈	—	4	1	no	—
82	D₁₈₈	—	4	1	no	—
83	D₁₈₈	—	4	1	no	—
84	C₈×C₄₇	—	2	2	no	—
85	C₈×C₄₇	—	4	1	no	—
86	C₄₇⋊C₈	—	2	2	no	—
87	C₄₇⋊C₈	—	4	1	no	—
88	C₂²×C₄₇	—	2	2	no	—
89	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
91	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
92	C₂×D₄₇	—	4	2	no	—
93	C₂×D₄₇	—	4	4	no	—
94	C₂×D₄₇	—	8	2	no	—
95	C₂×D₄₇	—	8	2	no	—
96	C₄×C₄₇	—	4	2	no	—
97	C₄×C₄₇	#121	1	1	yes	characteristic
98	C₄×C₄₇	#121	1	1	yes	characteristic
99	C₄×C₄₇	#124	1	1	yes	characteristic
100	C₄×C₄₇	#127	1	1	yes	characteristic
101	Dic₄₇	—	2	2	no	—
102	Dic₄₇	—	2	2	no	—
103	#1967	—	47	1	no	maximal, Sylow
104	C₂×C₄₇	—	2	1	no	—
105	C₂×C₄₇	—	2	1	no	—
106	C₂×C₄₇	#390	1	1	yes	characteristic
107	C₂×C₄₇	#413	1	2	yes	—
108	D₄₇	—	8	2	no	—
109	C₂×#124	—	47	1	no	—
110	#423	—	47	1	no	—
111	C₂×#101	—	47	1	no	—
112	C₂×#101	D₄₇	1	1	yes	characteristic, complemented
113	C₂×#100	—	47	1	no	—
114	C₂×#100	—	47	1	no	—
115	C₄×M₁₆	—	47	1	no	—
116	C₄₇	#1967	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
118	C₂²×D₄	—	47	1	no	—
119	C₂×M₁₆	—	47	1	no	—
120	C₂×M₁₆	—	47	1	no	—
121	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
122	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
123	#124	—	94	1	no	—
124	#124	—	94	1	no	—
125	#124	—	94	2	no	—
126	C₂×C₂²⋊C₄	—	47	4	no	—
127	C₂×C₄²	—	47	1	no	—
128	C₄⋊C₈	—	94	2	no	—
129	#101	—	2	2	no	—
130	#101	—	94	2	no	—
131	#100	—	94	2	no	—
132	#100	—	94	2	no	—
133	#97	—	94	1	no	—
134	C₄×C₈	—	94	1	no	—
135	C₂⁴	—	47	2	no	—
136	C₂×Q₈	—	2	1	no	—
137	C₂×Q₈	—	2	1	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	—	2	1	no	—
142	C₂×D₄	—	94	1	no	—
143	C₂×D₄	—	94	1	no	—
144	C₂×D₄	—	94	1	no	—
145	C₂×D₄	—	94	1	no	—
146	C₂×D₄	—	94	1	no	—
147	C₂×D₄	—	94	1	no	—
148	C₂²×C₄	—	47	2	no	—
149	C₂²×C₄	C₄×D₄₇	1	2	yes	—
150	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
151	M₁₆	—	2	1	no	—
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	47	2	no	—
155	M₁₆	—	94	1	no	—
156	M₁₆	—	94	1	no	—
157	M₁₆	—	94	1	no	—
158	M₁₆	C₂²⋊D₄₇	1	2	yes	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	94	1	no	—
162	C₂×C₈	—	94	1	no	—
163	C₂²⋊C₄	—	94	4	no	—
164	C₂²⋊C₄	—	94	4	no	—
165	C₄²	—	47	1	no	—
166	C₄²	—	47	1	no	—
167	C₄²	—	47	2	no	—
168	C₂³	—	94	2	no	—
169	C₂³	—	94	2	no	—
170	C₂³	—	94	2	no	—
171	C₂³	—	188	4	no	—
172	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
173	Q₈	—	4	1	no	—
174	Q₈	—	4	1	no	—
175	Q₈	—	4	1	no	—
176	Q₈	—	4	1	no	—
177	D₄	—	188	1	no	—
178	D₄	—	188	1	no	—
179	D₄	—	188	1	no	—
180	D₄	—	188	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	47	2	no	—
185	C₂×C₄	—	47	2	no	—
186	C₂×C₄	—	188	2	no	—
187	C₂×C₄	#10625350	1	1	yes	characteristic
188	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
189	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
190	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
191	C₂×C₄	#10625352	1	1	yes	characteristic
192	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₈	—	2	2	no	—
194	C₈	—	4	1	no	—
195	C₈	—	94	2	no	—
196	C₈	—	188	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	376	2	no	—
199	C₂²	—	376	2	no	—
200	C₂²	—	188	2	no	—
201	C₂²	—	188	4	no	—
202	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
203	C₂²	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
204	C₂²	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
205	C₄	—	4	2	no	—
206	C₄	—	94	2	no	—
207	C₄	—	94	2	no	—
208	C₄	#13093996	1	1	yes	characteristic
209	C₄	#13093997	1	1	yes	characteristic
210	C₄	#13094010	1	1	yes	characteristic
211	C₄	#13094020	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	—	376	2	no	—
215	C₂	#15689023	1	1	yes	characteristic
216	C₂	#15689106	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094013

C₂

yes

maximal, characteristic

#15689146

C₂

yes

maximal, characteristic, complemented

C₂×#13093943

C₂

yes

maximal, characteristic, complemented

C₂×#13093940

C₂

yes

maximal, characteristic

C₂×#13093940

C₂

yes

maximal, characteristic

M₁₆×Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#101×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

—

—

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

—

—

—

—

C₈×Dic₄₇

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#101×C₄₇

—

C₂³×D₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

—

—

—

—

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

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

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

100

C₄×C₄₇

#127

yes

characteristic

101

Dic₄₇

—

102

Dic₄₇

—

103

#1967

—

maximal, Sylow

104

C₂×C₄₇

—

105

C₂×C₄₇

—

106

C₂×C₄₇

#390

yes

characteristic

107

C₂×C₄₇

#413

yes

—

108

D₄₇

—

109

C₂×#124

—

110

#423

—

111

C₂×#101

—

112

C₂×#101

D₄₇

yes

characteristic, complemented

113

C₂×#100

—

114

C₂×#100

—

115

C₄×M₁₆

—

116

C₄₇

#1967

yes

characteristic, complemented, Sylow

117

C₂²×Q₈

C₂×D₄₇

yes

characteristic

118

C₂²×D₄

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

C₂×D₄₇

yes

characteristic

122

C₂×M₁₆

C₂×D₄₇

yes

characteristic

123

#124

—

124

#124

—

125

#124

—

126

C₂×C₂²⋊C₄

—

127

C₂×C₄²

—

128

C₄⋊C₈

—

129

#101

—

130

#101

—

131

#100

—

132

#100

—

133

#97

—

134

C₄×C₈

—

135

C₂⁴

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

—

145

C₂×D₄

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂²×C₄

—

149

C₂²×C₄

C₄×D₄₇

yes

—

150

C₂²×C₄

C₂²×D₄₇

yes

characteristic

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

M₁₆

—

155

M₁₆

—

156

M₁₆

—

157

M₁₆

—

158

M₁₆

C₂²⋊D₄₇

yes

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂²⋊C₄

—

164

C₂²⋊C₄

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

—

188

—

172

C₂³

C₂×C₄×D₄₇

yes

characteristic

173

Q₈

—

174

Q₈

—

175

Q₈

—

176

Q₈

—

177

D₄

—

188

—

178

D₄

—

188

—

179

D₄

—

188

—

180

D₄

—

188

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

188

—

187

C₂×C₄

#10625350

yes

characteristic

188

C₂×C₄

D₄×D₄₇

yes

characteristic

189

C₂×C₄

D₄×D₄₇

yes

characteristic

190

C₂×C₄

D₄×D₄₇

yes

characteristic

191

C₂×C₄

#10625352

yes

characteristic

192

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

188

—

197

C₂²

—

198

C₂²

—

376

—

199

C₂²

—

376

—

200

C₂²

—

188

—

201

C₂²

—

188

—

202

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

203

C₂²

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

204

C₂²

C₄×C₂²⋊D₄₇

yes

characteristic

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

#13093996

yes

characteristic

209

C₄

#13093997

yes

characteristic

210

C₄

#13094010

yes

characteristic

211

C₄

#13094020

yes

characteristic

212

C₂

—

213

C₂

—

214

C₂

—

376

—

215

C₂

#15689023

yes

characteristic

216

C₂

#15689106

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#101×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×#101	D₄₇
hypocenter	C₄₇	#1967
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#101×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
2. #15689146	C₂	semidirect
3. C₂×#13093943	C₂	semidirect
6. M₁₆×Dic₄₇	C₂	semidirect
7. C₂×#101×C₄₇	C₂	semidirect
112. C₂×#101	D₄₇	semidirect
116. C₄₇	#1967	semidirect

H ≅ G/N

type

2. #15689146

C₂

semidirect

3. C₂×#13093943

C₂

semidirect

6. M₁₆×Dic₄₇

C₂

semidirect

7. C₂×#101×C₄₇

C₂

semidirect

112. C₂×#101

D₄₇

semidirect

116. C₄₇

#1967

semidirect

Subgroups — #20244236

Conjugacy classes of subgroups

Complements