Group #20244258 — subgroups

Subgroups — #20244258

Order 6016.

304 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 99 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×#10625337	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093955	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093948	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093947	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093935	C₂	1	1	yes	maximal, characteristic
6	#15689029	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
9	C₂×#10625337	—	2	2	no	—
10	C₂×#10625337	—	2	2	no	—
11	C₂×#10625337	C₄	1	4	yes	complemented
12	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	#13093955	—	2	2	no	—
16	#13093948	—	2	2	no	—
17	#13093947	—	2	2	no	—
18	#13093935	—	2	2	no	—
19	C₄₇⋊#95	—	2	2	no	—
20	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	#95×C₄₇	—	2	2	no	—
24	C₂×Q₈×C₄₇	—	2	1	no	—
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	—	2	1	no	—
27	C₂×Q₈×C₄₇	—	2	1	no	—
28	C₂×Q₈×C₄₇	—	2	1	no	—
29	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₄₇	—	2	1	no	—
31	C₂²×C₄×C₄₇	—	2	1	no	—
32	C₂²×C₄×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²×Dic₄₇	—	2	1	no	—
35	C₂²×Dic₄₇	—	2	1	no	—
36	C₂×Dic₉₄	—	2	1	no	—
37	C₂×Dic₉₄	—	2	1	no	—
38	C₂×Dic₉₄	—	2	1	no	—
39	C₂×Dic₉₄	—	2	1	no	—
40	C₂×Dic₉₄	—	2	1	no	—
41	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
42	C₄⋊C₄×C₄₇	—	2	1	no	—
43	C₄⋊C₄×C₄₇	—	2	1	no	—
44	C₄²×C₄₇	—	2	1	no	—
45	C₄²×C₄₇	—	2	1	no	—
46	#10625337	—	2	8	no	—
47	C₄⋊Dic₄₇	—	2	1	no	—
48	C₄⋊Dic₄₇	—	2	1	no	—
49	C₂×C₄₇⋊C₈	—	2	1	no	—
50	C₂×C₄₇⋊C₈	—	2	1	no	—
51	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
52	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
53	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
54	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
55	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
56	Q₈×C₄₇	—	2	2	no	—
57	Q₈×C₄₇	—	2	2	no	—
58	Q₈×C₄₇	—	4	1	no	—
59	Q₈×C₄₇	—	4	1	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	2	2	no	—
63	C₂×C₄×C₄₇	—	2	2	no	—
64	C₂×C₄×C₄₇	—	4	1	no	—
65	C₂×C₄×C₄₇	—	4	1	no	—
66	C₂×C₄×C₄₇	—	4	1	no	—
67	C₂×C₄×C₄₇	—	4	2	no	—
68	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
73	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
74	C₂×Dic₄₇	—	2	2	no	—
75	C₂×Dic₄₇	—	4	1	no	—
76	C₂×Dic₄₇	—	4	1	no	—
77	C₂×Dic₄₇	—	4	1	no	—
78	C₂×Dic₄₇	—	4	2	no	—
79	Dic₉₄	—	2	2	no	—
80	Dic₉₄	—	2	2	no	—
81	Dic₉₄	—	4	1	no	—
82	Dic₉₄	—	4	1	no	—
83	C₄₇⋊C₈	—	2	2	no	—
84	C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₄₇	#121	1	1	yes	characteristic
87	C₂²×C₄₇	#127	1	1	yes	characteristic
88	C₂²×C₄₇	#137	1	2	yes	—
89	C₂²×C₄₇	#137	1	2	yes	—
90	C₄×C₄₇	—	4	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	—	8	1	no	—
93	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
95	C₄×C₄₇	#121	1	1	yes	characteristic
96	C₄×C₄₇	#124	1	1	yes	characteristic
97	Dic₄₇	—	4	2	no	—
98	Dic₄₇	—	8	1	no	—
99	#1971	—	47	1	no	maximal, Sylow
100	C₂×C₄₇	#390	1	1	yes	characteristic
101	C₂×C₄₇	#441	1	2	yes	—
102	C₂×C₄₇	#470	1	2	yes	—
103	C₂×C₄₇	#497	1	1	yes	characteristic
104	C₂×C₄₇	#497	1	1	yes	characteristic
105	C₂²×Q₁₆	—	47	1	no	—
106	C₂×#106	—	47	1	no	—
107	C₂×#103	—	47	1	no	—
108	C₂×#103	—	47	1	no	—
109	C₂×#97	—	47	1	no	—
110	#391	—	47	1	no	—
111	#391	D₄₇	1	1	yes	characteristic
112	C₄₇	#1971	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	47	1	no	—
114	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
115	C₂×Q₁₆	—	47	4	no	—
116	C₂×Q₁₆	—	94	2	no	—
117	C₂×Q₁₆	—	94	2	no	—
118	C₂²×C₈	—	47	1	no	—
119	C₂²×C₈	—	47	1	no	—
120	C₂×C₄⋊C₄	—	47	1	no	—
121	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
122	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
123	#106	—	94	2	no	—
124	#103	—	94	2	no	—
125	#103	—	94	2	no	—
126	#97	—	94	2	no	—
127	#95	—	2	2	no	—
128	#95	—	94	2	no	—
129	C₂×Q₈	—	2	1	no	—
130	C₂×Q₈	—	2	1	no	—
131	C₂×Q₈	—	2	1	no	—
132	C₂×Q₈	—	2	1	no	—
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	47	2	no	—
135	C₂×Q₈	—	94	1	no	—
136	C₂×Q₈	—	94	1	no	—
137	C₂×Q₈	—	94	1	no	—
138	C₂×Q₈	—	94	1	no	—
139	C₂×Q₈	—	94	1	no	—
140	C₂×Q₈	C₄×D₄₇	1	2	yes	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	94	1	no	—
144	C₂²×C₄	—	94	1	no	—
145	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
146	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
147	Q₁₆	—	94	8	no	—
148	C₂×C₈	—	47	2	no	—
149	C₂×C₈	—	47	2	no	—
150	C₂×C₈	—	47	2	no	—
151	C₂×C₈	—	47	2	no	—
152	C₂×C₈	—	94	1	no	—
153	C₂×C₈	—	94	1	no	—
154	C₄⋊C₄	—	2	1	no	—
155	C₄⋊C₄	—	2	1	no	—
156	C₄⋊C₄	—	94	1	no	—
157	C₄⋊C₄	—	94	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	1	no	—
160	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
161	Q₈	—	2	2	no	—
162	Q₈	—	2	2	no	—
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	94	2	no	—
166	Q₈	—	94	2	no	—
167	Q₈	—	188	1	no	—
168	Q₈	—	188	1	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	94	2	no	—
178	C₂×C₄	—	188	1	no	—
179	C₂×C₄	—	188	2	no	—
180	C₂×C₄	—	188	1	no	—
181	C₂×C₄	—	188	1	no	—
182	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
183	C₂×C₄	#10625350	1	1	yes	characteristic
184	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
185	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
186	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
187	C₂×C₄	#10625352	1	1	yes	characteristic
188	C₈	—	47	4	no	—
189	C₈	—	94	2	no	—
190	C₂²	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
191	C₂²	#13093996	1	1	yes	characteristic
192	C₂²	#13094020	1	1	yes	characteristic
193	C₂²	#13094054	1	2	yes	—
194	C₂²	#13094055	1	2	yes	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	8	1	no	—
198	C₄	—	376	1	no	—
199	C₄	—	188	2	no	—
200	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
201	C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
202	C₄	#13093997	1	1	yes	characteristic
203	C₄	#13094010	1	1	yes	characteristic
204	C₂	#15689023	1	1	yes	characteristic
205	C₂	#15689228	1	2	yes	—
206	C₂	#15689379	1	2	yes	—
207	C₂	#15689515	1	1	yes	characteristic
208	C₂	#15689518	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625337

C₂

yes

maximal, characteristic

C₂×#13093955

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal, characteristic

C₂×#13093947

C₂

yes

maximal, characteristic

C₂×#13093935

C₂

yes

maximal, characteristic

#15689029

C₂

yes

maximal, characteristic

C₄₇×#391

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

—

—

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

#10625337

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×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

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₄₇

C₄×D₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#137

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

C₄×D₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1971

—

maximal, Sylow

100

C₂×C₄₇

#390

yes

characteristic

101

C₂×C₄₇

#441

yes

—

102

C₂×C₄₇

#470

yes

—

103

C₂×C₄₇

#497

yes

characteristic

104

C₂×C₄₇

#497

yes

characteristic

105

C₂²×Q₁₆

—

106

C₂×#106

—

107

C₂×#103

—

108

C₂×#103

—

109

C₂×#97

—

110

#391

—

111

#391

D₄₇

yes

characteristic

112

C₄₇

#1971

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×Q₈

C₂×D₄₇

yes

characteristic

115

C₂×Q₁₆

—

116

C₂×Q₁₆

—

117

C₂×Q₁₆

—

118

C₂²×C₈

—

119

C₂²×C₈

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

122

C₂×C₄²

C₂×D₄₇

yes

characteristic

123

#106

—

124

#103

—

125

#103

—

126

#97

—

127

#95

—

128

#95

—

129

C₂×Q₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

—

137

C₂×Q₈

—

138

C₂×Q₈

—

139

C₂×Q₈

—

140

C₂×Q₈

C₄×D₄₇

yes

—

141

C₂²×C₄

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

C₂²⋊D₄₇

yes

—

146

C₂²×C₄

C₂²×D₄₇

yes

characteristic

147

Q₁₆

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄²

—

159

C₄²

—

160

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

161

Q₈

—

162

Q₈

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

167

Q₈

—

188

—

168

Q₈

—

188

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

188

—

179

C₂×C₄

—

188

—

180

C₂×C₄

—

188

—

181

C₂×C₄

—

188

—

182

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

183

C₂×C₄

#10625350

yes

characteristic

184

C₂×C₄

D₄×D₄₇

yes

characteristic

185

C₂×C₄

D₄×D₄₇

yes

characteristic

186

C₂×C₄

D₄×D₄₇

yes

characteristic

187

C₂×C₄

#10625352

yes

characteristic

188

C₈

—

189

C₈

—

190

C₂²

C₄×C₂²⋊D₄₇

yes

characteristic

191

C₂²

#13093996

yes

characteristic

192

C₂²

#13094020

yes

characteristic

193

C₂²

#13094054

yes

—

194

C₂²

#13094055

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

376

—

199

C₄

—

188

—

200

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

201

C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

202

C₄

#13093997

yes

characteristic

203

C₄

#13094010

yes

characteristic

204

C₂

#15689023

yes

characteristic

205

C₂

#15689228

yes

—

206

C₂

#15689379

yes

—

207

C₂

#15689515

yes

characteristic

208

C₂

#15689518

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#391	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#391	D₄₇
hypocenter	C₄₇	#1971
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#391

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
11. C₂×#10625337	C₄	semidirect
112. C₄₇	#1971	semidirect

H ≅ G/N

type

11. C₂×#10625337

C₄

semidirect

112. C₄₇

#1971

semidirect

Subgroups — #20244258

Conjugacy classes of subgroups

Complements