Group #20243711 — subgroups

Subgroups — #20243711

Order 6016.

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

normal subgroups only

This group has 412 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	C₂×#13093983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093945	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688885	C₂	1	2	yes	maximal, complemented
5	C₂×#121×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#13093983	—	2	4	no	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
10	C₂×C₄×Dic₄₇	—	2	1	no	—
11	#13093945	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
13	C₂²×D₄×C₄₇	—	2	1	no	—
14	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	#121×C₄₇	—	2	2	no	—
17	#121×C₄₇	C₄	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
20	C₂⁴×C₄₇	—	2	1	no	—
21	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
22	C₂×D₄×C₄₇	—	2	4	no	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	4	4	no	—
27	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	—	2	1	no	—
31	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	D₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂²×Dic₄₇	—	2	2	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₄₇	—	2	4	no	—
39	C₂²⋊Dic₄₇	—	2	4	no	—
40	C₄⋊Dic₄₇	—	2	2	no	—
41	C₄⋊Dic₄₇	D₄	1	4	yes	complemented
42	C₄×Dic₄₇	—	2	4	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂³×C₄₇	—	2	1	no	—
47	C₂³×C₄₇	—	2	1	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	—	2	2	no	—
51	C₂³×C₄₇	—	4	1	no	—
52	C₂³×C₄₇	—	4	1	no	—
53	C₂³×C₄₇	—	4	2	no	—
54	C₂³×C₄₇	—	4	2	no	—
55	C₂³×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
57	D₄×C₄₇	—	2	4	no	—
58	D₄×C₄₇	—	4	8	no	—
59	D₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	1	no	—
62	C₂×C₄×C₄₇	—	2	4	no	—
63	C₂×C₄×C₄₇	—	2	4	no	—
64	C₂×C₄×C₄₇	—	2	2	no	—
65	C₂×C₄×C₄₇	—	4	2	no	—
66	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₄₇	#41	1	2	yes	—
71	C₂×Dic₄₇	—	2	4	no	—
72	C₂×Dic₄₇	—	4	2	no	—
73	C₂×Dic₄₇	—	4	2	no	—
74	C₄₇⋊C₈	—	4	2	no	—
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	—	2	2	no	—
77	C₂²×C₄₇	—	2	2	no	—
78	C₂²×C₄₇	—	4	2	no	—
79	C₂²×C₄₇	—	4	2	no	—
80	C₂²×C₄₇	—	4	2	no	—
81	C₂²×C₄₇	—	4	2	no	—
82	C₂²×C₄₇	—	4	2	no	—
83	C₂²×C₄₇	—	8	2	no	—
84	C₂²×C₄₇	—	8	2	no	—
85	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×C₄₇	#120	1	1	yes	characteristic
87	C₂²×C₄₇	#123	1	1	yes	characteristic
88	C₂²×C₄₇	#135	1	2	yes	—
89	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
90	C₄×C₄₇	—	2	2	no	—
91	C₄×C₄₇	—	4	2	no	—
92	C₄×C₄₇	C₄×D₄	1	2	yes	—
93	C₄×C₄₇	#121	1	2	yes	—
94	Dic₄₇	—	4	4	no	—
95	#1875	—	47	1	no	maximal, Sylow
96	C₂×C₄₇	—	4	2	no	—
97	C₂×C₄₇	—	8	2	no	—
98	C₂×C₄₇	#386	1	1	yes	characteristic
99	C₂×C₄₇	#416	1	1	yes	characteristic
100	C₂×C₄₇	#418	1	1	yes	characteristic
101	C₂×C₄₇	#449	1	2	yes	—
102	C₂×C₄₇	#482	1	2	yes	—
103	C₂×#121	D₄₇	1	1	yes	characteristic
104	C₂×#117	—	47	1	no	—
105	C₂×#102	—	47	2	no	—
106	C₂×C₂²⋊C₈	—	47	1	no	—
107	#340	—	47	2	no	—
108	C₄₇	#1875	1	1	yes	characteristic, complemented, Sylow
109	C₂²×D₄	—	2	1	no	—
110	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
111	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
112	C₂²×C₈	—	47	2	no	—
113	#121	—	2	2	no	—
114	#121	Dic₄₇	1	4	yes	complemented
115	#117	—	94	4	no	—
116	C₂×C₄⋊C₄	—	47	2	no	—
117	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	—	2	1	no	—
119	C₂×C₂²⋊C₄	—	94	1	no	—
120	C₂×C₄²	—	94	1	no	—
121	#102	—	94	4	no	—
122	C₂²⋊C₈	—	94	2	no	—
123	C₂⁴	—	2	1	no	—
124	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂×D₄	—	2	4	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	2	1	no	—
129	C₂×D₄	—	4	4	no	—
130	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	94	2	no	—
135	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
136	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
137	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
138	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
139	C₂×C₈	—	94	2	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₄⋊C₄	—	2	1	no	—
143	C₄⋊C₄	—	47	4	no	—
144	C₄⋊C₄	—	94	2	no	—
145	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
146	C₂²⋊C₄	—	2	4	no	—
147	C₂²⋊C₄	—	94	4	no	—
148	C₄²	—	94	4	no	—
149	C₂³	—	2	1	no	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	2	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	—	4	1	no	—
155	C₂³	—	4	1	no	—
156	C₂³	—	4	2	no	—
157	C₂³	—	4	2	no	—
158	C₂³	C₂²⋊Dic₄₇	1	2	yes	—
159	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
160	D₄	—	2	4	no	—
161	D₄	—	4	8	no	—
162	D₄	—	4	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	94	4	no	—
170	C₂×C₄	—	188	2	no	—
171	C₂×C₄	—	188	2	no	—
172	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
173	C₂×C₄	D₄×D₄₇	1	2	yes	—
174	C₂×C₄	#10625352	1	2	yes	—
175	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
176	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
177	C₈	—	188	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₂²	#13093992	1	1	yes	characteristic
190	C₂²	#13094005	1	1	yes	characteristic
191	C₂²	#13094044	1	2	yes	—
192	C₂²	#13094050	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	188	4	no	—
196	C₄	D₄×Dic₄₇	1	2	yes	—
197	C₄	#13093996	1	2	yes	—
198	C₂	—	4	2	no	—
199	C₂	—	8	2	no	—
200	C₂	#15689003	1	1	yes	characteristic
201	C₂	#15689119	1	1	yes	characteristic
202	C₂	#15689128	1	1	yes	characteristic
203	C₂	#15689266	1	2	yes	—
204	C₂	#15689444	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal, characteristic, complemented

C₂×#13093945

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#15688885

C₂

yes

maximal, complemented

C₂×#121×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093983

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

—

#13093945

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#121×C₄₇

—

#121×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

complemented

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

—

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×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

#1875

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

100

C₂×C₄₇

#418

yes

characteristic

101

C₂×C₄₇

#449

yes

—

102

C₂×C₄₇

#482

yes

—

103

C₂×#121

D₄₇

yes

characteristic

104

C₂×#117

—

105

C₂×#102

—

106

C₂×C₂²⋊C₈

—

107

#340

—

108

C₄₇

#1875

yes

characteristic, complemented, Sylow

109

C₂²×D₄

—

110

C₂²×D₄

C₂×D₄₇

yes

characteristic

111

C₂³×C₄

C₂×D₄₇

yes

characteristic

112

C₂²×C₈

—

113

#121

—

114

#121

Dic₄₇

yes

complemented

115

#117

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

118

C₂×C₂²⋊C₄

—

119

C₂×C₂²⋊C₄

—

120

C₂×C₄²

—

121

#102

—

122

C₂²⋊C₈

—

123

C₂⁴

—

124

C₂⁴

C₂²⋊D₄₇

yes

characteristic

125

C₂×D₄

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂×D₄

—

130

C₂×D₄

C₂×Dic₄₇

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

—

135

C₂²×C₄

C₂×Dic₄₇

yes

—

136

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

137

C₂²×C₄

C₂²⋊D₄₇

yes

—

138

C₂²×C₄

C₂²×D₄₇

yes

characteristic

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

C₂×Dic₄₇

yes

—

146

C₂²⋊C₄

—

147

C₂²⋊C₄

—

148

C₄²

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

152

C₂³

—

153

C₂³

—

154

C₂³

—

155

C₂³

—

156

C₂³

—

157

C₂³

—

158

C₂³

C₂²⋊Dic₄₇

yes

—

159

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

160

D₄

—

161

D₄

—

162

D₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

188

—

171

C₂×C₄

—

188

—

172

C₂×C₄

C₂²⋊Dic₄₇

yes

—

173

C₂×C₄

D₄×D₄₇

yes

—

174

C₂×C₄

#10625352

yes

—

175

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

176

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

177

C₈

—

188

—

178

C₂²

—

179

C₂²

—

180

C₂²

—

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

189

C₂²

#13093992

yes

characteristic

190

C₂²

#13094005

yes

characteristic

191

C₂²

#13094044

yes

—

192

C₂²

#13094050

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

188

—

196

C₄

D₄×Dic₄₇

yes

—

197

C₄

#13093996

yes

—

198

C₂

—

199

C₂

—

200

C₂

#15689003

yes

characteristic

201

C₂

#15689119

yes

characteristic

202

C₂

#15689128

yes

characteristic

203

C₂

#15689266

yes

—

204

C₂

#15689444

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#121×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×#121	D₄₇
hypocenter	C₄₇	#1875
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#121×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#121

N	H ≅ G/N	type
1. C₂×#13093983	C₂	semidirect
2. C₂×#13093945	C₂	semidirect
3. C₂×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
4. #15688885	C₂	semidirect
17. #121×C₄₇	C₄	semidirect
41. C₄⋊Dic₄₇	D₄	semidirect
108. C₄₇	#1875	semidirect
114. #121	Dic₄₇	semidirect

H ≅ G/N

type

1. C₂×#13093983

C₂

semidirect

2. C₂×#13093945

C₂

semidirect

3. C₂×C₂²⋊(C₄₇⋊C₈)

C₂

semidirect

4. #15688885

C₂

semidirect

17. #121×C₄₇

C₄

semidirect

41. C₄⋊Dic₄₇

D₄

semidirect

108. C₄₇

#1875

semidirect

114. #121

Dic₄₇

semidirect

Subgroups — #20243711

Conjugacy classes of subgroups

Complements