Group #20252918 — subgroups

Subgroups — #20252918

Order 6016.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13094055	C₂	1	1	yes	maximal, characteristic
2	C₂²×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689699	C₂	1	1	yes	maximal, characteristic
4	#15689345	C₂	1	4	yes	maximal, complemented
5	#15689333	C₂	1	4	yes	maximal, complemented
6	#15689150	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689131	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689117	C₂	1	1	yes	maximal, characteristic
9	C₄₇×#519	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10625350	C₂²	1	1	yes	characteristic
11	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
12	#13094055	—	2	4	no	—
13	C₂×#10625337	—	2	4	no	—
14	C₂×#10625337	C₂²	1	2	yes	—
15	C₂×#10625337	C₂²	1	4	yes	—
16	C₂×#10625336	C₂²	1	2	yes	—
17	C₂×#10625329	C₂²	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	#13094024	C₂²	1	2	yes	—
20	#13094012	C₂²	1	2	yes	—
21	#13094008	—	2	2	no	—
22	#13094004	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
24	#13093947	C₂²	1	2	yes	—
25	#13093947	C₂²	1	4	yes	—
26	#13093944	C₂²	1	2	yes	—
27	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₄₇	—	2	2	no	—
29	C₄×Q₈×C₄₇	C₂²	1	4	yes	—
30	#117×C₄₇	—	2	2	no	—
31	#117×C₄₇	C₂²	1	1	yes	characteristic
32	#117×C₄₇	C₂²	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	—	2	1	no	—
34	C₂×Q₈×C₄₇	—	2	2	no	—
35	C₂×Q₈×C₄₇	D₄	1	4	yes	—
36	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
37	C₂²×C₄×C₄₇	—	2	1	no	—
38	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₂²⋊D₄₇	—	2	1	no	—
40	C₂²×Dic₄₇	—	2	1	no	—
41	#10625350	—	2	2	no	—
42	#10625350	—	2	2	no	—
43	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₄₇	—	2	1	no	—
45	C₂×Dic₉₄	—	2	2	no	—
46	C₂×Dic₉₄	—	2	2	no	—
47	C₂×Dic₉₄	—	2	1	no	—
48	C₂×Dic₉₄	C₂³	1	2	yes	—
49	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
50	C₄⋊C₄×C₄₇	—	2	4	no	—
51	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
52	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
53	C₂²⋊C₄×C₄₇	—	2	1	no	—
54	C₂²⋊C₄×C₄₇	D₄	1	2	yes	—
55	C₄²×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	—	2	1	no	—
57	C₄²×C₄₇	C₂³	1	2	yes	—
58	#10625337	—	2	8	no	—
59	#10625337	—	2	4	no	—
60	#10625336	—	2	4	no	—
61	(C₂×D₄₇)⋊C₄	—	2	4	no	—
62	C₄⋊Dic₄₇	—	2	4	no	—
63	#10625329	—	2	2	no	—
64	C₂×C₄₇⋊C₈	—	2	2	no	—
65	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
66	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
67	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₄₇	—	2	1	no	—
69	Q₈×C₄₇	—	2	4	no	—
70	Q₈×C₄₇	—	2	2	no	—
71	Q₈×C₄₇	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₄₇	—	2	4	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	2	2	no	—
75	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#41	1	2	yes	—
79	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
80	C₂²⋊D₄₇	—	4	2	no	—
81	C₂×Dic₄₇	—	2	1	no	—
82	C₂×Dic₄₇	—	2	2	no	—
83	C₂×Dic₄₇	—	4	2	no	—
84	D₁₈₈	—	2	1	no	—
85	D₁₈₈	—	2	1	no	—
86	C₄×D₄₇	—	4	2	no	—
87	Dic₉₄	—	2	2	no	—
88	Dic₉₄	—	2	1	no	—
89	Dic₉₄	—	2	2	no	—
90	Dic₉₄	—	2	1	no	—
91	Dic₉₄	—	4	2	no	—
92	C₄₇⋊C₈	—	2	2	no	—
93	C₄₇⋊C₈	—	2	2	no	—
94	C₂²×C₄₇	—	2	2	no	—
95	C₂²×C₄₇	#121	1	2	yes	—
96	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
97	C₂×D₄₇	—	4	1	no	—
98	C₂×D₄₇	—	4	2	no	—
99	C₄×C₄₇	—	2	4	no	—
100	C₄×C₄₇	—	2	4	no	—
101	C₄×C₄₇	—	4	2	no	—
102	C₄×C₄₇	#121	1	2	yes	—
103	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
105	Dic₄₇	—	4	2	no	—
106	Dic₄₇	—	4	1	no	—
107	#3044	—	47	1	no	maximal, Sylow
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	C₂×#121	1	1	yes	characteristic
110	C₂×C₄₇	#578	1	2	yes	—
111	D₄₇	—	8	1	no	—
112	C₂×#137	—	47	1	no	—
113	C₂²×Q₁₆	—	47	1	no	—
114	#536	—	47	1	no	—
115	#519	D₄₇	1	1	yes	characteristic, complemented
116	#464	—	47	4	no	—
117	#462	—	47	4	no	—
118	#424	—	47	1	no	—
119	#419	—	47	1	no	—
120	#416	—	47	1	no	—
121	C₄₇	#3044	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	—	47	1	no	—
123	C₂²×Q₈	—	47	1	no	—
124	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
125	#137	—	94	4	no	—
126	C₂×Q₁₆	—	47	4	no	—
127	C₂×Q₁₆	—	47	2	no	—
128	C₂×Q₁₆	—	94	4	no	—
129	C₂×SD₁₆	—	47	2	no	—
130	C₂×M₁₆	—	47	1	no	—
131	C₂²×C₈	—	47	1	no	—
132	C₄⋊Q₈	—	47	2	no	—
133	#124	—	47	2	no	—
134	#123	—	94	2	no	—
135	C₂²⋊Q₈	—	94	2	no	—
136	C₄×Q₈	—	2	2	no	—
137	C₄×Q₈	C₂×D₄₇	1	4	yes	—
138	#117	—	2	2	no	—
139	#117	C₂×D₄₇	1	1	yes	characteristic
140	#117	C₂×D₄₇	1	1	yes	characteristic
141	C₄⋊C₈	—	47	4	no	—
142	#103	—	47	2	no	—
143	#103	—	47	4	no	—
144	#102	—	47	2	no	—
145	#41	—	94	2	no	—
146	#41	—	94	2	no	—
147	C₂×Q₈	—	2	2	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	47	2	no	—
150	C₂×Q₈	—	47	1	no	—
151	C₂×Q₈	—	94	1	no	—
152	C₂×Q₈	—	94	2	no	—
153	C₂×Q₈	—	94	2	no	—
154	C₂×Q₈	D₁₈₈	1	4	yes	—
155	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
156	C₂×D₄	—	47	1	no	—
157	C₂×D₄	—	94	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	94	1	no	—
160	C₂²×C₄	—	94	1	no	—
161	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
162	Q₁₆	—	94	8	no	—
163	Q₁₆	—	94	4	no	—
164	SD₁₆	—	94	4	no	—
165	M₁₆	—	94	2	no	—
166	C₂×C₈	—	47	2	no	—
167	C₂×C₈	—	47	2	no	—
168	C₂×C₈	—	94	2	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	94	4	no	—
171	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
172	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
173	C₂²⋊C₄	—	2	1	no	—
174	C₂²⋊C₄	—	94	4	no	—
175	C₂²⋊C₄	C₂²⋊D₄₇	1	2	yes	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	C₂²×D₄₇	1	2	yes	—
179	C₂³	—	94	1	no	—
180	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
181	Q₈	—	2	4	no	—
182	Q₈	—	2	2	no	—
183	Q₈	—	94	2	no	—
184	Q₈	—	94	1	no	—
185	Q₈	—	94	2	no	—
186	Q₈	—	94	1	no	—
187	Q₈	—	188	2	no	—
188	Q₈	C₂×D₁₈₈	1	4	yes	—
189	D₄	—	94	1	no	—
190	D₄	—	94	1	no	—
191	D₄	—	188	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	94	2	no	—
196	C₂×C₄	—	94	1	no	—
197	C₂×C₄	—	188	2	no	—
198	C₂×C₄	—	188	2	no	—
199	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
200	C₂×C₄	#10625350	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
202	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
204	C₈	—	94	2	no	—
205	C₈	—	94	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	188	1	no	—
208	C₂²	—	188	2	no	—
209	C₂²	#13093998	1	2	yes	—
210	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
211	C₄	—	2	4	no	—
212	C₄	—	2	4	no	—
213	C₄	—	4	2	no	—
214	C₄	—	188	1	no	—
215	C₄	—	188	2	no	—
216	C₄	#13093998	1	2	yes	—
217	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
218	C₄	C₂×#10625350	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	—	376	1	no	—
221	C₂	C₂×#13093998	1	1	yes	characteristic
222	C₂	#15689938	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094055

C₂

yes

maximal, characteristic

C₂²×#10625337

C₂

yes

maximal, characteristic, complemented

#15689699

C₂

yes

maximal, characteristic

#15689345

C₂

yes

maximal, complemented

#15689333

C₂

yes

maximal, complemented

#15689150

C₂

yes

maximal, characteristic, complemented

#15689131

C₂

yes

maximal, characteristic, complemented

#15689117

C₂

yes

maximal, characteristic

C₄₇×#519

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

C₂²×Dic₉₄

C₂²

yes

—

—

C₂²

yes

—

C₂×#10625337

C₂²

yes

—

C₂×#10625336

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

#13094024

C₂²

yes

—

#13094012

C₂²

yes

—

#13094008

—

#13094004

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093947

C₂²

yes

—

#13093947

C₂²

yes

—

#13093944

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

—

#117×C₄₇

—

#117×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

C₂³

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

C₂³

yes

—

C₂×Dic₉₄

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

#10625337

—

#10625337

—

#10625336

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

100

C₄×C₄₇

—

101

C₄×C₄₇

—

102

C₄×C₄₇

#121

yes

—

103

C₄×C₄₇

C₂²×D₄

yes

characteristic

104

C₄×C₄₇

C₂×#41

yes

characteristic

105

Dic₄₇

—

106

Dic₄₇

—

107

#3044

—

maximal, Sylow

108

C₂×C₄₇

—

109

C₂×C₄₇

C₂×#121

yes

characteristic

110

C₂×C₄₇

#578

yes

—

111

D₄₇

—

112

C₂×#137

—

113

C₂²×Q₁₆

—

114

#536

—

115

#519

D₄₇

yes

characteristic, complemented

116

#464

—

117

#462

—

118

#424

—

119

#419

—

120

#416

—

121

C₄₇

#3044

yes

characteristic, complemented, Sylow

122

C₂×#41

—

123

C₂²×Q₈

—

124

C₂²×Q₈

C₂×D₄₇

yes

characteristic

125

#137

—

126

C₂×Q₁₆

—

127

C₂×Q₁₆

—

128

C₂×Q₁₆

—

129

C₂×SD₁₆

—

130

C₂×M₁₆

—

131

C₂²×C₈

—

132

C₄⋊Q₈

—

133

#124

—

134

#123

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

C₂×D₄₇

yes

—

138

#117

—

139

#117

C₂×D₄₇

yes

characteristic

140

#117

C₂×D₄₇

yes

characteristic

141

C₄⋊C₈

—

142

#103

—

143

#103

—

144

#102

—

145

#41

—

146

#41

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂×Q₈

D₁₈₈

yes

—

155

C₂×Q₈

C₂²×D₄₇

yes

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

C₂²×D₄₇

yes

characteristic

162

Q₁₆

—

163

Q₁₆

—

164

SD₁₆

—

165

M₁₆

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

C₂²⋊D₄₇

yes

—

172

C₄⋊C₄

C₂²×D₄₇

yes

—

173

C₂²⋊C₄

—

174

C₂²⋊C₄

—

175

C₂²⋊C₄

C₂²⋊D₄₇

yes

—

176

C₄²

—

177

C₄²

—

178

C₄²

C₂²×D₄₇

yes

—

179

C₂³

—

180

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

181

Q₈

—

182

Q₈

—

183

Q₈

—

184

Q₈

—

185

Q₈

—

186

Q₈

—

187

Q₈

—

188

—

188

Q₈

C₂×D₁₈₈

yes

—

189

D₄

—

190

D₄

—

191

D₄

—

188

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

188

—

198

C₂×C₄

—

188

—

199

C₂×C₄

C₂×D₁₈₈

yes

—

200

C₂×C₄

#10625350

yes

—

201

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

202

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

203

C₂×C₄

C₂³×D₄₇

yes

characteristic

204

C₈

—

205

C₈

—

206

C₂²

—

207

C₂²

—

188

—

208

C₂²

—

188

—

209

C₂²

#13093998

yes

—

210

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

188

—

215

C₄

—

188

—

216

C₄

#13093998

yes

—

217

C₄

C₂²×D₁₈₈

yes

characteristic

218

C₄

C₂×#10625350

yes

characteristic

219

C₂

—

220

C₂

—

376

—

221

C₂

C₂×#13093998

yes

characteristic

222

C₂

#15689938

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#519	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#519	D₄₇
hypocenter	C₄₇	#3044
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#519

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #519

N	H ≅ G/N	type
2. C₂²×#10625337	C₂	semidirect
4. #15689345	C₂	semidirect
5. #15689333	C₂	semidirect
6. #15689150	C₂	semidirect
7. #15689131	C₂	semidirect
9. C₄₇×#519	C₂	semidirect
115. #519	D₄₇	semidirect
121. C₄₇	#3044	semidirect

H ≅ G/N

type

2. C₂²×#10625337

C₂

semidirect

4. #15689345

C₂

semidirect

5. #15689333

C₂

semidirect

6. #15689150

C₂

semidirect

7. #15689131

C₂

semidirect

9. C₄₇×#519

C₂

semidirect

115. #519

D₄₇

semidirect

121. C₄₇

#3044

semidirect

Subgroups — #20252918

Conjugacy classes of subgroups

Complements