Group #26116237 — subgroups

Subgroups — #26116237

Order 12032.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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	#20253902	C₂	1	1	yes	maximal, characteristic
2	#20243526	C₂	1	1	yes	maximal, characteristic
3	#20241844	C₂	1	2	yes	maximal
4	#20241844	C₂	1	2	yes	maximal
5	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	2	yes	—
7	#15689528	C₄	1	1	yes	characteristic
8	#15689528	C₄	1	1	yes	characteristic
9	#15689525	—	2	2	no	—
10	#15689521	C₄	1	2	yes	—
11	#15689161	—	2	2	no	—
12	#15689147	C₂²	1	2	yes	—
13	#15688888	—	2	2	no	—
14	#15688855	—	2	4	no	—
15	#15688855	—	2	4	no	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄₇×#322	—	2	2	no	—
20	C₂×#10625329	—	2	2	no	—
21	#13094024	C₂×C₄	1	2	yes	—
22	#13094024	C₂×C₄	1	2	yes	—
23	#13094016	—	2	2	no	—
24	#13094003	—	2	2	no	—
25	#13094003	—	2	2	no	—
26	C₄×Dic₉₄	—	2	2	no	—
27	C₄×Dic₉₄	—	2	2	no	—
28	#13093983	—	2	2	no	—
29	#13093958	—	2	2	no	—
30	#13093958	—	2	2	no	—
31	#13093956	—	2	4	no	—
32	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
34	C₂²×C₈×C₄₇	—	2	1	no	—
35	C₂²×C₈×C₄₇	—	2	1	no	—
36	C₂²×C₈×C₄₇	D₄	1	2	yes	—
37	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₄₇	—	2	1	no	—
39	C₄×C₈×C₄₇	—	2	1	no	—
40	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₇	D₄	1	2	yes	—
43	C₄×C₈×C₄₇	D₄	1	2	yes	—
44	C₄×C₈×C₄₇	D₄	1	2	yes	—
45	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
48	C₂×Dic₉₄	—	2	2	no	—
49	C₂×Dic₉₄	—	2	2	no	—
50	C₂×C₈×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	4	no	—
53	C₂×C₈×C₄₇	—	2	2	no	—
54	C₂×C₈×C₄₇	—	4	2	no	—
55	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²⋊Dic₄₇	—	4	2	no	—
62	C₄⋊Dic₄₇	—	2	2	no	—
63	C₄⋊Dic₄₇	—	2	2	no	—
64	C₄⋊Dic₄₇	—	2	2	no	—
65	C₄⋊Dic₄₇	—	2	2	no	—
66	Dic₄₇⋊C₄	—	4	2	no	—
67	Dic₄₇⋊C₄	—	4	2	no	—
68	C₄×Dic₄₇	—	4	2	no	—
69	#10625329	—	4	4	no	—
70	C₂×C₄₇⋊C₈	—	4	2	no	—
71	C₂³×C₄₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₄₇	#137	1	4	yes	—
81	C₂×Dic₄₇	—	4	2	no	—
82	C₂×Dic₄₇	—	4	2	no	—
83	Dic₉₄	—	4	4	no	—
84	Dic₉₄	—	4	4	no	—
85	C₈×C₄₇	—	2	4	no	—
86	C₈×C₄₇	—	2	2	no	—
87	C₈×C₄₇	—	2	2	no	—
88	C₈×C₄₇	—	4	2	no	—
89	C₄₇⋊C₈	—	8	2	no	—
90	#9859	—	47	1	no	maximal, Sylow
91	C₂²×C₄₇	—	2	1	no	—
92	C₂²×C₄₇	—	2	1	no	—
93	C₂²×C₄₇	#390	1	1	yes	characteristic
94	C₂²×C₄₇	#491	1	2	yes	—
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	#419	1	2	yes	—
98	C₄×C₄₇	#443	1	2	yes	—
99	C₄×C₄₇	#444	1	2	yes	—
100	C₄×C₄₇	#497	1	2	yes	—
101	Dic₄₇	—	8	2	no	—
102	Dic₄₇	—	8	2	no	—
103	#3160	—	47	1	no	—
104	#1837	—	47	1	no	—
105	#1493	—	47	2	no	—
106	#1493	—	47	2	no	—
107	C₂×#322	D₄₇	1	1	yes	characteristic
108	C₂×C₄₇	—	2	2	no	—
109	C₂×C₄₇	#1960	1	1	yes	characteristic
110	C₂×C₄₇	#1974	1	1	yes	characteristic
111	C₂×C₄₇	#1975	1	1	yes	characteristic
112	#533	—	47	2	no	—
113	#500	—	47	1	no	—
114	#500	—	47	1	no	—
115	#499	—	94	2	no	—
116	#498	—	47	2	no	—
117	#427	—	94	2	no	—
118	#423	—	47	2	no	—
119	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
122	#341	—	94	2	no	—
123	Q₈⋊C₈	—	94	4	no	—
124	Q₈⋊C₈	—	94	4	no	—
125	#322	—	2	2	no	—
126	C₄₇	#9859	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	94	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	D₁₈₈	1	2	yes	—
131	C₄⋊Q₈	—	47	2	no	—
132	C₄⋊Q₈	—	47	2	no	—
133	#125	—	94	2	no	—
134	C₂²⋊Q₈	—	94	2	no	—
135	C₂²⋊Q₈	—	94	2	no	—
136	C₄×Q₈	—	94	2	no	—
137	C₄×Q₈	—	94	2	no	—
138	#117	—	94	2	no	—
139	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
140	#107	—	94	2	no	—
141	#107	—	94	2	no	—
142	#106	—	94	4	no	—
143	C₄⋊C₈	—	94	2	no	—
144	C₄⋊C₈	—	94	2	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	—	2	1	no	—
147	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
148	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
149	C₄×C₈	D₁₈₈	1	2	yes	—
150	C₄×C₈	D₁₈₈	1	2	yes	—
151	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
152	C₂×Q₈	—	94	2	no	—
153	C₂×Q₈	—	94	2	no	—
154	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
155	C₂²×C₄	#10625350	1	1	yes	characteristic
156	C₂²×C₄	#10625350	1	1	yes	characteristic
157	M₁₆	—	188	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	188	2	no	—
164	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
165	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
166	C₄⋊C₄	—	94	2	no	—
167	C₄⋊C₄	—	94	2	no	—
168	C₄⋊C₄	—	94	2	no	—
169	C₄⋊C₄	—	94	2	no	—
170	C₄⋊C₄	—	188	2	no	—
171	C₄⋊C₄	—	188	2	no	—
172	C₂²⋊C₄	—	188	2	no	—
173	C₄²	—	188	2	no	—
174	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
175	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
176	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
178	C₂³	#13094012	1	1	yes	characteristic
179	Q₈	—	188	4	no	—
180	Q₈	—	188	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	188	2	no	—
184	C₂×C₄	—	188	2	no	—
185	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
186	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
187	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
188	C₂×C₄	#13093998	1	1	yes	characteristic
189	C₂×C₄	#13093998	1	1	yes	characteristic
190	C₂×C₄	#13094021	1	1	yes	characteristic
191	C₂×C₄	#13094057	1	4	yes	—
192	C₈	—	2	4	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	2	no	—
195	C₈	—	4	2	no	—
196	C₈	—	376	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	#15689025	1	1	yes	characteristic
200	C₂²	#15689492	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	376	2	no	—
204	C₄	—	376	2	no	—
205	C₄	#15689133	1	2	yes	—
206	C₄	#15689241	1	2	yes	—
207	C₄	#15689246	1	2	yes	—
208	C₄	#15689517	1	2	yes	—
209	C₂	—	2	2	no	—
210	C₂	#20244208	1	1	yes	characteristic
211	C₂	#20244276	1	1	yes	characteristic
212	C₂	#20244281	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20253902

C₂

yes

maximal, characteristic

#20243526

C₂

yes

maximal, characteristic

#20241844

C₂

yes

maximal

#20241844

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689528

C₄

yes

characteristic

#15689528

C₄

yes

characteristic

#15689525

—

#15689521

C₄

yes

—

#15689161

—

#15689147

C₂²

yes

—

#15688888

—

#15688855

—

#15688855

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₂×#10625329

—

#13094024

C₂×C₄

yes

—

#13094024

C₂×C₄

yes

—

#13094016

—

#13094003

—

#13094003

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

—

—

—

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#9859

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#491

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#419

yes

—

C₄×C₄₇

#443

yes

—

C₄×C₄₇

#444

yes

—

100

C₄×C₄₇

#497

yes

—

101

Dic₄₇

—

102

Dic₄₇

—

103

#3160

—

104

#1837

—

105

#1493

—

106

#1493

—

107

C₂×#322

D₄₇

yes

characteristic

108

C₂×C₄₇

—

109

C₂×C₄₇

#1960

yes

characteristic

110

C₂×C₄₇

#1974

yes

characteristic

111

C₂×C₄₇

#1975

yes

characteristic

112

#533

—

113

#500

—

114

#500

—

115

#499

—

116

#498

—

117

#427

—

118

#423

—

119

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

120

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

121

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

122

#341

—

123

Q₈⋊C₈

—

124

Q₈⋊C₈

—

125

#322

—

126

C₄₇

#9859

yes

characteristic, complemented, Sylow

127

C₂×M₁₆

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

D₁₈₈

yes

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

133

#125

—

134

C₂²⋊Q₈

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

C₄×Q₈

—

138

#117

—

139

C₂×C₄²

C₂²×D₄₇

yes

characteristic

140

#107

—

141

#107

—

142

#106

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

C₄×D₄₇

yes

characteristic

148

C₄×C₈

C₄×D₄₇

yes

characteristic

149

C₄×C₈

D₁₈₈

yes

—

150

C₄×C₈

D₁₈₈

yes

—

151

C₄×C₈

C₂²⋊D₄₇

yes

—

152

C₂×Q₈

—

153

C₂×Q₈

—

154

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

155

C₂²×C₄

#10625350

yes

characteristic

156

C₂²×C₄

#10625350

yes

characteristic

157

M₁₆

—

188

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

188

—

164

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

165

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

188

—

171

C₄⋊C₄

—

188

—

172

C₂²⋊C₄

—

188

—

173

C₄²

—

188

—

174

C₄²

C₂×C₄×D₄₇

yes

characteristic

175

C₄²

C₂×D₁₈₈

yes

characteristic

176

C₄²

C₂×D₁₈₈

yes

characteristic

177

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

178

C₂³

#13094012

yes

characteristic

179

Q₈

—

188

—

180

Q₈

—

188

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

188

—

184

C₂×C₄

—

188

—

185

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

186

C₂×C₄

C₄×D₁₈₈

yes

characteristic

187

C₂×C₄

C₄×D₁₈₈

yes

characteristic

188

C₂×C₄

#13093998

yes

characteristic

189

C₂×C₄

#13093998

yes

characteristic

190

C₂×C₄

#13094021

yes

characteristic

191

C₂×C₄

#13094057

yes

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

376

—

197

C₂²

—

198

C₂²

—

199

C₂²

#15689025

yes

characteristic

200

C₂²

#15689492

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

376

—

204

C₄

—

376

—

205

C₄

#15689133

yes

—

206

C₄

#15689241

yes

—

207

C₄

#15689246

yes

—

208

C₄

#15689517

yes

—

209

C₂

—

210

C₂

#20244208

yes

characteristic

211

C₂

#20244276

yes

characteristic

212

C₂

#20244281

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9859
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#322

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
126. C₄₇	#9859	semidirect

H ≅ G/N

type

126. C₄₇

#9859

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20253902	2
#20243526	2
#20241844	2
C₂×C₄₇×#322	2
#9859	47

subgroup

index

C₂×C₄₇×#322

Subgroups — #26116237

Conjugacy classes of subgroups

Complements

Maximal subgroups