Group #26116012 — subgroups

Subgroups — #26116012

Order 12032.

526 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 211 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×#15689521	C₂	1	1	yes	maximal, characteristic
2	#20244349	C₂	1	2	yes	maximal
3	#20244190	C₂	1	2	yes	maximal
4	#20243545	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689521	—	2	2	no	—
8	#15689521	C₄	1	4	yes	complemented
9	C₂×#13093956	—	2	2	no	—
10	C₂×#13093956	—	2	2	no	—
11	C₂×#13093949	—	2	4	no	—
12	#15688993	C₂²	1	1	yes	characteristic
13	#15688993	C₂²	1	1	yes	characteristic
14	#15688886	—	2	4	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₈²×C₄₇	—	2	2	no	—
18	C₂²×Dic₉₄	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	#13094024	—	2	2	no	—
21	#13094024	—	2	2	no	—
22	#13094024	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	C₂×C₄⋊Dic₄₇	—	2	1	no	—
28	C₂×C₄⋊Dic₄₇	—	2	1	no	—
29	#13093956	—	2	8	no	—
30	#13093956	—	4	4	no	—
31	#13093949	—	4	8	no	—
32	C₂²×C₈×C₄₇	D₄	1	4	yes	—
33	C₂²×C₈×C₄₇	D₄	1	2	yes	—
34	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₄×C₈×C₄₇	—	2	2	no	—
37	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₄₇	D₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
41	C₂²×Dic₄₇	—	4	2	no	—
42	C₂²×Dic₄₇	—	4	1	no	—
43	C₂²×Dic₄₇	—	4	1	no	—
44	C₂×Dic₉₄	—	2	4	no	—
45	C₂×Dic₉₄	—	4	4	no	—
46	C₂×Dic₉₄	—	4	4	no	—
47	C₂×Dic₉₄	—	4	2	no	—
48	C₂×C₈×C₄₇	—	2	4	no	—
49	C₂×C₈×C₄₇	—	2	4	no	—
50	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
51	C₂×C₈×C₄₇	SD₁₆	1	8	yes	—
52	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
53	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
54	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₄₇	#41	1	2	yes	—
57	C₄⋊Dic₄₇	—	2	4	no	—
58	C₄⋊Dic₄₇	—	2	4	no	—
59	C₄⋊Dic₄₇	—	4	2	no	—
60	C₄⋊Dic₄₇	—	4	2	no	—
61	C₄⋊Dic₄₇	—	4	4	no	—
62	C₄⋊Dic₄₇	—	4	2	no	—
63	C₄⋊Dic₄₇	—	4	2	no	—
64	C₄⋊Dic₄₇	—	4	4	no	—
65	C₂³×C₄₇	#127	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₄₇	#121	1	2	yes	—
69	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
72	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
73	C₂×C₄×C₄₇	#135	1	4	yes	—
74	C₂×Dic₄₇	—	4	4	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	C₂×Dic₄₇	—	8	2	no	—
77	C₂×Dic₄₇	—	8	1	no	—
78	C₂×Dic₄₇	—	8	2	no	—
79	C₂×Dic₄₇	—	8	1	no	—
80	C₂×Dic₄₇	—	8	2	no	—
81	Dic₉₄	—	4	8	no	—
82	Dic₉₄	—	8	4	no	—
83	C₈×C₄₇	—	2	8	no	—
84	C₈×C₄₇	#103	1	8	yes	complemented
85	#9819	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	#390	1	1	yes	characteristic
87	C₂²×C₄₇	#492	1	1	yes	characteristic
88	C₂²×C₄₇	#492	1	2	yes	—
89	C₂²×C₄₇	#494	1	1	yes	characteristic
90	C₂²×C₄₇	#495	1	2	yes	—
91	C₄×C₄₇	C₂×#103	1	2	yes	—
92	C₄×C₄₇	C₄×SD₁₆	1	4	yes	—
93	C₄×C₄₇	#465	1	4	yes	—
94	C₄×C₄₇	#487	1	2	yes	—
95	Dic₄₇	—	8	4	no	—
96	Dic₄₇	—	16	1	no	—
97	Dic₄₇	—	16	1	no	—
98	C₂×#498	—	47	1	no	—
99	#1990	—	47	2	no	—
100	#1956	—	47	2	no	—
101	#1844	—	47	1	no	—
102	C₂×C₈²	D₄₇	1	1	yes	characteristic
103	C₂×C₄₇	#1705	1	1	yes	characteristic
104	C₂×C₄₇	#1707	1	2	yes	—
105	C₂×C₄₇	#1708	1	1	yes	characteristic
106	C₂×C₄₇	#1956	1	1	yes	characteristic
107	C₂×C₄₇	#1968	1	2	yes	—
108	C₂×C₄⋊Q₈	—	47	2	no	—
109	#498	—	47	4	no	—
110	#498	—	94	2	no	—
111	C₂×#106	—	94	2	no	—
112	C₂×#106	—	94	2	no	—
113	C₂×#103	—	94	4	no	—
114	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
115	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
116	#384	—	47	1	no	—
117	#384	—	47	1	no	—
118	#340	—	94	4	no	—
119	C₈²	—	2	2	no	—
120	C₄₇	#9819	1	1	yes	characteristic, complemented, Sylow
121	C₂²×Q₈	—	94	2	no	—
122	C₂²×C₈	D₁₈₈	1	4	yes	—
123	C₂²×C₈	D₁₈₈	1	2	yes	—
124	C₄⋊Q₈	—	47	4	no	—
125	C₄⋊Q₈	—	94	2	no	—
126	C₄⋊Q₈	—	94	2	no	—
127	C₄⋊Q₈	—	94	2	no	—
128	C₂×C₄⋊C₄	—	94	2	no	—
129	C₂×C₄⋊C₄	—	94	2	no	—
130	C₂×C₄⋊C₄	—	94	2	no	—
131	C₂×C₄⋊C₄	—	94	2	no	—
132	C₂×C₄⋊C₄	—	94	1	no	—
133	C₂×C₄⋊C₄	—	94	1	no	—
134	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
135	#106	—	94	8	no	—
136	#106	—	188	4	no	—
137	#103	—	188	8	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	2	2	no	—
140	C₄×C₈	C₄×D₄₇	1	2	yes	—
141	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
142	C₂×Q₈	—	94	4	no	—
143	C₂×Q₈	—	188	4	no	—
144	C₂×Q₈	—	188	4	no	—
145	C₂×Q₈	—	188	2	no	—
146	C₂²×C₄	—	188	1	no	—
147	C₂²×C₄	—	188	1	no	—
148	C₂²×C₄	—	188	2	no	—
149	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	#10625325	1	8	yes	—
154	C₂×C₈	#10625325	1	4	yes	—
155	C₂×C₈	Dic₁₈₈	1	4	yes	—
156	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
157	C₄⋊C₄	—	94	4	no	—
158	C₄⋊C₄	—	94	4	no	—
159	C₄⋊C₄	—	188	4	no	—
160	C₄⋊C₄	—	188	2	no	—
161	C₄⋊C₄	—	188	2	no	—
162	C₄⋊C₄	—	188	4	no	—
163	C₄⋊C₄	—	188	2	no	—
164	C₄⋊C₄	—	188	2	no	—
165	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
166	C₄²	#10625350	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
168	C₂³	#13094021	1	1	yes	characteristic
169	Q₈	—	376	4	no	—
170	Q₈	—	188	8	no	—
171	C₂×C₄	—	376	2	no	—
172	C₂×C₄	—	376	2	no	—
173	C₂×C₄	—	376	2	no	—
174	C₂×C₄	—	376	1	no	—
175	C₂×C₄	—	376	1	no	—
176	C₂×C₄	—	376	2	no	—
177	C₂×C₄	—	188	4	no	—
178	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
179	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
180	C₂×C₄	#13093998	1	2	yes	—
181	C₂×C₄	#13094012	1	1	yes	characteristic
182	C₂×C₄	C₂×#10625325	1	2	yes	—
183	C₂×C₄	C₂×#10625325	1	4	yes	—
184	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
185	C₂×C₄	#13094046	1	4	yes	—
186	C₈	—	2	8	no	—
187	C₈	#13093949	1	8	yes	complemented
188	C₂²	#15689025	1	1	yes	characteristic
189	C₂²	#15689496	1	2	yes	—
190	C₂²	#15689496	1	1	yes	characteristic
191	C₂²	#15689503	1	1	yes	characteristic
192	C₂²	#15689507	1	2	yes	—
193	C₄	—	376	4	no	—
194	C₄	—	752	1	no	—
195	C₄	—	752	1	no	—
196	C₄	C₂×#13093949	1	2	yes	—
197	C₄	C₄×#10625325	1	4	yes	—
198	C₄	#15689352	1	4	yes	—
199	C₄	#15689474	1	2	yes	—
200	C₂	#20242975	1	1	yes	characteristic
201	C₂	#20242985	1	2	yes	—
202	C₂	#20242989	1	1	yes	characteristic
203	C₂	#20244190	1	1	yes	characteristic
204	C₂	#20244242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689521

C₂

yes

maximal, characteristic

#20244349

C₂

yes

maximal

#20244190

C₂

yes

maximal

#20243545

C₂

yes

maximal, characteristic

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

#15689521

—

#15689521

C₄

yes

—

—

—

C₂²

yes

characteristic

#15688993

C₂²

yes

characteristic

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

—

C₂²×Dic₉₄

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

#13093956

—

#13093949

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#494

yes

characteristic

C₂²×C₄₇

#495

yes

—

C₄×C₄₇

C₂×#103

yes

—

C₄×C₄₇

C₄×SD₁₆

yes

—

C₄×C₄₇

#465

yes

—

C₄×C₄₇

#487

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×#498

—

#1990

—

100

#1956

—

101

#1844

—

102

C₂×C₈²

D₄₇

yes

characteristic

103

C₂×C₄₇

#1705

yes

characteristic

104

C₂×C₄₇

#1707

yes

—

105

C₂×C₄₇

#1708

yes

characteristic

106

C₂×C₄₇

#1956

yes

characteristic

107

C₂×C₄₇

#1968

yes

—

108

C₂×C₄⋊Q₈

—

109

#498

—

110

#498

—

111

C₂×#106

—

112

C₂×#106

—

113

C₂×#103

—

114

C₂×C₄×C₈

C₂×D₄₇

yes

—

115

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

116

#384

—

117

#384

—

118

#340

—

119

C₈²

—

120

C₄₇

#9819

yes

characteristic, complemented, Sylow

121

C₂²×Q₈

—

122

C₂²×C₈

D₁₈₈

yes

—

123

C₂²×C₈

D₁₈₈

yes

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄²

C₂²×D₄₇

yes

characteristic

135

#106

—

136

#106

—

188

—

137

#103

—

188

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

C₄×D₄₇

yes

—

141

C₄×C₈

C₂²⋊D₄₇

yes

—

142

C₂×Q₈

—

143

C₂×Q₈

—

188

—

144

C₂×Q₈

—

188

—

145

C₂×Q₈

—

188

—

146

C₂²×C₄

—

188

—

147

C₂²×C₄

—

188

—

148

C₂²×C₄

—

188

—

149

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

150

C₂²×C₄

C₂×D₁₈₈

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#10625325

yes

—

154

C₂×C₈

#10625325

yes

—

155

C₂×C₈

Dic₁₈₈

yes

—

156

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

188

—

160

C₄⋊C₄

—

188

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

—

188

—

164

C₄⋊C₄

—

188

—

165

C₄²

C₂×C₄×D₄₇

yes

characteristic

166

C₄²

#10625350

yes

—

167

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

168

C₂³

#13094021

yes

characteristic

169

Q₈

—

376

—

170

Q₈

—

188

—

171

C₂×C₄

—

376

—

172

C₂×C₄

—

376

—

173

C₂×C₄

—

376

—

174

C₂×C₄

—

376

—

175

C₂×C₄

—

376

—

176

C₂×C₄

—

376

—

177

C₂×C₄

—

188

—

178

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

179

C₂×C₄

C₄×D₁₈₈

yes

—

180

C₂×C₄

#13093998

yes

—

181

C₂×C₄

#13094012

yes

characteristic

182

C₂×C₄

C₂×#10625325

yes

—

183

C₂×C₄

C₂×#10625325

yes

—

184

C₂×C₄

C₂×Dic₁₈₈

yes

—

185

C₂×C₄

#13094046

yes

—

186

C₈

—

187

C₈

#13093949

yes

complemented

188

C₂²

#15689025

yes

characteristic

189

C₂²

#15689496

yes

—

190

C₂²

#15689496

yes

characteristic

191

C₂²

#15689503

yes

characteristic

192

C₂²

#15689507

yes

—

193

C₄

—

376

—

194

C₄

—

752

—

195

C₄

—

752

—

196

C₄

C₂×#13093949

yes

—

197

C₄

C₄×#10625325

yes

—

198

C₄

#15689352

yes

—

199

C₄

#15689474

yes

—

200

C₂

#20242975

yes

characteristic

201

C₂

#20242985

yes

—

202

C₂

#20242989

yes

characteristic

203

C₂

#20244190

yes

characteristic

204

C₂

#20244242

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#9819
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₈²×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₂×C₈²

D₄₇

hypocenter

C₄₇

#9819

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. #15689521	C₄	semidirect
84. C₈×C₄₇	#103	semidirect
120. C₄₇	#9819	semidirect
187. C₈	#13093949	semidirect

H ≅ G/N

type

8. #15689521

C₄

semidirect

84. C₈×C₄₇

#103

semidirect

120. C₄₇

#9819

semidirect

187. C₈

#13093949

semidirect

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

subgroup	index
C₂×#15689521	2
#20244349	2
#20244190	2
#20243545	2
C₂×C₈²×C₄₇	2
#9819	47

subgroup

index

C₂×C₈²×C₄₇

Subgroups — #26116012

Conjugacy classes of subgroups

Complements

Maximal subgroups