Group #26121326 — subgroups

Subgroups — #26121326

Order 12032.

526 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 153 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	#20244194	C₂	1	1	yes	maximal, characteristic
2	#20244193	C₂	1	2	yes	maximal
3	#20243547	C₂	1	2	yes	maximal, complemented
4	C₄₇⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094021	C₂²	1	1	yes	characteristic
7	C₂×#13093957	—	2	4	no	—
8	C₂×#13093946	—	2	2	no	—
9	C₂×#13093944	—	2	4	no	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15688993	C₂²	1	1	yes	characteristic
12	C₂×#107×C₄₇	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₂²×D₁₈₈	—	2	2	no	—
16	C₂²×D₁₈₈	—	2	1	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
18	#13094021	—	2	2	no	—
19	#13094021	—	2	1	no	—
20	#13094021	Q₈	1	2	yes	—
21	C₂×C₄⋊Dic₄₇	—	2	1	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	#13093957	—	4	8	no	—
24	#13093946	—	4	4	no	—
25	#13093944	—	4	8	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	#107×C₄₇	—	4	4	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	C₂³×D₄₇	—	4	1	no	—
36	C₂²×C₄×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₇	—	4	1	no	—
40	C₂×D₁₈₈	—	2	2	no	—
41	C₂×D₁₈₈	—	2	4	no	—
42	C₂×D₁₈₈	—	4	4	no	—
43	C₂×D₁₈₈	—	4	2	no	—
44	C₂×D₁₈₈	—	4	2	no	—
45	C₂×D₁₈₈	—	4	4	no	—
46	C₂×D₁₈₈	—	4	1	no	—
47	C₂×D₁₈₈	—	4	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	D₈	1	4	yes	—
51	C₄⋊C₄×C₄₇	—	4	4	no	—
52	C₄⋊C₄×C₄₇	—	4	4	no	—
53	C₄⋊C₄×C₄₇	—	4	4	no	—
54	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₄₇	#41	1	2	yes	—
56	C₄²×C₄₇	#41	1	1	yes	characteristic
57	C₄⋊Dic₄₇	—	4	2	no	—
58	C₄⋊Dic₄₇	—	4	4	no	—
59	C₂×C₄₇⋊C₈	—	2	4	no	—
60	C₂×C₄₇⋊C₈	—	2	4	no	—
61	C₂×C₄₇⋊C₈	D₈	1	8	yes	—
62	C₂³×C₄₇	#127	1	1	yes	characteristic
63	C₂²×D₄₇	—	4	2	no	—
64	C₂²×D₄₇	—	8	2	no	—
65	C₂²×D₄₇	—	8	1	no	—
66	C₂²×D₄₇	—	8	1	no	—
67	C₂²×D₄₇	—	8	2	no	—
68	C₂×C₄×C₄₇	—	8	2	no	—
69	C₂×C₄×C₄₇	—	8	2	no	—
70	C₂×C₄×C₄₇	—	8	2	no	—
71	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
74	C₂×C₄×C₄₇	#123	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×D₈	1	4	yes	—
76	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
77	C₂×C₄×C₄₇	#137	1	2	yes	—
78	C₂×C₄×C₄₇	#137	1	4	yes	—
79	C₂×Dic₄₇	—	8	2	no	—
80	C₂×Dic₄₇	—	8	1	no	—
81	D₁₈₈	—	4	4	no	—
82	D₁₈₈	—	4	8	no	—
83	D₁₈₈	—	8	2	no	—
84	D₁₈₈	—	8	4	no	—
85	C₈×C₄₇	—	2	4	no	—
86	C₄₇⋊C₈	—	2	8	no	—
87	#10673	—	47	1	no	maximal, Sylow
88	C₂²×C₄₇	#399	1	1	yes	characteristic
89	C₂²×C₄₇	#493	1	2	yes	—
90	C₂²×C₄₇	#493	1	1	yes	characteristic
91	C₂²×C₄₇	#497	1	1	yes	characteristic
92	C₂²×C₄₇	#497	1	2	yes	—
93	C₂×D₄₇	—	8	4	no	—
94	C₂×D₄₇	—	8	2	no	—
95	C₂×D₄₇	—	16	1	no	—
96	C₂×D₄₇	—	16	2	no	—
97	C₂×D₄₇	—	16	1	no	—
98	C₄×C₄₇	—	16	2	no	—
99	C₄×C₄₇	#474	1	2	yes	—
100	C₄×C₄₇	#474	1	4	yes	—
101	C₄×C₄₇	#480	1	2	yes	—
102	C₄×C₄₇	#480	1	4	yes	—
103	Dic₄₇	—	16	1	no	—
104	#1957	—	47	1	no	—
105	#1957	—	47	2	no	—
106	#1845	—	47	2	no	—
107	#1845	D₄₇	1	1	yes	characteristic, complemented
108	#1273	—	47	1	no	—
109	C₂×C₄₇	#1712	1	1	yes	characteristic
110	C₂×C₄₇	#1722	1	2	yes	—
111	C₂×C₄₇	#1722	1	1	yes	characteristic
112	C₂×C₄₇	#2093	1	2	yes	—
113	C₂×C₄₇	#2093	1	1	yes	characteristic
114	D₄₇	—	16	2	no	—
115	C₂×#127	—	47	1	no	—
116	C₂×#107	—	2	2	no	—
117	C₂×#107	—	94	4	no	—
118	C₂×#102	—	94	4	no	—
119	C₂×#102	—	94	2	no	—
120	C₂×C₄×C₈	—	47	2	no	—
121	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
122	#384	—	47	1	no	—
123	#384	C₂×D₄₇	1	2	yes	—
124	C₄₇	#10673	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₄	—	94	2	no	—
126	C₂²×D₄	—	94	1	no	—
127	C₂²×C₈	—	47	4	no	—
128	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
129	#127	—	47	2	no	—
130	#127	—	94	2	no	—
131	#127	—	94	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	94	2	no	—
136	C₂×C₄⋊C₄	—	94	1	no	—
137	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
138	#107	—	4	4	no	—
139	#107	—	188	8	no	—
140	#102	—	188	8	no	—
141	#102	—	188	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	94	2	no	—
144	C₄×C₈	—	94	2	no	—
145	C₂⁴	—	188	1	no	—
146	C₂×D₄	—	94	2	no	—
147	C₂×D₄	—	94	4	no	—
148	C₂×D₄	—	188	1	no	—
149	C₂×D₄	—	188	2	no	—
150	C₂×D₄	—	188	4	no	—
151	C₂×D₄	—	188	2	no	—
152	C₂×D₄	—	188	2	no	—
153	C₂×D₄	—	188	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	188	1	no	—
156	C₂²×C₄	D₄×D₄₇	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	47	8	no	—
161	C₂×C₈	—	94	4	no	—
162	C₂×C₈	—	94	4	no	—
163	C₂×C₈	#10625334	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	188	4	no	—
168	C₄⋊C₄	—	188	2	no	—
169	C₄²	#10625350	1	2	yes	—
170	C₄²	Q₈×D₄₇	1	1	yes	characteristic
171	C₄²	#10625354	1	1	yes	characteristic
172	C₂³	—	376	2	no	—
173	C₂³	—	376	1	no	—
174	C₂³	—	376	1	no	—
175	C₂³	—	376	2	no	—
176	C₂³	—	188	2	no	—
177	C₂³	#13094020	1	1	yes	characteristic
178	D₄	—	376	2	no	—
179	D₄	—	376	4	no	—
180	D₄	—	188	4	no	—
181	D₄	—	188	8	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	376	2	no	—
186	C₂×C₄	—	376	1	no	—
187	C₂×C₄	#13094000	1	1	yes	characteristic
188	C₂×C₄	#13094002	1	2	yes	—
189	C₂×C₄	#13094007	1	1	yes	characteristic
190	C₂×C₄	#13094009	1	2	yes	—
191	C₂×C₄	D₈×D₄₇	1	4	yes	—
192	C₂×C₄	C₂×#10625334	1	2	yes	—
193	C₂×C₄	#13094055	1	2	yes	—
194	C₂×C₄	#13094056	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	94	8	no	—
197	C₂²	—	376	4	no	—
198	C₂²	—	376	2	no	—
199	C₂²	—	752	1	no	—
200	C₂²	—	752	2	no	—
201	C₂²	—	752	1	no	—
202	C₂²	#15689058	1	1	yes	characteristic
203	C₂²	#15689498	1	2	yes	—
204	C₂²	#15689499	1	1	yes	characteristic
205	C₂²	#15689515	1	1	yes	characteristic
206	C₂²	#15689516	1	2	yes	—
207	C₄	—	16	2	no	—
208	C₄	—	752	1	no	—
209	C₄	#15689400	1	2	yes	—
210	C₄	#15689402	1	4	yes	—
211	C₄	#15689436	1	2	yes	—
212	C₄	#15689438	1	4	yes	—
213	C₂	—	752	2	no	—
214	C₂	#20243002	1	1	yes	characteristic
215	C₂	#20243058	1	2	yes	—
216	C₂	#20243060	1	1	yes	characteristic
217	C₂	#20244913	1	2	yes	—
218	C₂	#20244917	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244194

C₂

yes

maximal, characteristic

#20244193

C₂

yes

maximal

#20243547

C₂

yes

maximal, complemented

C₄₇⋊#1273

C₂

yes

maximal, characteristic, complemented

C₄₇×#1845

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

—

—

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

characteristic

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂²×C₄₇⋊C₈

D₄

yes

—

#13094021

—

#13094021

—

#13094021

Q₈

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093957

—

#13093946

—

#13093944

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₄×C₈×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#474

yes

—

100

C₄×C₄₇

#474

yes

—

101

C₄×C₄₇

#480

yes

—

102

C₄×C₄₇

#480

yes

—

103

Dic₄₇

—

104

#1957

—

105

#1957

—

106

#1845

—

107

#1845

D₄₇

yes

characteristic, complemented

108

#1273

—

109

C₂×C₄₇

#1712

yes

characteristic

110

C₂×C₄₇

#1722

yes

—

111

C₂×C₄₇

#1722

yes

characteristic

112

C₂×C₄₇

#2093

yes

—

113

C₂×C₄₇

#2093

yes

characteristic

114

D₄₇

—

115

C₂×#127

—

116

C₂×#107

—

117

C₂×#107

—

118

C₂×#102

—

119

C₂×#102

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

122

#384

—

123

#384

C₂×D₄₇

yes

—

124

C₄₇

#10673

yes

characteristic, complemented, Sylow

125

C₂²×D₄

—

126

C₂²×D₄

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₄₇

yes

—

129

#127

—

130

#127

—

131

#127

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

C₂²×D₄₇

yes

characteristic

138

#107

—

139

#107

—

188

—

140

#102

—

188

—

141

#102

—

188

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₂⁴

—

188

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

188

—

149

C₂×D₄

—

188

—

150

C₂×D₄

—

188

—

151

C₂×D₄

—

188

—

152

C₂×D₄

—

188

—

153

C₂×D₄

—

188

—

154

C₂²×C₄

—

155

C₂²×C₄

—

188

—

156

C₂²×C₄

D₄×D₄₇

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

#10625334

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

188

—

168

C₄⋊C₄

—

188

—

169

C₄²

#10625350

yes

—

170

C₄²

Q₈×D₄₇

yes

characteristic

171

C₄²

#10625354

yes

characteristic

172

C₂³

—

376

—

173

C₂³

—

376

—

174

C₂³

—

376

—

175

C₂³

—

376

—

176

C₂³

—

188

—

177

C₂³

#13094020

yes

characteristic

178

D₄

—

376

—

179

D₄

—

376

—

180

D₄

—

188

—

181

D₄

—

188

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

376

—

186

C₂×C₄

—

376

—

187

C₂×C₄

#13094000

yes

characteristic

188

C₂×C₄

#13094002

yes

—

189

C₂×C₄

#13094007

yes

characteristic

190

C₂×C₄

#13094009

yes

—

191

C₂×C₄

D₈×D₄₇

yes

—

192

C₂×C₄

C₂×#10625334

yes

—

193

C₂×C₄

#13094055

yes

—

194

C₂×C₄

#13094056

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

376

—

198

C₂²

—

376

—

199

C₂²

—

752

—

200

C₂²

—

752

—

201

C₂²

—

752

—

202

C₂²

#15689058

yes

characteristic

203

C₂²

#15689498

yes

—

204

C₂²

#15689499

yes

characteristic

205

C₂²

#15689515

yes

characteristic

206

C₂²

#15689516

yes

—

207

C₄

—

208

C₄

—

752

—

209

C₄

#15689400

yes

—

210

C₄

#15689402

yes

—

211

C₄

#15689436

yes

—

212

C₄

#15689438

yes

—

213

C₂

—

752

—

214

C₂

#20243002

yes

characteristic

215

C₂

#20243058

yes

—

216

C₂

#20243060

yes

characteristic

217

C₂

#20244913

yes

—

218

C₂

#20244917

yes

characteristic

	subgroup	G/H
center	C₂³	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1845	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#10673
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13094020

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1845

C₂

C₂³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
3. #20243547	C₂	semidirect
4. C₄₇⋊#1273	C₂	semidirect
5. C₄₇×#1845	C₂	semidirect
107. #1845	D₄₇	semidirect
124. C₄₇	#10673	semidirect

H ≅ G/N

type

3. #20243547

C₂

semidirect

4. C₄₇⋊#1273

C₂

semidirect

5. C₄₇×#1845

C₂

semidirect

107. #1845

D₄₇

semidirect

124. C₄₇

#10673

semidirect

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

subgroup	index
#20244194	2
#20244193	2
#20243547	2
C₄₇⋊#1273	2
C₄₇×#1845	2
#10673	47

subgroup

index

#20244194

#20244193

#20243547

C₄₇⋊#1273

C₄₇×#1845

#10673

Subgroups — #26121326

Conjugacy classes of subgroups

Complements

Maximal subgroups