Group #26116398 — subgroups

Subgroups — #26116398

Order 12032.

430 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20251751	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#20241272	C₂	1	4	yes	maximal
4	C₄₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	—	2	1	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	#15689021	—	2	2	no	—
9	#15689021	C₄	1	4	yes	—
10	#15688993	C₂²	1	2	yes	—
11	#15688989	—	2	1	no	—
12	#15688878	—	2	1	no	—
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#336	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	—	2	2	no	—
17	#13094016	—	4	4	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
21	C₂×Dic₄₇⋊C₄	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	Q₈	1	4	yes	—
24	C₄×C₄₇⋊C₈	—	4	2	no	—
25	C₄₇⋊#95	—	2	4	no	—
26	C₂²×C₈×C₄₇	—	2	2	no	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	2	yes	—
30	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	4	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	4	4	no	—
41	C₄²×C₄₇	—	2	4	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	4	2	no	—
44	C₄²×C₄₇	SD₁₆	1	2	yes	—
45	C₄²×C₄₇	Q₁₆	1	2	yes	—
46	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
47	C₄²×C₄₇	#41	1	2	yes	—
48	C₄⋊Dic₄₇	—	4	4	no	—
49	C₄⋊Dic₄₇	—	4	4	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	Dic₄₇⋊C₄	—	4	8	no	—
52	C₂×C₄₇⋊C₈	—	2	4	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	2	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	#103	1	4	yes	—
63	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
65	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
68	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₄₇	#135	1	4	yes	—
70	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
71	C₂×Dic₄₇	—	4	4	no	—
72	C₂×Dic₄₇	—	4	4	no	—
73	C₂×Dic₄₇	—	4	4	no	—
74	C₈×C₄₇	—	8	2	no	—
75	C₄₇⋊C₈	—	4	4	no	—
76	#9897	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	#391	1	1	yes	characteristic
78	C₂²×C₄₇	#413	1	2	yes	—
79	C₂²×C₄₇	C₂×#103	1	1	yes	characteristic
80	C₂²×C₄₇	#416	1	2	yes	—
81	C₂²×C₄₇	#418	1	1	yes	characteristic
82	C₄×C₄₇	—	2	4	no	—
83	C₄×C₄₇	—	2	2	no	—
84	C₄×C₄₇	—	4	2	no	—
85	C₄×C₄₇	#476	1	2	yes	—
86	C₄×C₄₇	#477	1	2	yes	—
87	C₄×C₄₇	#488	1	2	yes	—
88	C₄×C₄₇	#499	1	2	yes	—
89	Dic₄₇	—	8	4	no	—
90	#2859	—	47	1	no	—
91	#1831	D₄₇	1	1	yes	characteristic
92	C₄×C₄⋊C₈	—	47	1	no	—
93	#1296	—	47	4	no	—
94	C₂×C₄₇	#1542	1	2	yes	—
95	C₂×C₄₇	#1553	1	2	yes	—
96	C₂×C₄₇	#1987	1	1	yes	characteristic
97	C₂×C₄₇	#1991	1	1	yes	characteristic
98	C₂×C₄₇	#1998	1	1	yes	characteristic
99	C₂×#125	—	94	1	no	—
100	C₂×C₄⋊C₈	—	47	2	no	—
101	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
102	C₂×C₄×C₈	—	94	1	no	—
103	#389	—	47	4	no	—
104	#389	—	94	2	no	—
105	#384	—	47	2	no	—
106	#382	—	94	1	no	—
107	C₄³	C₂×D₄₇	1	1	yes	characteristic
108	#336	—	2	2	no	—
109	#336	—	94	1	no	—
110	C₄₇	#9897	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	94	2	no	—
113	#125	—	188	4	no	—
114	C₂×C₄⋊C₄	—	47	4	no	—
115	C₂×C₄⋊C₄	—	94	4	no	—
116	C₂×C₄⋊C₄	—	94	2	no	—
117	C₂×C₄⋊C₄	—	94	2	no	—
118	C₂×C₄²	C₄×D₄₇	1	2	yes	—
119	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
120	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
121	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
122	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₄⋊C₈	—	2	4	no	—
124	C₄⋊C₈	—	47	4	no	—
125	C₄⋊C₈	—	94	2	no	—
126	C₄×C₈	—	188	2	no	—
127	#95	—	94	4	no	—
128	C₂²×C₄	—	94	4	no	—
129	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
130	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
132	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	94	4	no	—
137	C₂×C₈	—	188	2	no	—
138	C₂×C₈	—	188	2	no	—
139	C₄⋊C₄	—	188	4	no	—
140	C₄⋊C₄	—	188	8	no	—
141	C₄⋊C₄	—	188	4	no	—
142	C₄⋊C₄	—	188	4	no	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	#10625325	1	2	yes	—
147	C₄²	Dic₁₈₈	1	2	yes	—
148	C₄²	#10625352	1	2	yes	—
149	C₄²	Q₈×D₄₇	1	2	yes	—
150	C₂³	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	188	4	no	—
158	C₂×C₄	—	188	4	no	—
159	C₂×C₄	—	188	4	no	—
160	C₂×C₄	#13093949	1	4	yes	—
161	C₂×C₄	#13093989	1	2	yes	—
162	C₂×C₄	#13094004	1	2	yes	—
163	C₂×C₄	#13094011	1	1	yes	characteristic
164	C₂×C₄	#13094022	1	1	yes	characteristic
165	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₁₈₈	1	1	yes	characteristic
167	C₂×C₄	#13094044	1	4	yes	—
168	C₂×C₄	#13094052	1	2	yes	—
169	C₈	—	8	2	no	—
170	C₈	—	188	4	no	—
171	C₂²	#15689028	1	1	yes	characteristic
172	C₂²	#15689105	1	2	yes	—
173	C₂²	C₂×#13093949	1	1	yes	characteristic
174	C₂²	#15689117	1	2	yes	—
175	C₂²	#15689129	1	1	yes	characteristic
176	C₄	—	2	4	no	—
177	C₄	—	2	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	376	4	no	—
180	C₄	#15689414	1	2	yes	—
181	C₄	#15689419	1	2	yes	—
182	C₄	#15689477	1	2	yes	—
183	C₄	#15689522	1	2	yes	—
184	C₂	#20242043	1	2	yes	—
185	C₂	#20242094	1	2	yes	—
186	C₂	#20244335	1	1	yes	characteristic
187	C₂	#20244352	1	1	yes	characteristic
188	C₂	#20244380	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20251751

C₂

yes

maximal, characteristic

C₄×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#20241272

C₂

yes

maximal

C₄₇×#1831

C₂

yes

maximal, characteristic

C₂×#13094016

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

—

#15689021

—

#15689021

C₄

yes

—

#15688993

C₂²

yes

—

#15688989

—

#15688878

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×Dic₄₇⋊C₄

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

Q₈

yes

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×#103

yes

characteristic

C₂²×C₄₇

#416

yes

—

C₂²×C₄₇

#418

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

#477

yes

—

C₄×C₄₇

#488

yes

—

C₄×C₄₇

#499

yes

—

Dic₄₇

—

#2859

—

#1831

D₄₇

yes

characteristic

C₄×C₄⋊C₈

—

#1296

—

C₂×C₄₇

#1542

yes

—

C₂×C₄₇

#1553

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

102

C₂×C₄×C₈

—

103

#389

—

104

#389

—

105

#384

—

106

#382

—

107

C₄³

C₂×D₄₇

yes

characteristic

108

#336

—

109

#336

—

110

C₄₇

#9897

yes

characteristic, complemented, Sylow

111

C₂²×C₈

—

112

C₂²×C₈

—

113

#125

—

188

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄²

C₄×D₄₇

yes

—

119

C₂×C₄²

D₁₈₈

yes

characteristic

120

C₂×C₄²

D₁₈₈

yes

characteristic

121

C₂×C₄²

C₂²⋊D₄₇

yes

—

122

C₂×C₄²

C₂²×D₄₇

yes

characteristic

123

C₄⋊C₈

—

124

C₄⋊C₈

—

125

C₄⋊C₈

—

126

C₄×C₈

—

188

—

127

#95

—

128

C₂²×C₄

—

129

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

130

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

131

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

132

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

133

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

188

—

138

C₂×C₈

—

188

—

139

C₄⋊C₄

—

188

—

140

C₄⋊C₄

—

188

—

141

C₄⋊C₄

—

188

—

142

C₄⋊C₄

—

188

—

143

C₄²

—

144

C₄²

—

145

C₄²

—

146

C₄²

#10625325

yes

—

147

C₄²

Dic₁₈₈

yes

—

148

C₄²

#10625352

yes

—

149

C₄²

Q₈×D₄₇

yes

—

150

C₂³

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

188

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

#13093949

yes

—

161

C₂×C₄

#13093989

yes

—

162

C₂×C₄

#13094004

yes

—

163

C₂×C₄

#13094011

yes

characteristic

164

C₂×C₄

#13094022

yes

characteristic

165

C₂×C₄

C₂×#10625325

yes

characteristic

166

C₂×C₄

C₂×Dic₁₈₈

yes

characteristic

167

C₂×C₄

#13094044

yes

—

168

C₂×C₄

#13094052

yes

—

169

C₈

—

170

C₈

—

188

—

171

C₂²

#15689028

yes

characteristic

172

C₂²

#15689105

yes

—

173

C₂²

C₂×#13093949

yes

characteristic

174

C₂²

#15689117

yes

—

175

C₂²

#15689129

yes

characteristic

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

—

376

—

180

C₄

#15689414

yes

—

181

C₄

#15689419

yes

—

182

C₄

#15689477

yes

—

183

C₄

#15689522

yes

—

184

C₂

#20242043

yes

—

185

C₂

#20242094

yes

—

186

C₂

#20244335

yes

characteristic

187

C₂

#20244352

yes

characteristic

188

C₂

#20244380

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1831	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1831	D₄₇
hypocenter	C₄₇	#9897
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂³

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1831

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
110. C₄₇	#9897	semidirect

H ≅ G/N

type

110. C₄₇

#9897

semidirect

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

subgroup	index
#20251751	2
C₄×C₄⋊(C₄₇⋊C₈)	2
#20241272	2
C₄₇×#1831	2
#9897	47

subgroup

index

#20251751

C₄×C₄⋊(C₄₇⋊C₈)

#20241272

C₄₇×#1831

#9897

Subgroups — #26116398

Conjugacy classes of subgroups

Complements

Maximal subgroups