Group #26109304 — subgroups

Subgroups — #26109304

Order 12032.

404 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20242147	C₂	1	2	yes	maximal, complemented
2	#20242123	C₂	1	2	yes	maximal
3	C₄₇⋊#1309	C₂	1	2	yes	maximal, complemented
4	C₄₇×#2931	C₂	1	1	yes	maximal, characteristic
5	#15689147	C₂²	1	4	yes	—
6	#15688876	—	2	2	no	—
7	#15688876	—	2	2	no	—
8	#15688872	—	2	2	no	—
9	#15688867	—	2	2	no	—
10	C₂×#131×C₄₇	C₄	1	2	yes	—
11	C₄₇×#532	C₂²	1	1	yes	characteristic
12	C₄₇×#436	—	2	2	no	—
13	C₈×D₄×C₄₇	—	2	2	no	—
14	C₄₇×#423	C₂²	1	1	yes	characteristic
15	C₄₇×#408	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×#10625329	—	2	4	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₂×#41×C₄₇	C₂×C₄	1	2	yes	—
21	#131×C₄₇	—	4	4	no	—
22	C₂×M₁₆×C₄₇	—	2	2	no	—
23	C₂×M₁₆×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₄₇	—	2	2	no	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊Q₈×C₄₇	D₄	1	1	yes	characteristic
28	#127×C₄₇	D₄	1	1	yes	characteristic
29	#124×C₄₇	Q₈	1	2	yes	—
30	#121×C₄₇	—	2	2	no	—
31	C₄×D₄×C₄₇	—	2	2	no	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	—	2	2	no	—
34	C₂²⋊C₈×C₄₇	—	2	4	no	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	#41×C₄₇	—	4	2	no	—
38	#41×C₄₇	—	4	2	no	—
39	C₂×Q₈×C₄₇	C₄⋊C₄	1	2	yes	—
40	C₂×D₄×C₄₇	—	2	2	no	—
41	C₂×D₄×C₄₇	—	2	2	no	—
42	C₂×D₄×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₄₇	—	2	2	no	—
44	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₄₇	#41	1	2	yes	—
46	M₁₆×C₄₇	—	4	4	no	—
47	M₁₆×C₄₇	—	4	2	no	—
48	C₂×C₈×C₄₇	—	2	2	no	—
49	C₂×C₈×C₄₇	—	2	2	no	—
50	C₂×C₈×C₄₇	—	4	1	no	—
51	C₂×C₈×C₄₇	—	4	4	no	—
52	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₄₇	—	2	2	no	—
56	C₂²⋊C₄×C₄₇	—	2	4	no	—
57	C₄²×C₄₇	C₂×D₄	1	2	yes	—
58	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	#10625329	—	8	4	no	—
61	C₂×C₄₇⋊C₈	—	4	4	no	—
62	C₂³×C₄₇	—	2	2	no	—
63	C₂³×C₄₇	#117	1	1	yes	characteristic
64	Q₈×C₄₇	—	4	2	no	—
65	D₄×C₄₇	—	4	2	no	—
66	D₄×C₄₇	—	4	4	no	—
67	D₄×C₄₇	—	4	2	no	—
68	D₄×C₄₇	—	4	2	no	—
69	C₂×C₄×C₄₇	—	2	1	no	—
70	C₂×C₄×C₄₇	—	2	1	no	—
71	C₂×C₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	—	4	2	no	—
75	C₂×C₄×C₄₇	—	4	2	no	—
76	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#121	1	2	yes	—
79	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
80	C₈×C₄₇	—	2	1	no	—
81	C₈×C₄₇	—	2	2	no	—
82	C₈×C₄₇	—	4	2	no	—
83	C₈×C₄₇	#100	1	2	yes	—
84	C₄₇⋊C₈	—	8	4	no	—
85	#8638	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	1	no	—
87	C₂²×C₄₇	—	2	2	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	—	4	4	no	—
90	C₂²×C₄₇	#380	1	1	yes	characteristic
91	C₄×C₄₇	—	2	1	no	—
92	C₄×C₄₇	—	2	2	no	—
93	C₄×C₄₇	—	4	2	no	—
94	C₄×C₄₇	C₂×#100	1	1	yes	characteristic
95	C₄×C₄₇	#413	1	1	yes	characteristic
96	C₄×C₄₇	#472	1	2	yes	—
97	C₄×C₄₇	#473	1	2	yes	—
98	#2931	D₄₇	1	1	yes	characteristic
99	#1567	—	47	2	no	—
100	#1560	—	47	2	no	—
101	#1309	—	47	2	no	—
102	C₂×C₄₇	—	4	1	no	—
103	C₂×C₄₇	—	8	2	no	—
104	C₂×C₄₇	#1800	1	1	yes	characteristic
105	C₂×C₄₇	#1816	1	2	yes	—
106	C₂×#131	Dic₄₇	1	2	yes	—
107	#532	C₂×D₄₇	1	1	yes	characteristic
108	#436	—	2	2	no	—
109	C₈×D₄	—	2	2	no	—
110	#423	—	47	4	no	—
111	#423	C₂×D₄₇	1	1	yes	characteristic
112	#408	Dic₄₇	1	2	yes	—
113	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
114	#335	—	94	2	no	—
115	#335	—	94	2	no	—
116	#333	—	94	2	no	—
117	#331	—	94	2	no	—
118	C₄₇	#8638	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	C₂×Dic₄₇	1	2	yes	—
120	#131	—	4	4	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	—	94	4	no	—
123	C₂×M₁₆	C₂×Dic₄₇	1	2	yes	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
126	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
127	C₄⋊Q₈	D₁₈₈	1	1	yes	characteristic
128	#127	D₁₈₈	1	1	yes	characteristic
129	#124	Dic₉₄	1	2	yes	—
130	#121	—	2	2	no	—
131	C₄×D₄	—	2	2	no	—
132	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	94	4	no	—
135	C₄⋊C₈	—	94	4	no	—
136	C₂²⋊C₈	—	2	4	no	—
137	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
138	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
139	#41	—	4	2	no	—
140	#41	—	4	2	no	—
141	C₂×Q₈	C₄⋊Dic₄₇	1	2	yes	—
142	C₂×D₄	—	2	2	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₄⋊Dic₄₇	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	#10625350	1	2	yes	—
147	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
148	M₁₆	—	4	4	no	—
149	M₁₆	—	4	2	no	—
150	M₁₆	—	376	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	188	4	no	—
156	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
157	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
162	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
163	C₄²	C₂×C₂²⋊D₄₇	1	2	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	#13093983	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	2	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	2	no	—
170	D₄	—	4	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
180	C₂×C₄	#13093998	1	2	yes	—
181	C₂×C₄	#13094003	1	2	yes	—
182	C₈	—	2	1	no	—
183	C₈	—	2	2	no	—
184	C₈	—	4	2	no	—
185	C₈	—	376	4	no	—
186	C₈	#13093941	1	2	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	4	no	—
191	C₂²	#15688983	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₂×#13093941	1	1	yes	characteristic
196	C₄	#15689107	1	1	yes	characteristic
197	C₄	#15689393	1	2	yes	—
198	C₄	#15689398	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	#20243380	1	1	yes	characteristic
202	C₂	#20243452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242147

C₂

yes

maximal, complemented

#20242123

C₂

yes

maximal

C₄₇⋊#1309

C₂

yes

maximal, complemented

C₄₇×#2931

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

—

—

—

—

C₂×#131×C₄₇

C₄

yes

—

C₄₇×#532

C₂²

yes

characteristic

C₄₇×#436

—

C₈×D₄×C₄₇

—

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#408

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂×#41×C₄₇

C₂×C₄

yes

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₄₇

D₄

yes

characteristic

#127×C₄₇

D₄

yes

characteristic

#124×C₄₇

Q₈

yes

—

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₄⋊C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

#117

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#100

yes

—

C₄₇⋊C₈

—

#8638

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#380

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#473

yes

—

#2931

D₄₇

yes

characteristic

#1567

—

100

#1560

—

101

#1309

—

102

C₂×C₄₇

—

103

C₂×C₄₇

—

104

C₂×C₄₇

#1800

yes

characteristic

105

C₂×C₄₇

#1816

yes

—

106

C₂×#131

Dic₄₇

yes

—

107

#532

C₂×D₄₇

yes

characteristic

108

#436

—

109

C₈×D₄

—

110

#423

—

111

#423

C₂×D₄₇

yes

characteristic

112

#408

Dic₄₇

yes

—

113

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

114

#335

—

115

#335

—

116

#333

—

117

#331

—

118

C₄₇

#8638

yes

characteristic, complemented, Sylow

119

C₂×#41

C₂×Dic₄₇

yes

—

120

#131

—

121

C₂×M₁₆

—

122

C₂×M₁₆

—

123

C₂×M₁₆

C₂×Dic₄₇

yes

—

124

C₂²×C₈

—

125

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

126

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

127

C₄⋊Q₈

D₁₈₈

yes

characteristic

128

#127

D₁₈₈

yes

characteristic

129

#124

Dic₉₄

yes

—

130

#121

—

131

C₄×D₄

—

132

C₂×C₄²

C₂²×D₄₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₂²⋊C₈

—

137

C₄×C₈

C₂²⋊D₄₇

yes

—

138

C₄×C₈

C₂²⋊D₄₇

yes

—

139

#41

—

140

#41

—

141

C₂×Q₈

C₄⋊Dic₄₇

yes

—

142

C₂×D₄

—

143

C₂×D₄

—

144

C₂×D₄

C₄⋊Dic₄₇

yes

—

145

C₂²×C₄

—

146

C₂²×C₄

#10625350

yes

—

147

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

376

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

188

—

156

C₂×C₈

C₂²⋊Dic₄₇

yes

—

157

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

158

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

C₂×Dic₉₄

yes

characteristic

162

C₄²

C₂×D₁₈₈

yes

characteristic

163

C₄²

C₂×C₂²⋊D₄₇

yes

—

164

C₂³

—

165

C₂³

#13093983

yes

characteristic

166

Q₈

—

167

D₄

—

168

D₄

—

169

D₄

—

170

D₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

179

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

180

C₂×C₄

#13093998

yes

—

181

C₂×C₄

#13094003

yes

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

376

—

186

C₈

#13093941

yes

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₂²

—

191

C₂²

#15688983

yes

characteristic

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₂×#13093941

yes

characteristic

196

C₄

#15689107

yes

characteristic

197

C₄

#15689393

yes

—

198

C₄

#15689398

yes

—

199

C₂

—

200

C₂

—

201

C₂

#20243380

yes

characteristic

202

C₂

#20243452

yes

—

	subgroup	G/H
center	C₂²	#15688983
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2931	C₂
socle	C₂²×C₄₇	#380
prime radical	C₄₇×#532	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2931	D₄₇
hypocenter	C₄₇	#8638
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

subgroup

G/H

center

C₂²

#15688983

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2931

C₂

socle

C₂²×C₄₇

#380

prime radical

C₄₇×#532

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

N	H ≅ G/N	type
1. #20242147	C₂	semidirect
3. C₄₇⋊#1309	C₂	semidirect
118. C₄₇	#8638	semidirect

H ≅ G/N

type

1. #20242147

C₂

semidirect

3. C₄₇⋊#1309

C₂

semidirect

118. C₄₇

#8638

semidirect

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

subgroup	index
#20242147	2
#20242123	2
C₄₇⋊#1309	2
C₄₇×#2931	2
#8638	47

subgroup

index

#20242147

#20242123

C₄₇⋊#1309

C₄₇×#2931

#8638

Subgroups — #26109304

Conjugacy classes of subgroups

Complements

Maximal subgroups