Group #26113893 — subgroups

Subgroups — #26113893

Order 12032.

452 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 111 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	#20252490	C₂	1	1	yes	maximal, characteristic, complemented
2	#20242126	C₂	1	1	yes	maximal, characteristic, complemented
3	#20242056	C₂	1	2	yes	maximal
4	#20241278	C₂	1	2	yes	maximal, complemented
5	C₄₇×#3138	C₂	1	1	yes	maximal, characteristic
6	#15689254	—	2	2	no	—
7	Q₈×C₄₇⋊C₈	—	2	2	no	—
8	#15689184	—	2	2	no	—
9	#15689176	C₂²	1	2	yes	—
10	#15689147	C₂²	1	2	yes	—
11	#15688869	—	2	2	no	—
12	#15688867	—	2	2	no	—
13	#15688867	—	2	2	no	—
14	C₄₇×#533	C₂²	1	1	yes	characteristic
15	C₂×#127×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#490	C₄	1	4	yes	—
17	C₄₇×#486	—	2	2	no	—
18	C₄₇×#418	—	2	2	no	—
19	C₄×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×#10625329	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
25	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
26	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
27	#13093935	—	2	2	no	—
28	C₄×C₄₇⋊C₈	—	2	2	no	—
29	C₄×C₄₇⋊C₈	—	2	2	no	—
30	C₂²×D₄×C₄₇	—	2	1	no	—
31	C₂²×D₄×C₄₇	—	2	2	no	—
32	C₂×M₁₆×C₄₇	—	2	1	no	—
33	C₄⋊Q₈×C₄₇	D₄	1	2	yes	—
34	#127×C₄₇	—	2	1	no	—
35	#127×C₄₇	—	2	2	no	—
36	#127×C₄₇	C₂×C₄	1	2	yes	—
37	#125×C₄₇	C₂×C₄	1	2	yes	—
38	C₂²⋊Q₈×C₄₇	—	2	2	no	—
39	C₄×Q₈×C₄₇	—	2	2	no	—
40	#117×C₄₇	D₄	1	2	yes	—
41	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	#102×C₄₇	—	2	8	no	—
43	#97×C₄₇	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₄₇	—	2	1	no	—
45	C₂⁴×C₄₇	—	4	1	no	—
46	C₂×Q₈×C₄₇	—	2	2	no	—
47	C₂×D₄×C₄₇	—	2	2	no	—
48	C₂×D₄×C₄₇	—	2	4	no	—
49	C₂×D₄×C₄₇	—	4	4	no	—
50	C₂×D₄×C₄₇	—	4	1	no	—
51	C₂×D₄×C₄₇	—	4	2	no	—
52	C₂×D₄×C₄₇	—	8	2	no	—
53	C₂×D₄×C₄₇	—	8	2	no	—
54	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	#41	1	2	yes	—
56	M₁₆×C₄₇	—	4	2	no	—
57	C₂×C₈×C₄₇	—	2	2	no	—
58	C₄⋊C₄×C₄₇	—	2	4	no	—
59	C₄⋊C₄×C₄₇	—	2	2	no	—
60	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
61	C₂²⋊C₄×C₄₇	—	2	2	no	—
62	C₄²×C₄₇	—	2	2	no	—
63	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₄₇	C₂×D₄	1	2	yes	—
66	#10625329	—	8	2	no	—
67	C₂×C₄₇⋊C₈	—	2	2	no	—
68	C₂×C₄₇⋊C₈	—	2	2	no	—
69	C₂×C₄₇⋊C₈	—	4	2	no	—
70	C₂³×C₄₇	—	4	2	no	—
71	C₂³×C₄₇	—	8	2	no	—
72	C₂³×C₄₇	—	8	4	no	—
73	C₂³×C₄₇	#123	1	1	yes	characteristic
74	Q₈×C₄₇	—	4	2	no	—
75	Q₈×C₄₇	—	4	2	no	—
76	D₄×C₄₇	—	4	4	no	—
77	D₄×C₄₇	—	8	4	no	—
78	D₄×C₄₇	—	8	2	no	—
79	D₄×C₄₇	—	8	2	no	—
80	D₄×C₄₇	—	8	2	no	—
81	C₂×C₄×C₄₇	—	2	2	no	—
82	C₂×C₄×C₄₇	—	2	2	no	—
83	C₂×C₄×C₄₇	—	2	2	no	—
84	C₂×C₄×C₄₇	—	2	4	no	—
85	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
88	C₂×C₄×C₄₇	#121	1	2	yes	—
89	C₈×C₄₇	—	4	2	no	—
90	C₄₇⋊C₈	—	4	2	no	—
91	C₄₇⋊C₈	—	4	2	no	—
92	C₄₇⋊C₈	—	8	2	no	—
93	#9438	—	47	1	no	maximal, Sylow
94	C₂²×C₄₇	—	2	1	no	—
95	C₂²×C₄₇	—	2	2	no	—
96	C₂²×C₄₇	—	8	2	no	—
97	C₂²×C₄₇	—	8	4	no	—
98	C₂²×C₄₇	—	16	4	no	—
99	C₂²×C₄₇	#386	1	1	yes	characteristic
100	C₄×C₄₇	—	2	1	no	—
101	C₄×C₄₇	—	2	2	no	—
102	C₄×C₄₇	—	4	4	no	—
103	C₄×C₄₇	#454	1	2	yes	—
104	C₄×C₄₇	#471	1	2	yes	—
105	C₄×C₄₇	#472	1	2	yes	—
106	#3138	D₄₇	1	1	yes	characteristic
107	#2986	—	47	1	no	—
108	#1561	—	47	1	no	—
109	#1545	—	47	2	no	—
110	#1299	—	47	2	no	—
111	C₂×C₄₇	—	4	1	no	—
112	C₂×C₄₇	—	16	2	no	—
113	C₂×C₄₇	#1888	1	1	yes	characteristic
114	C₂×C₄₇	#1907	1	2	yes	—
115	#533	C₂×D₄₇	1	1	yes	characteristic
116	C₂×#127	C₂×D₄₇	1	1	yes	characteristic
117	#490	Dic₄₇	1	4	yes	—
118	#486	—	2	2	no	—
119	#446	—	94	2	no	—
120	C₈×Q₈	—	94	2	no	—
121	#433	—	94	2	no	—
122	#431	—	47	2	no	—
123	#423	—	47	2	no	—
124	#418	—	2	2	no	—
125	C₄×M₁₆	C₂×D₄₇	1	1	yes	characteristic
126	#332	—	94	2	no	—
127	#331	—	94	2	no	—
128	#331	—	94	2	no	—
129	C₄₇	#9438	1	1	yes	characteristic, complemented, Sylow
130	C₂²×D₄	—	2	2	no	—
131	C₂²×D₄	—	2	1	no	—
132	C₂×M₁₆	—	2	1	no	—
133	C₂×M₁₆	—	94	2	no	—
134	C₄⋊Q₈	C₂²⋊D₄₇	1	2	yes	—
135	#127	—	2	2	no	—
136	#127	—	2	1	no	—
137	#127	C₂×Dic₄₇	1	2	yes	—
138	#125	C₂×Dic₄₇	1	2	yes	—
139	C₂²⋊Q₈	—	2	2	no	—
140	C₄×Q₈	—	2	2	no	—
141	#117	C₂²⋊D₄₇	1	2	yes	—
142	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
143	C₄⋊C₈	—	47	2	no	—
144	C₄⋊C₈	—	47	2	no	—
145	C₄⋊C₈	—	94	4	no	—
146	C₄⋊C₈	—	94	2	no	—
147	C₄⋊C₈	—	94	2	no	—
148	#102	—	2	8	no	—
149	C₂²⋊C₈	—	94	2	no	—
150	#97	—	94	2	no	—
151	#97	C₂×Dic₄₇	1	2	yes	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	—	94	2	no	—
154	C₄×C₈	—	94	2	no	—
155	C₂⁴	—	4	1	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	8	2	no	—
163	C₂×D₄	—	8	2	no	—
164	C₂²×C₄	#10625352	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
166	M₁₆	—	4	2	no	—
167	M₁₆	—	376	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	94	2	no	—
170	C₂×C₈	—	94	2	no	—
171	C₂×C₈	—	188	2	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	C₂²⋊Dic₄₇	1	4	yes	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₄²	—	2	2	no	—
177	C₄²	D₄×D₄₇	1	2	yes	—
178	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
179	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
180	C₂³	—	4	2	no	—
181	C₂³	—	8	2	no	—
182	C₂³	—	8	4	no	—
183	C₂³	#13094005	1	1	yes	characteristic
184	Q₈	—	4	2	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	4	4	no	—
187	D₄	—	8	4	no	—
188	D₄	—	8	2	no	—
189	D₄	—	8	2	no	—
190	D₄	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
196	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
197	C₂×C₄	#13093992	1	1	yes	characteristic
198	C₂×C₄	#13093996	1	2	yes	—
199	C₈	—	4	2	no	—
200	C₈	—	376	2	no	—
201	C₈	—	188	2	no	—
202	C₈	—	188	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	2	no	—
205	C₂²	—	8	2	no	—
206	C₂²	—	8	4	no	—
207	C₂²	—	16	4	no	—
208	C₂²	#15689003	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	2	no	—
211	C₄	—	4	4	no	—
212	C₄	#15689295	1	2	yes	—
213	C₄	#15689384	1	2	yes	—
214	C₄	#15689391	1	2	yes	—
215	C₂	—	4	1	no	—
216	C₂	—	16	2	no	—
217	C₂	#20243792	1	1	yes	characteristic
218	C₂	#20243898	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252490

C₂

yes

maximal, characteristic, complemented

#20242126

C₂

yes

maximal, characteristic, complemented

#20242056

C₂

yes

maximal

#20241278

C₂

yes

maximal, complemented

C₄₇×#3138

C₂

yes

maximal, characteristic

#15689254

—

Q₈×C₄₇⋊C₈

—

#15689184

—

#15689176

C₂²

yes

—

#15689147

C₂²

yes

—

#15688869

—

#15688867

—

#15688867

—

C₄₇×#533

C₂²

yes

characteristic

C₂×#127×C₄₇

C₂²

yes

characteristic

C₄₇×#490

C₄

yes

—

C₄₇×#486

—

C₄₇×#418

—

C₄×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

D₄

yes

—

#127×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₂×C₄

yes

—

#125×C₄₇

C₂×C₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#102×C₄₇

—

#97×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

—

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9438

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#386

yes

characteristic

100

C₄×C₄₇

—

101

C₄×C₄₇

—

102

C₄×C₄₇

—

103

C₄×C₄₇

#454

yes

—

104

C₄×C₄₇

#471

yes

—

105

C₄×C₄₇

#472

yes

—

106

#3138

D₄₇

yes

characteristic

107

#2986

—

108

#1561

—

109

#1545

—

110

#1299

—

111

C₂×C₄₇

—

112

C₂×C₄₇

—

113

C₂×C₄₇

#1888

yes

characteristic

114

C₂×C₄₇

#1907

yes

—

115

#533

C₂×D₄₇

yes

characteristic

116

C₂×#127

C₂×D₄₇

yes

characteristic

117

#490

Dic₄₇

yes

—

118

#486

—

119

#446

—

120

C₈×Q₈

—

121

#433

—

122

#431

—

123

#423

—

124

#418

—

125

C₄×M₁₆

C₂×D₄₇

yes

characteristic

126

#332

—

127

#331

—

128

#331

—

129

C₄₇

#9438

yes

characteristic, complemented, Sylow

130

C₂²×D₄

—

131

C₂²×D₄

—

132

C₂×M₁₆

—

133

C₂×M₁₆

—

134

C₄⋊Q₈

C₂²⋊D₄₇

yes

—

135

#127

—

136

#127

—

137

#127

C₂×Dic₄₇

yes

—

138

#125

C₂×Dic₄₇

yes

—

139

C₂²⋊Q₈

—

140

C₄×Q₈

—

141

#117

C₂²⋊D₄₇

yes

—

142

C₂×C₄²

C₂²×D₄₇

yes

characteristic

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

—

148

#102

—

149

C₂²⋊C₈

—

150

#97

—

151

#97

C₂×Dic₄₇

yes

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₂⁴

—

156

C₂×Q₈

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

—

162

C₂×D₄

—

163

C₂×D₄

—

164

C₂²×C₄

#10625352

yes

—

165

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

166

M₁₆

—

167

M₁₆

—

376

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

188

—

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄⋊C₄

C₂²⋊Dic₄₇

yes

—

175

C₂²⋊C₄

—

176

C₄²

—

177

C₄²

D₄×D₄₇

yes

—

178

C₄²

C₂²×Dic₄₇

yes

characteristic

179

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

180

C₂³

—

181

C₂³

—

182

C₂³

—

183

C₂³

#13094005

yes

characteristic

184

Q₈

—

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

D₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

196

C₂×C₄

D₄×Dic₄₇

yes

—

197

C₂×C₄

#13093992

yes

characteristic

198

C₂×C₄

#13093996

yes

—

199

C₈

—

200

C₈

—

376

—

201

C₈

—

188

—

202

C₈

—

188

—

203

C₂²

—

204

C₂²

—

205

C₂²

—

206

C₂²

—

207

C₂²

—

208

C₂²

#15689003

yes

characteristic

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#15689295

yes

—

213

C₄

#15689384

yes

—

214

C₄

#15689391

yes

—

215

C₂

—

216

C₂

—

217

C₂

#20243792

yes

characteristic

218

C₂

#20243898

yes

—

	subgroup	G/H
center	C₂²	#15689003
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3138	C₂
socle	C₂²×C₄₇	#386
prime radical	C₂×#127×C₄₇	C₂²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3138	D₄₇
hypocenter	C₄₇	#9438
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

subgroup

G/H

center

C₂²

#15689003

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3138

C₂

socle

C₂²×C₄₇

#386

prime radical

C₂×#127×C₄₇

C₂²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3138

N	H ≅ G/N	type
1. #20252490	C₂	semidirect
2. #20242126	C₂	semidirect
4. #20241278	C₂	semidirect
129. C₄₇	#9438	semidirect

H ≅ G/N

type

1. #20252490

C₂

semidirect

2. #20242126

C₂

semidirect

4. #20241278

C₂

semidirect

129. C₄₇

#9438

semidirect

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

subgroup	index
#20252490	2
#20242126	2
#20242056	2
#20241278	2
C₄₇×#3138	2
#9438	47

subgroup

index

C₄₇×#3138

Subgroups — #26113893

Conjugacy classes of subgroups

Complements

Maximal subgroups