Group #26373955 — subgroups

Subgroups — #26373955

Order 12032.

528 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 165 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	Q₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245611	C₂	1	3	yes	maximal, complemented
3	#20245535	C₂	1	3	yes	maximal, complemented
4	#20245425	C₂	1	3	yes	maximal, complemented
5	Q₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13093968	C₂	1	3	yes	maximal, complemented
7	C₄₇×#3391	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₉₄	—	2	1	no	—
9	C₂×#13093968	C₂²	1	3	yes	—
10	#15689528	C₂²	1	3	yes	—
11	#15689419	—	2	3	no	—
12	C₄×Dic₁₈₈	C₂²	1	3	yes	—
13	#15688950	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	3	yes	—
15	#15688926	C₂²	1	1	yes	characteristic
16	#15688924	C₂²	1	6	yes	—
17	#15688924	C₂²	1	3	yes	—
18	C₄×C₄₇⋊C₁₆	C₂²	1	3	yes	—
19	C₄₇×#550	—	2	1	no	—
20	C₄₇×#493	C₂²	1	3	yes	—
21	C₄₇×#459	—	2	3	no	—
22	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₄₇	C₂²	1	3	yes	—
24	C₂×Dic₁₈₈	C₂³	1	1	yes	characteristic
25	#13094024	—	2	3	no	—
26	#13094023	—	4	3	no	—
27	Q₈×Dic₄₇	—	4	1	no	—
28	C₄×Dic₉₄	—	2	3	no	—
29	#13093968	—	2	6	no	—
30	C₂×C₄₇⋊C₁₆	C₂³	1	3	yes	—
31	C₂×C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
32	#13093958	—	2	3	no	—
33	#13093958	C₂³	1	3	yes	—
34	#13093949	—	2	3	no	—
35	C₂×#41×C₄₇	—	4	1	no	—
36	C₂×D₈×C₄₇	—	2	3	no	—
37	C₂×D₈×C₄₇	C₂³	1	3	yes	—
38	#127×C₄₇	—	2	3	no	—
39	#121×C₄₇	—	4	3	no	—
40	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
41	C₄×D₄×C₄₇	—	2	3	no	—
42	#107×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	D₄	1	3	yes	—
44	#102×C₄₇	—	2	3	no	—
45	C₄×C₈×C₄₇	C₂³	1	3	yes	—
46	#41×C₄₇	—	8	4	no	—
47	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
48	C₂×D₄×C₄₇	—	2	3	no	—
49	C₂×D₄×C₄₇	—	4	3	no	—
50	C₂×D₄×C₄₇	—	4	3	no	—
51	C₂²×C₄×C₄₇	—	4	3	no	—
52	C₂×Dic₉₄	—	2	1	no	—
53	C₂×Dic₉₄	—	4	3	no	—
54	D₈×C₄₇	—	2	3	no	—
55	D₈×C₄₇	—	2	3	no	—
56	D₈×C₄₇	—	8	3	no	—
57	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₄₇	—	2	1	no	—
60	C₄⋊C₄×C₄₇	D₈	1	3	yes	—
61	C₂²⋊C₄×C₄₇	—	4	3	no	—
62	C₄²×C₄₇	C₂×D₄	1	3	yes	—
63	C₄⋊Dic₄₇	—	2	3	no	—
64	C₄⋊Dic₄₇	—	4	3	no	—
65	Dic₄₇⋊C₄	—	4	3	no	—
66	C₄×Dic₄₇	—	4	3	no	—
67	Dic₁₈₈	—	2	1	no	—
68	Dic₁₈₈	#41	1	2	yes	complemented
69	C₄₇⋊C₁₆	—	2	3	no	—
70	C₄₇⋊C₁₆	—	2	1	no	—
71	C₂³×C₄₇	—	4	3	no	—
72	Q₈×C₄₇	SD₃₂	1	4	yes	complemented
73	D₄×C₄₇	—	4	3	no	—
74	D₄×C₄₇	—	4	3	no	—
75	D₄×C₄₇	—	8	6	no	—
76	D₄×C₄₇	—	16	3	no	—
77	C₂×C₄×C₄₇	—	4	1	no	—
78	C₂×C₄×C₄₇	—	8	6	no	—
79	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
80	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
81	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
82	C₂×Dic₄₇	—	4	1	no	—
83	C₂×Dic₄₇	—	4	3	no	—
84	Dic₉₄	—	2	2	no	—
85	Dic₉₄	—	4	1	no	—
86	Dic₉₄	—	16	3	no	—
87	C₈×C₄₇	—	2	3	no	—
88	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
89	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
90	#33929	—	47	1	no	maximal, Sylow
91	C₂²×C₄₇	—	8	3	no	—
92	C₂²×C₄₇	—	8	3	no	—
93	C₂²×C₄₇	—	8	3	no	—
94	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
95	C₄×C₄₇	—	2	3	no	—
96	C₄×C₄₇	—	8	1	no	—
97	C₄×C₄₇	C₂×SD₃₂	1	6	yes	—
98	C₄×C₄₇	#548	1	1	yes	characteristic
99	C₄×C₄₇	#577	1	1	yes	characteristic
100	Dic₄₇	—	4	2	no	—
101	Dic₄₇	—	8	3	no	—
102	#3391	D₄₇	1	1	yes	characteristic
103	Q₈×Q₁₆	—	47	1	no	—
104	#2240	—	47	3	no	—
105	#2226	—	47	3	no	—
106	#2207	—	47	3	no	—
107	Q₈×C₁₆	—	47	1	no	—
108	C₄×SD₃₂	—	47	3	no	—
109	C₂×C₄₇	—	16	3	no	—
110	C₂×C₄₇	#3358	1	1	yes	characteristic
111	C₂×C₄₇	C₂²×SD₃₂	1	1	yes	characteristic
112	C₂×C₄₇	#3415	1	1	yes	characteristic
113	Q₈²	—	94	1	no	—
114	#550	—	2	1	no	—
115	C₂×SD₃₂	—	47	3	no	—
116	#500	—	47	3	no	—
117	#493	C₂×D₄₇	1	3	yes	—
118	#477	—	94	3	no	—
119	#459	—	2	3	no	—
120	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
121	C₄×Q₁₆	—	47	3	no	—
122	C₄×D₈	C₂×D₄₇	1	3	yes	—
123	#367	—	47	3	no	—
124	C₄⋊C₁₆	—	47	3	no	—
125	#358	—	47	1	no	—
126	#357	—	47	3	no	—
127	#357	—	47	6	no	—
128	C₄×C₁₆	—	47	3	no	—
129	C₄₇	#33929	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	4	1	no	—
131	C₂×Q₁₆	—	47	1	no	—
132	C₂×D₈	—	2	3	no	—
133	C₂×D₈	C₂²×D₄₇	1	3	yes	—
134	C₄⋊Q₈	—	94	3	no	—
135	C₄⋊Q₈	—	188	3	no	—
136	#127	—	2	3	no	—
137	#121	—	4	3	no	—
138	C₄×Q₈	—	94	3	no	—
139	C₄×Q₈	—	188	1	no	—
140	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
141	C₄×D₄	—	2	3	no	—
142	SD₃₂	—	94	6	no	—
143	C₂×C₁₆	—	47	1	no	—
144	C₂×C₁₆	—	47	3	no	—
145	#107	—	47	3	no	—
146	#107	—	94	3	no	—
147	#107	C₂²×D₄₇	1	1	yes	characteristic
148	C₄⋊C₈	C₂²⋊D₄₇	1	3	yes	—
149	#103	—	94	3	no	—
150	#102	—	2	3	no	—
151	C₄×C₈	C₂²×D₄₇	1	3	yes	—
152	#41	—	8	4	no	—
153	C₂×Q₈	—	94	1	no	—
154	C₂×Q₈	—	188	3	no	—
155	C₂×Q₈	#10625334	1	1	yes	characteristic
156	C₂×D₄	—	2	3	no	—
157	C₂×D₄	—	4	3	no	—
158	C₂×D₄	—	4	3	no	—
159	C₂²×C₄	—	4	3	no	—
160	Q₁₆	—	47	2	no	—
161	Q₁₆	—	94	1	no	—
162	D₈	—	2	3	no	—
163	D₈	—	2	3	no	—
164	D₈	—	8	3	no	—
165	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
166	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	94	3	no	—
169	C₄⋊C₄	—	188	3	no	—
170	C₄⋊C₄	—	188	3	no	—
171	C₄⋊C₄	#10625334	1	3	yes	—
172	C₂²⋊C₄	—	4	3	no	—
173	C₄²	—	188	3	no	—
174	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
175	C₁₆	—	94	3	no	—
176	C₁₆	—	94	1	no	—
177	C₂³	—	4	3	no	—
178	Q₈	—	94	2	no	—
179	Q₈	—	188	1	no	—
180	Q₈	—	752	3	no	—
181	Q₈	#13093968	1	4	yes	complemented
182	D₄	—	4	3	no	—
183	D₄	—	4	3	no	—
184	D₄	—	8	6	no	—
185	D₄	—	16	3	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	8	6	no	—
188	C₂×C₄	—	188	3	no	—
189	C₂×C₄	—	188	1	no	—
190	C₂×C₄	C₂×#10625334	1	3	yes	—
191	C₂×C₄	C₂×#10625334	1	3	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
193	C₈	—	2	3	no	—
194	C₈	#41×D₄₇	1	1	yes	characteristic
195	C₈	#13094071	1	1	yes	characteristic
196	C₂²	—	8	3	no	—
197	C₂²	—	8	3	no	—
198	C₂²	—	8	3	no	—
199	C₂²	C₂²×#10625334	1	1	yes	characteristic
200	C₄	—	2	3	no	—
201	C₄	—	8	1	no	—
202	C₄	—	376	3	no	—
203	C₄	—	188	2	no	—
204	C₄	C₂×#13093968	1	6	yes	—
205	C₄	#15689780	1	1	yes	characteristic
206	C₄	#15689934	1	1	yes	characteristic
207	C₂	—	16	3	no	—
208	C₂	#20256019	1	1	yes	characteristic
209	C₂	C₂²×#13093968	1	1	yes	characteristic
210	C₂	#20256527	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₈₈

C₂

yes

maximal, characteristic, complemented

#20245611

C₂

yes

maximal, complemented

#20245535

C₂

yes

maximal, complemented

#20245425

C₂

yes

maximal, complemented

Q₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄×#13093968

C₂

yes

maximal, complemented

C₄₇×#3391

C₂

yes

maximal, characteristic

Q₈×Dic₉₄

—

C₂×#13093968

C₂²

yes

—

#15689528

C₂²

yes

—

#15689419

—

C₄×Dic₁₈₈

C₂²

yes

—

#15688950

C₂²

yes

—

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

—

#15688926

C₂²

yes

characteristic

#15688924

C₂²

yes

—

#15688924

C₂²

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#550

—

C₄₇×#493

C₂²

yes

—

C₄₇×#459

—

C₈×Q₈×C₄₇

C₂²

yes

characteristic

C₄×D₈×C₄₇

C₂²

yes

—

C₂×Dic₁₈₈

C₂³

yes

characteristic

#13094024

—

#13094023

—

Q₈×Dic₄₇

—

C₄×Dic₉₄

—

#13093968

—

C₂×C₄₇⋊C₁₆

C₂³

yes

—

C₂×C₄₇⋊C₁₆

C₂³

yes

characteristic

#13093958

—

#13093958

C₂³

yes

—

#13093949

—

C₂×#41×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

C₂³

yes

—

#127×C₄₇

—

#121×C₄₇

—

C₄×Q₈×C₄₇

D₄

yes

characteristic

C₄×D₄×C₄₇

—

#107×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

D₄

yes

—

#102×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

—

#41×C₄₇

—

C₂×Q₈×C₄₇

D₈

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₈

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

#41

yes

complemented

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

Q₈×C₄₇

SD₃₂

yes

complemented

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂²×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×#41

yes

characteristic

C₈×C₄₇

2₋¹⁺⁴

yes

characteristic

#33929

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₃₂

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

100

Dic₄₇

—

101

Dic₄₇

—

102

#3391

D₄₇

yes

characteristic

103

Q₈×Q₁₆

—

104

#2240

—

105

#2226

—

106

#2207

—

107

Q₈×C₁₆

—

108

C₄×SD₃₂

—

109

C₂×C₄₇

—

110

C₂×C₄₇

#3358

yes

characteristic

111

C₂×C₄₇

C₂²×SD₃₂

yes

characteristic

112

C₂×C₄₇

#3415

yes

characteristic

113

Q₈²

—

114

#550

—

115

C₂×SD₃₂

—

116

#500

—

117

#493

C₂×D₄₇

yes

—

118

#477

—

119

#459

—

120

C₈×Q₈

C₂×D₄₇

yes

characteristic

121

C₄×Q₁₆

—

122

C₄×D₈

C₂×D₄₇

yes

—

123

#367

—

124

C₄⋊C₁₆

—

125

#358

—

126

#357

—

127

#357

—

128

C₄×C₁₆

—

129

C₄₇

#33929

yes

characteristic, complemented, Sylow

130

C₂×#41

—

131

C₂×Q₁₆

—

132

C₂×D₈

—

133

C₂×D₈

C₂²×D₄₇

yes

—

134

C₄⋊Q₈

—

135

C₄⋊Q₈

—

188

—

136

#127

—

137

#121

—

138

C₄×Q₈

—

139

C₄×Q₈

—

188

—

140

C₄×Q₈

C₂²⋊D₄₇

yes

characteristic

141

C₄×D₄

—

142

SD₃₂

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

#107

—

146

#107

—

147

#107

C₂²×D₄₇

yes

characteristic

148

C₄⋊C₈

C₂²⋊D₄₇

yes

—

149

#103

—

150

#102

—

151

C₄×C₈

C₂²×D₄₇

yes

—

152

#41

—

153

C₂×Q₈

—

154

C₂×Q₈

—

188

—

155

C₂×Q₈

#10625334

yes

characteristic

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

Q₁₆

—

161

Q₁₆

—

162

D₈

—

163

D₈

—

164

D₈

—

165

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

166

C₂×C₈

C₂³×D₄₇

yes

characteristic

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

188

—

170

C₄⋊C₄

—

188

—

171

C₄⋊C₄

#10625334

yes

—

172

C₂²⋊C₄

—

173

C₄²

—

188

—

174

C₄²

C₂×C₂²⋊D₄₇

yes

—

175

C₁₆

—

176

C₁₆

—

177

C₂³

—

178

Q₈

—

179

Q₈

—

188

—

180

Q₈

—

752

—

181

Q₈

#13093968

yes

complemented

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

188

—

189

C₂×C₄

—

188

—

190

C₂×C₄

C₂×#10625334

yes

—

191

C₂×C₄

C₂×#10625334

yes

—

192

C₂×C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

193

C₈

—

194

C₈

#41×D₄₇

yes

characteristic

195

C₈

#13094071

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

C₂²×#10625334

yes

characteristic

200

C₄

—

201

C₄

—

202

C₄

—

376

—

203

C₄

—

188

—

204

C₄

C₂×#13093968

yes

—

205

C₄

#15689780

yes

characteristic

206

C₄

#15689934

yes

characteristic

207

C₂

—

208

C₂

#20256019

yes

characteristic

209

C₂

C₂²×#13093968

yes

characteristic

210

C₂

#20256527

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting, prime radical	C₄₇×#3391	C₂
socle	C₂²×C₄₇	C₂²×D₈
hypercenter	#3391	D₄₇
hypocenter	C₄₇	#33929
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

subgroup

G/H

center

C₂²

C₂²×#10625334

commutator, cosocle

C₂×C₈×C₄₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₇

Fitting, prime radical

C₄₇×#3391

C₂

socle

C₂²×C₄₇

C₂²×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3391

N	H ≅ G/N	type
1. Q₈×Dic₁₈₈	C₂	semidirect
2. #20245611	C₂	semidirect
3. #20245535	C₂	semidirect
4. #20245425	C₂	semidirect
5. Q₈×C₄₇⋊C₁₆	C₂	semidirect
6. C₄×#13093968	C₂	semidirect
68. Dic₁₈₈	#41	semidirect
72. Q₈×C₄₇	SD₃₂	semidirect
129. C₄₇	#33929	semidirect
181. Q₈	#13093968	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₈₈

C₂

semidirect

2. #20245611

C₂

semidirect

3. #20245535

C₂

semidirect

4. #20245425

C₂

semidirect

5. Q₈×C₄₇⋊C₁₆

C₂

semidirect

6. C₄×#13093968

C₂

semidirect

68. Dic₁₈₈

#41

semidirect

72. Q₈×C₄₇

SD₃₂

semidirect

129. C₄₇

#33929

semidirect

181. Q₈

#13093968

semidirect

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

subgroup	index
Q₈×Dic₁₈₈	2
#20245611	2
#20245535	2
#20245425	2
Q₈×C₄₇⋊C₁₆	2
C₄×#13093968	2
C₄₇×#3391	2
#33929	47

subgroup

index

Q₈×Dic₁₈₈

Q₈×C₄₇⋊C₁₆

C₄₇×#3391

Subgroups — #26373955

Conjugacy classes of subgroups

Complements

Maximal subgroups