Group #26373962 — subgroups

Subgroups — #26373962

Order 12032.

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

normal subgroups only

This group has 472 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
2	#20245615	C₂	1	3	yes	maximal
3	#20245540	C₂	1	3	yes	maximal
4	#20245435	C₂	1	3	yes	maximal
5	Q₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₄×#13093970	C₂	1	3	yes	maximal
7	C₄₇×#3395	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₉₄	—	2	1	no	—
9	C₂×#13093970	C₂²	1	3	yes	—
10	#15689528	C₂²	1	3	yes	—
11	#15689419	—	2	3	no	—
12	C₄×Dic₁₈₈	C₂²	1	3	yes	—
13	#15688948	C₂²	1	3	yes	—
14	C₄⋊(C₄₇⋊C₁₆)	C₂²	1	3	yes	—
15	#15688927	C₂²	1	6	yes	—
16	#15688927	C₂²	1	3	yes	—
17	#15688926	C₂²	1	1	yes	characteristic
18	C₄×C₄₇⋊C₁₆	C₂²	1	3	yes	—
19	C₄₇×#557	—	2	1	no	—
20	C₄₇×#494	C₂²	1	3	yes	—
21	C₄₇×#462	—	2	3	no	—
22	C₈×Q₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₄×Q₁₆×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	#13093970	—	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×Q₁₆×C₄₇	—	2	3	no	—
36	C₂×Q₁₆×C₄₇	C₂³	1	3	yes	—
37	C₄⋊Q₈×C₄₇	—	2	3	no	—
38	#125×C₄₇	—	4	3	no	—
39	C₄×Q₈×C₄₇	—	2	3	no	—
40	C₄×Q₈×C₄₇	—	4	1	no	—
41	C₄×Q₈×C₄₇	D₄	1	1	yes	characteristic
42	#107×C₄₇	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₄₇	D₄	1	3	yes	—
44	#103×C₄₇	—	2	3	no	—
45	C₄×C₈×C₄₇	C₂³	1	3	yes	—
46	C₂×Q₈×C₄₇	—	2	3	no	—
47	C₂×Q₈×C₄₇	D₈	1	1	yes	characteristic
48	C₂×Dic₉₄	—	2	1	no	—
49	C₂×Dic₉₄	—	4	3	no	—
50	Q₁₆×C₄₇	—	2	3	no	—
51	Q₁₆×C₄₇	—	2	3	no	—
52	Q₁₆×C₄₇	—	8	3	no	—
53	C₂×C₈×C₄₇	C₂×D₄	1	3	yes	—
54	C₂×C₈×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₄×C₄₇	—	2	1	no	—
56	C₄⋊C₄×C₄₇	—	4	3	no	—
57	C₄⋊C₄×C₄₇	—	4	3	no	—
58	C₄⋊C₄×C₄₇	—	4	3	no	—
59	C₄⋊C₄×C₄₇	D₈	1	3	yes	—
60	C₄²×C₄₇	—	4	3	no	—
61	C₄²×C₄₇	C₂×D₄	1	3	yes	—
62	C₄⋊Dic₄₇	—	2	3	no	—
63	C₄⋊Dic₄₇	—	4	3	no	—
64	Dic₄₇⋊C₄	—	4	3	no	—
65	C₄×Dic₄₇	—	4	3	no	—
66	Dic₁₈₈	—	2	1	no	—
67	Dic₁₈₈	#41	1	2	yes	—
68	C₄₇⋊C₁₆	—	2	3	no	—
69	C₄₇⋊C₁₆	—	2	1	no	—
70	Q₈×C₄₇	—	4	3	no	—
71	Q₈×C₄₇	—	4	3	no	—
72	Q₈×C₄₇	Q₃₂	1	4	yes	complemented
73	C₂×C₄×C₄₇	—	4	1	no	—
74	C₂×C₄×C₄₇	—	4	3	no	—
75	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
76	C₂×C₄×C₄₇	C₂×D₈	1	3	yes	—
77	C₂×C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×Dic₄₇	—	4	1	no	—
79	C₂×Dic₄₇	—	4	3	no	—
80	Dic₉₄	—	2	2	no	—
81	Dic₉₄	—	4	1	no	—
82	Dic₉₄	—	16	3	no	—
83	C₈×C₄₇	—	2	3	no	—
84	C₈×C₄₇	C₂×#41	1	1	yes	characteristic
85	C₈×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
86	#33930	—	47	1	no	maximal, Sylow
87	C₂²×C₄₇	C₂²×D₈	1	1	yes	characteristic
88	C₄×C₄₇	—	2	3	no	—
89	C₄×C₄₇	—	8	3	no	—
90	C₄×C₄₇	—	8	1	no	—
91	C₄×C₄₇	C₂×Q₃₂	1	6	yes	—
92	C₄×C₄₇	#548	1	1	yes	characteristic
93	C₄×C₄₇	#577	1	1	yes	characteristic
94	Dic₄₇	—	4	2	no	—
95	Dic₄₇	—	8	3	no	—
96	#3395	D₄₇	1	1	yes	characteristic
97	Q₈×Q₁₆	—	47	1	no	—
98	#2241	—	47	3	no	—
99	#2227	—	47	3	no	—
100	#2209	—	47	3	no	—
101	Q₈×C₁₆	—	47	1	no	—
102	C₄×Q₃₂	—	47	3	no	—
103	C₂×C₄₇	#3358	1	1	yes	characteristic
104	C₂×C₄₇	C₂²×Q₃₂	1	1	yes	characteristic
105	C₂×C₄₇	#3416	1	1	yes	characteristic
106	Q₈²	—	94	1	no	—
107	#557	—	2	1	no	—
108	C₂×Q₃₂	—	47	3	no	—
109	#500	—	47	3	no	—
110	#494	C₂×D₄₇	1	3	yes	—
111	#477	—	94	3	no	—
112	#462	—	2	3	no	—
113	C₈×Q₈	C₂×D₄₇	1	1	yes	characteristic
114	C₄×Q₁₆	—	47	3	no	—
115	C₄×Q₁₆	C₂×D₄₇	1	3	yes	—
116	#366	—	47	3	no	—
117	C₄⋊C₁₆	—	47	3	no	—
118	#358	—	47	3	no	—
119	#358	—	47	6	no	—
120	#358	—	47	1	no	—
121	C₄×C₁₆	—	47	3	no	—
122	C₄₇	#33930	1	1	yes	characteristic, complemented, Sylow
123	C₂×Q₁₆	—	2	3	no	—
124	C₂×Q₁₆	—	47	1	no	—
125	C₂×Q₁₆	C₂²×D₄₇	1	3	yes	—
126	C₄⋊Q₈	—	2	3	no	—
127	C₄⋊Q₈	—	94	3	no	—
128	C₄⋊Q₈	—	188	3	no	—
129	#125	—	4	3	no	—
130	C₄×Q₈	—	2	3	no	—
131	C₄×Q₈	—	4	1	no	—
132	C₄×Q₈	—	94	3	no	—
133	C₄×Q₈	—	188	1	no	—
134	C₄×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
135	Q₃₂	—	94	6	no	—
136	C₂×C₁₆	—	47	1	no	—
137	C₂×C₁₆	—	47	3	no	—
138	#107	—	47	3	no	—
139	#107	—	94	3	no	—
140	#107	C₂²×D₄₇	1	1	yes	characteristic
141	C₄⋊C₈	C₂²⋊D₄₇	1	3	yes	—
142	#103	—	2	3	no	—
143	#103	—	94	3	no	—
144	C₄×C₈	C₂²×D₄₇	1	3	yes	—
145	C₂×Q₈	—	2	3	no	—
146	C₂×Q₈	—	94	1	no	—
147	C₂×Q₈	—	188	3	no	—
148	C₂×Q₈	#10625334	1	1	yes	characteristic
149	Q₁₆	—	2	3	no	—
150	Q₁₆	—	2	3	no	—
151	Q₁₆	—	8	3	no	—
152	Q₁₆	—	47	2	no	—
153	Q₁₆	—	94	1	no	—
154	C₂×C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
155	C₂×C₈	C₂³×D₄₇	1	1	yes	characteristic
156	C₄⋊C₄	—	2	1	no	—
157	C₄⋊C₄	—	4	3	no	—
158	C₄⋊C₄	—	4	3	no	—
159	C₄⋊C₄	—	4	3	no	—
160	C₄⋊C₄	—	94	3	no	—
161	C₄⋊C₄	—	188	3	no	—
162	C₄⋊C₄	—	188	3	no	—
163	C₄⋊C₄	#10625334	1	3	yes	—
164	C₄²	—	4	3	no	—
165	C₄²	—	188	3	no	—
166	C₄²	C₂×C₂²⋊D₄₇	1	3	yes	—
167	C₁₆	—	94	3	no	—
168	C₁₆	—	94	1	no	—
169	Q₈	—	4	3	no	—
170	Q₈	—	4	3	no	—
171	Q₈	—	94	2	no	—
172	Q₈	—	188	1	no	—
173	Q₈	—	752	3	no	—
174	Q₈	#13093970	1	4	yes	complemented
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	3	no	—
177	C₂×C₄	—	188	3	no	—
178	C₂×C₄	—	188	1	no	—
179	C₂×C₄	C₂×#10625334	1	3	yes	—
180	C₂×C₄	C₂×#10625334	1	3	yes	—
181	C₂×C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
182	C₈	—	2	3	no	—
183	C₈	#41×D₄₇	1	1	yes	characteristic
184	C₈	#13094071	1	1	yes	characteristic
185	C₂²	C₂²×#10625334	1	1	yes	characteristic
186	C₄	—	2	3	no	—
187	C₄	—	8	3	no	—
188	C₄	—	8	1	no	—
189	C₄	—	376	3	no	—
190	C₄	—	188	2	no	—
191	C₄	C₂×#13093970	1	6	yes	—
192	C₄	#15689780	1	1	yes	characteristic
193	C₄	#15689934	1	1	yes	characteristic
194	C₂	#20256019	1	1	yes	characteristic
195	C₂	C₂²×#13093970	1	1	yes	characteristic
196	C₂	#20256532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20245615

C₂

yes

maximal

#20245540

C₂

yes

maximal

#20245435

C₂

yes

maximal

Q₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

C₄×#13093970

C₂

yes

maximal

C₄₇×#3395

C₂

yes

maximal, characteristic

Q₈×Dic₉₄

—

C₂×#13093970

C₂²

yes

—

#15689528

C₂²

yes

—

#15689419

—

C₄×Dic₁₈₈

C₂²

yes

—

#15688948

C₂²

yes

—

C₄⋊(C₄₇⋊C₁₆)

C₂²

yes

—

#15688927

C₂²

yes

—

#15688927

C₂²

yes

—

#15688926

C₂²

yes

characteristic

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#557

—

C₄₇×#494

C₂²

yes

—

C₄₇×#462

—

C₈×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₁₆×C₄₇

C₂²

yes

—

C₂×Dic₁₈₈

C₂³

yes

characteristic

#13094024

—

#13094023

—

Q₈×Dic₄₇

—

C₄×Dic₉₄

—

#13093970

—

C₂×C₄₇⋊C₁₆

C₂³

yes

—

C₂×C₄₇⋊C₁₆

C₂³

yes

characteristic

#13093958

—

#13093958

C₂³

yes

—

#13093949

—

C₂×Q₁₆×C₄₇

—

C₂×Q₁₆×C₄₇

C₂³

yes

—

C₄⋊Q₈×C₄₇

—

#125×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

D₄

yes

characteristic

#107×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

D₄

yes

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₈

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

#41

yes

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₃₂

yes

complemented

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

#33930

—

maximal, Sylow

C₂²×C₄₇

C₂²×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₃₂

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#3395

D₄₇

yes

characteristic

Q₈×Q₁₆

—

#2241

—

#2227

—

100

#2209

—

101

Q₈×C₁₆

—

102

C₄×Q₃₂

—

103

C₂×C₄₇

#3358

yes

characteristic

104

C₂×C₄₇

C₂²×Q₃₂

yes

characteristic

105

C₂×C₄₇

#3416

yes

characteristic

106

Q₈²

—

107

#557

—

108

C₂×Q₃₂

—

109

#500

—

110

#494

C₂×D₄₇

yes

—

111

#477

—

112

#462

—

113

C₈×Q₈

C₂×D₄₇

yes

characteristic

114

C₄×Q₁₆

—

115

C₄×Q₁₆

C₂×D₄₇

yes

—

116

#366

—

117

C₄⋊C₁₆

—

118

#358

—

119

#358

—

120

#358

—

121

C₄×C₁₆

—

122

C₄₇

#33930

yes

characteristic, complemented, Sylow

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

C₂²×D₄₇

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

188

—

129

#125

—

130

C₄×Q₈

—

131

C₄×Q₈

—

132

C₄×Q₈

—

133

C₄×Q₈

—

188

—

134

C₄×Q₈

C₂²⋊D₄₇

yes

characteristic

135

Q₃₂

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

#107

—

139

#107

—

140

#107

C₂²×D₄₇

yes

characteristic

141

C₄⋊C₈

C₂²⋊D₄₇

yes

—

142

#103

—

143

#103

—

144

C₄×C₈

C₂²×D₄₇

yes

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

188

—

148

C₂×Q₈

#10625334

yes

characteristic

149

Q₁₆

—

150

Q₁₆

—

151

Q₁₆

—

152

Q₁₆

—

153

Q₁₆

—

154

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

155

C₂×C₈

C₂³×D₄₇

yes

characteristic

156

C₄⋊C₄

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

188

—

162

C₄⋊C₄

—

188

—

163

C₄⋊C₄

#10625334

yes

—

164

C₄²

—

165

C₄²

—

188

—

166

C₄²

C₂×C₂²⋊D₄₇

yes

—

167

C₁₆

—

168

C₁₆

—

169

Q₈

—

170

Q₈

—

171

Q₈

—

172

Q₈

—

188

—

173

Q₈

—

752

—

174

Q₈

#13093970

yes

complemented

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

188

—

178

C₂×C₄

—

188

—

179

C₂×C₄

C₂×#10625334

yes

—

180

C₂×C₄

C₂×#10625334

yes

—

181

C₂×C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

182

C₈

—

183

C₈

#41×D₄₇

yes

characteristic

184

C₈

#13094071

yes

characteristic

185

C₂²

C₂²×#10625334

yes

characteristic

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

376

—

190

C₄

—

188

—

191

C₄

C₂×#13093970

yes

—

192

C₄

#15689780

yes

characteristic

193

C₄

#15689934

yes

characteristic

194

C₂

#20256019

yes

characteristic

195

C₂

C₂²×#13093970

yes

characteristic

196

C₂

#20256532

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#10625334
commutator, cosocle	C₂×C₈×C₄₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₇
Fitting	C₄₇×#3395	C₂
socle, prime radical	C₂²×C₄₇	C₂²×D₈
hypercenter	#3395	D₄₇
hypocenter	C₄₇	#33930
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

subgroup

G/H

center

C₂²

C₂²×#10625334

commutator, cosocle

C₂×C₈×C₄₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₄₇

Fitting

C₄₇×#3395

C₂

socle, prime radical

C₂²×C₄₇

C₂²×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

N	H ≅ G/N	type
72. Q₈×C₄₇	Q₃₂	semidirect
122. C₄₇	#33930	semidirect
174. Q₈	#13093970	semidirect

H ≅ G/N

type

72. Q₈×C₄₇

Q₃₂

semidirect

122. C₄₇

#33930

semidirect

174. Q₈

#13093970

semidirect

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

subgroup	index
Q₈×Dic₁₈₈	2
#20245615	2
#20245540	2
#20245435	2
Q₈×C₄₇⋊C₁₆	2
C₄×#13093970	2
C₄₇×#3395	2
#33930	47

subgroup

index

Q₈×Dic₁₈₈

Q₈×C₄₇⋊C₁₆

C₄₇×#3395

Subgroups — #26373962

Conjugacy classes of subgroups

Complements

Maximal subgroups