Group #26121394 — subgroups

Subgroups — #26121394

Order 12032.

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

normal subgroups only

This group has 442 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	#20244349	C₂	1	2	yes	maximal
2	#20244190	C₂	1	1	yes	maximal, characteristic
3	#20243545	C₂	1	1	yes	maximal, characteristic
4	#20243545	C₂	1	2	yes	maximal
5	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	C₂×#13093956	—	2	2	no	—
8	C₂×#13093956	—	2	4	no	—
9	C₂×#13093949	—	2	2	no	—
10	#15688993	C₂²	1	2	yes	—
11	#15688993	C₂²	1	1	yes	characteristic
12	#15688886	—	2	4	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₄	—	2	1	no	—
16	#13094024	—	2	2	no	—
17	#13094024	—	2	1	no	—
18	#13094024	Q₈	1	2	yes	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×C₄⋊Dic₄₇	—	2	1	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×C₄⋊Dic₄₇	—	2	2	no	—
23	C₂×C₄⋊Dic₄₇	—	2	2	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	#13093956	—	4	4	no	—
26	#13093956	—	4	8	no	—
27	#13093949	—	4	4	no	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂²×C₈×C₄₇	D₄	1	4	yes	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₄×C₈×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
36	C₂²×Dic₄₇	—	4	1	no	—
37	C₂²×Dic₄₇	—	4	1	no	—
38	C₂²×Dic₄₇	—	4	2	no	—
39	C₂×Dic₉₄	—	2	2	no	—
40	C₂×Dic₉₄	—	4	2	no	—
41	C₂×Dic₉₄	—	4	2	no	—
42	C₂×Dic₉₄	—	4	1	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	4	no	—
45	C₂×C₈×C₄₇	—	2	2	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
48	C₂×C₈×C₄₇	SD₁₆	1	8	yes	—
49	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	2	yes	—
51	C₄²×C₄₇	#41	1	1	yes	characteristic
52	C₄⋊Dic₄₇	—	2	4	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	C₄⋊Dic₄₇	—	4	2	no	—
55	C₄⋊Dic₄₇	—	4	4	no	—
56	C₄⋊Dic₄₇	—	4	4	no	—
57	C₄⋊Dic₄₇	—	4	4	no	—
58	C₄⋊Dic₄₇	—	4	4	no	—
59	C₂³×C₄₇	#127	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#123	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×SD₁₆	1	4	yes	—
65	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
66	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	C₂×C₄×C₄₇	#137	1	4	yes	—
68	C₂×Dic₄₇	—	4	2	no	—
69	C₂×Dic₄₇	—	8	1	no	—
70	C₂×Dic₄₇	—	8	1	no	—
71	C₂×Dic₄₇	—	8	1	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	C₂×Dic₄₇	—	8	2	no	—
76	Dic₉₄	—	4	4	no	—
77	Dic₉₄	—	8	2	no	—
78	C₈×C₄₇	—	2	4	no	—
79	C₈×C₄₇	—	2	8	no	—
80	#10685	—	47	1	no	maximal, Sylow
81	C₂²×C₄₇	#399	1	1	yes	characteristic
82	C₂²×C₄₇	#492	1	1	yes	characteristic
83	C₂²×C₄₇	#492	1	2	yes	—
84	C₂²×C₄₇	#496	1	1	yes	characteristic
85	C₂²×C₄₇	#497	1	2	yes	—
86	C₄×C₄₇	#475	1	4	yes	—
87	C₄×C₄₇	#476	1	2	yes	—
88	C₄×C₄₇	#481	1	2	yes	—
89	C₄×C₄₇	#483	1	4	yes	—
90	Dic₄₇	—	8	2	no	—
91	Dic₄₇	—	16	1	no	—
92	Dic₄₇	—	16	2	no	—
93	#1990	—	47	2	no	—
94	#1956	—	47	1	no	—
95	#1844	—	47	2	no	—
96	#1844	—	47	1	no	—
97	#1273	D₄₇	1	1	yes	characteristic
98	C₂×C₄₇	#1705	1	1	yes	characteristic
99	C₂×C₄₇	#1714	1	2	yes	—
100	C₂×C₄₇	#1716	1	1	yes	characteristic
101	C₂×C₄₇	#2094	1	1	yes	characteristic
102	C₂×C₄₇	#2096	1	2	yes	—
103	C₂×C₄⋊Q₈	—	47	1	no	—
104	C₂×#106	—	94	2	no	—
105	C₂×#106	—	94	4	no	—
106	C₂×#103	—	94	2	no	—
107	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
109	#384	—	47	1	no	—
110	#384	—	47	2	no	—
111	#340	—	94	4	no	—
112	C₄₇	#10685	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	94	1	no	—
114	C₂²×C₈	D₁₈₈	1	4	yes	—
115	C₂²×C₈	D₁₈₈	1	2	yes	—
116	C₄⋊Q₈	—	47	2	no	—
117	C₄⋊Q₈	—	94	2	no	—
118	C₄⋊Q₈	—	94	1	no	—
119	C₂×C₄⋊C₄	—	94	2	no	—
120	C₂×C₄⋊C₄	—	94	2	no	—
121	C₂×C₄⋊C₄	—	94	1	no	—
122	C₂×C₄⋊C₄	—	94	2	no	—
123	C₂×C₄⋊C₄	—	94	2	no	—
124	C₂×C₄⋊C₄	—	94	2	no	—
125	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	#106	—	188	4	no	—
127	#106	—	188	8	no	—
128	#103	—	188	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	2	2	no	—
132	C₂×Q₈	—	94	2	no	—
133	C₂×Q₈	—	188	2	no	—
134	C₂×Q₈	—	188	2	no	—
135	C₂×Q₈	—	188	1	no	—
136	C₂²×C₄	—	188	1	no	—
137	C₂²×C₄	—	188	2	no	—
138	C₂²×C₄	—	188	1	no	—
139	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	4	no	—
145	C₂×C₈	#10625325	1	4	yes	—
146	C₂×C₈	#10625325	1	8	yes	—
147	C₄⋊C₄	—	94	4	no	—
148	C₄⋊C₄	—	188	2	no	—
149	C₄⋊C₄	—	188	4	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄⋊C₄	—	188	4	no	—
152	C₄⋊C₄	—	188	4	no	—
153	C₄⋊C₄	—	188	2	no	—
154	C₄²	#10625352	1	2	yes	—
155	C₄²	#10625352	1	1	yes	characteristic
156	C₄²	Q₈×D₄₇	1	1	yes	characteristic
157	C₂³	#13094021	1	1	yes	characteristic
158	Q₈	—	376	2	no	—
159	Q₈	—	188	4	no	—
160	C₂×C₄	—	376	1	no	—
161	C₂×C₄	—	376	1	no	—
162	C₂×C₄	—	376	2	no	—
163	C₂×C₄	—	376	1	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	376	2	no	—
166	C₂×C₄	—	376	2	no	—
167	C₂×C₄	—	188	2	no	—
168	C₂×C₄	#13094004	1	2	yes	—
169	C₂×C₄	#13094004	1	1	yes	characteristic
170	C₂×C₄	#13094008	1	1	yes	characteristic
171	C₂×C₄	#13094008	1	2	yes	—
172	C₂×C₄	C₂×#10625325	1	4	yes	—
173	C₂×C₄	C₂×#10625325	1	2	yes	—
174	C₂×C₄	#13094052	1	2	yes	—
175	C₂×C₄	#13094057	1	4	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	2	8	no	—
178	C₂²	#15689060	1	1	yes	characteristic
179	C₂²	#15689496	1	2	yes	—
180	C₂²	#15689496	1	1	yes	characteristic
181	C₂²	#15689512	1	1	yes	characteristic
182	C₂²	#15689517	1	2	yes	—
183	C₄	—	376	2	no	—
184	C₄	—	752	1	no	—
185	C₄	—	752	2	no	—
186	C₄	#15689409	1	4	yes	—
187	C₄	#15689414	1	2	yes	—
188	C₄	#15689442	1	2	yes	—
189	C₄	#15689454	1	4	yes	—
190	C₂	#20242975	1	1	yes	characteristic
191	C₂	#20243010	1	2	yes	—
192	C₂	#20243021	1	1	yes	characteristic
193	C₂	#20244920	1	1	yes	characteristic
194	C₂	#20244930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244349

C₂

yes

maximal

#20244190

C₂

yes

maximal, characteristic

#20243545

C₂

yes

maximal, characteristic

#20243545

C₂

yes

maximal

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

—

—

C₂²

yes

—

#15688993

C₂²

yes

characteristic

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

—

#13094024

—

#13094024

—

#13094024

Q₈

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

#13093956

—

#13093949

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#496

yes

characteristic

C₂²×C₄₇

#497

yes

—

C₄×C₄₇

#475

yes

—

C₄×C₄₇

#476

yes

—

C₄×C₄₇

#481

yes

—

C₄×C₄₇

#483

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#1716

yes

characteristic

101

C₂×C₄₇

#2094

yes

characteristic

102

C₂×C₄₇

#2096

yes

—

103

C₂×C₄⋊Q₈

—

104

C₂×#106

—

105

C₂×#106

—

106

C₂×#103

—

107

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

108

C₂×C₄×C₈

C₂×D₄₇

yes

—

109

#384

—

110

#384

—

111

#340

—

112

C₄₇

#10685

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

D₁₈₈

yes

—

115

C₂²×C₈

D₁₈₈

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₄₇

yes

characteristic

126

#106

—

188

—

127

#106

—

188

—

128

#103

—

188

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

188

—

134

C₂×Q₈

—

188

—

135

C₂×Q₈

—

188

—

136

C₂²×C₄

—

188

—

137

C₂²×C₄

—

188

—

138

C₂²×C₄

—

188

—

139

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

140

C₂²×C₄

C₂×D₁₈₈

yes

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

#10625325

yes

—

146

C₂×C₈

#10625325

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

188

—

149

C₄⋊C₄

—

188

—

150

C₄⋊C₄

—

188

—

151

C₄⋊C₄

—

188

—

152

C₄⋊C₄

—

188

—

153

C₄⋊C₄

—

188

—

154

C₄²

#10625352

yes

—

155

C₄²

#10625352

yes

characteristic

156

C₄²

Q₈×D₄₇

yes

characteristic

157

C₂³

#13094021

yes

characteristic

158

Q₈

—

376

—

159

Q₈

—

188

—

160

C₂×C₄

—

376

—

161

C₂×C₄

—

376

—

162

C₂×C₄

—

376

—

163

C₂×C₄

—

376

—

164

C₂×C₄

—

376

—

165

C₂×C₄

—

376

—

166

C₂×C₄

—

376

—

167

C₂×C₄

—

188

—

168

C₂×C₄

#13094004

yes

—

169

C₂×C₄

#13094004

yes

characteristic

170

C₂×C₄

#13094008

yes

characteristic

171

C₂×C₄

#13094008

yes

—

172

C₂×C₄

C₂×#10625325

yes

—

173

C₂×C₄

C₂×#10625325

yes

—

174

C₂×C₄

#13094052

yes

—

175

C₂×C₄

#13094057

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#15689060

yes

characteristic

179

C₂²

#15689496

yes

—

180

C₂²

#15689496

yes

characteristic

181

C₂²

#15689512

yes

characteristic

182

C₂²

#15689517

yes

—

183

C₄

—

376

—

184

C₄

—

752

—

185

C₄

—

752

—

186

C₄

#15689409

yes

—

187

C₄

#15689414

yes

—

188

C₄

#15689442

yes

—

189

C₄

#15689454

yes

—

190

C₂

#20242975

yes

characteristic

191

C₂

#20243010

yes

—

192

C₂

#20243021

yes

characteristic

193

C₂

#20244920

yes

characteristic

194

C₂

#20244930

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle, prime radical	C₂³×C₄₇	#127
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10685
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1273

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
112. C₄₇	#10685	semidirect

H ≅ G/N

type

112. C₄₇

#10685

semidirect

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

subgroup	index
#20244349	2
#20244190	2
#20243545	2
C₄₇×#1273	2
#10685	47

subgroup

index

C₄₇×#1273

Subgroups — #26121394

Conjugacy classes of subgroups

Complements

Maximal subgroups