Group #26121399 — subgroups

Subgroups — #26121399

Order 12032.

410 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20243516	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×#13093949	—	2	2	no	—
9	#15688993	C₂²	1	1	yes	characteristic
10	#15688993	C₂²	1	2	yes	—
11	#15688886	—	2	4	no	—
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	#13093949	—	4	4	no	—
27	C₂²×C₈×C₄₇	D₄	1	4	yes	—
28	C₂²×C₈×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
35	C₂²×Dic₄₇	—	4	1	no	—
36	C₂²×Dic₄₇	—	4	1	no	—
37	C₂²×Dic₄₇	—	4	2	no	—
38	C₂×Dic₉₄	—	2	2	no	—
39	C₂×Dic₉₄	—	4	2	no	—
40	C₂×Dic₉₄	—	4	2	no	—
41	C₂×Dic₉₄	—	4	1	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	4	no	—
45	C₂×C₈×C₄₇	—	2	4	no	—
46	C₂×C₈×C₄₇	—	2	4	no	—
47	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	2	yes	—
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	C₄⋊Dic₄₇	—	4	4	no	—
55	C₄⋊Dic₄₇	—	4	4	no	—
56	C₄⋊Dic₄₇	—	4	4	no	—
57	C₄⋊Dic₄₇	—	4	4	no	—
58	C₂³×C₄₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#123	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₄₇	#135	1	4	yes	—
65	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
66	C₂×C₄×C₄₇	#137	1	4	yes	—
67	C₂×Dic₄₇	—	4	2	no	—
68	C₂×Dic₄₇	—	8	1	no	—
69	C₂×Dic₄₇	—	8	1	no	—
70	C₂×Dic₄₇	—	8	1	no	—
71	C₂×Dic₄₇	—	8	2	no	—
72	C₂×Dic₄₇	—	8	2	no	—
73	C₂×Dic₄₇	—	8	2	no	—
74	C₂×Dic₄₇	—	8	2	no	—
75	Dic₉₄	—	4	4	no	—
76	Dic₉₄	—	8	2	no	—
77	C₈×C₄₇	—	2	4	no	—
78	C₈×C₄₇	—	4	4	no	—
79	#10686	—	47	1	no	maximal, Sylow
80	C₂²×C₄₇	#399	1	1	yes	characteristic
81	C₂²×C₄₇	#492	1	1	yes	characteristic
82	C₂²×C₄₇	#495	1	2	yes	—
83	C₂²×C₄₇	#496	1	1	yes	characteristic
84	C₂²×C₄₇	#497	1	2	yes	—
85	C₄×C₄₇	#476	1	2	yes	—
86	C₄×C₄₇	#479	1	4	yes	—
87	C₄×C₄₇	#481	1	2	yes	—
88	C₄×C₄₇	#485	1	4	yes	—
89	Dic₄₇	—	8	2	no	—
90	Dic₄₇	—	16	1	no	—
91	Dic₄₇	—	16	2	no	—
92	#1990	—	47	2	no	—
93	#1956	—	47	1	no	—
94	#1844	—	47	1	no	—
95	#1834	—	47	2	no	—
96	#1273	D₄₇	1	1	yes	characteristic
97	C₂×C₄₇	#1708	1	1	yes	characteristic
98	C₂×C₄₇	#1716	1	1	yes	characteristic
99	C₂×C₄₇	#1719	1	2	yes	—
100	C₂×C₄₇	#2094	1	1	yes	characteristic
101	C₂×C₄₇	#2097	1	2	yes	—
102	C₂×C₄⋊Q₈	—	47	1	no	—
103	C₂×#106	—	94	2	no	—
104	C₂×#103	—	94	2	no	—
105	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
106	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	#384	—	47	1	no	—
108	#384	—	47	2	no	—
109	#340	—	94	4	no	—
110	#340	—	94	4	no	—
111	C₄₇	#10686	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	94	1	no	—
113	C₂²×C₈	D₁₈₈	1	4	yes	—
114	C₂²×C₈	D₁₈₈	1	2	yes	—
115	C₄⋊Q₈	—	47	2	no	—
116	C₄⋊Q₈	—	94	2	no	—
117	C₄⋊Q₈	—	94	1	no	—
118	C₂×C₄⋊C₄	—	94	2	no	—
119	C₂×C₄⋊C₄	—	94	2	no	—
120	C₂×C₄⋊C₄	—	94	2	no	—
121	C₂×C₄⋊C₄	—	94	2	no	—
122	C₂×C₄⋊C₄	—	94	1	no	—
123	C₂×C₄⋊C₄	—	94	2	no	—
124	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	#106	—	188	4	no	—
126	#103	—	188	4	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	2	no	—
130	C₂×Q₈	—	94	2	no	—
131	C₂×Q₈	—	188	2	no	—
132	C₂×Q₈	—	188	2	no	—
133	C₂×Q₈	—	188	1	no	—
134	C₂²×C₄	—	188	1	no	—
135	C₂²×C₄	—	188	2	no	—
136	C₂²×C₄	—	188	1	no	—
137	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	#10625325	1	4	yes	—
145	C₄⋊C₄	—	94	4	no	—
146	C₄⋊C₄	—	188	2	no	—
147	C₄⋊C₄	—	188	4	no	—
148	C₄⋊C₄	—	188	4	no	—
149	C₄⋊C₄	—	188	4	no	—
150	C₄⋊C₄	—	188	4	no	—
151	C₄⋊C₄	—	188	2	no	—
152	C₄²	#10625352	1	1	yes	characteristic
153	C₄²	#10625352	1	2	yes	—
154	C₄²	Q₈×D₄₇	1	1	yes	characteristic
155	C₂³	#13094021	1	1	yes	characteristic
156	Q₈	—	376	2	no	—
157	Q₈	—	188	4	no	—
158	C₂×C₄	—	376	1	no	—
159	C₂×C₄	—	376	1	no	—
160	C₂×C₄	—	376	1	no	—
161	C₂×C₄	—	376	2	no	—
162	C₂×C₄	—	376	2	no	—
163	C₂×C₄	—	376	2	no	—
164	C₂×C₄	—	376	2	no	—
165	C₂×C₄	—	188	2	no	—
166	C₂×C₄	#13094004	1	1	yes	characteristic
167	C₂×C₄	#13094004	1	2	yes	—
168	C₂×C₄	#13094008	1	1	yes	characteristic
169	C₂×C₄	#13094008	1	2	yes	—
170	C₂×C₄	C₂×#10625325	1	2	yes	—
171	C₂×C₄	#13094046	1	4	yes	—
172	C₂×C₄	#13094052	1	2	yes	—
173	C₂×C₄	#13094057	1	4	yes	—
174	C₈	—	2	4	no	—
175	C₈	—	4	4	no	—
176	C₂²	#15689060	1	1	yes	characteristic
177	C₂²	#15689496	1	1	yes	characteristic
178	C₂²	#15689507	1	2	yes	—
179	C₂²	#15689512	1	1	yes	characteristic
180	C₂²	#15689517	1	2	yes	—
181	C₄	—	376	2	no	—
182	C₄	—	752	1	no	—
183	C₄	—	752	2	no	—
184	C₄	#15689414	1	2	yes	—
185	C₄	#15689431	1	4	yes	—
186	C₄	#15689442	1	2	yes	—
187	C₄	#15689465	1	4	yes	—
188	C₂	#20242989	1	1	yes	characteristic
189	C₂	#20243021	1	1	yes	characteristic
190	C₂	#20243041	1	2	yes	—
191	C₂	#20244920	1	1	yes	characteristic
192	C₂	#20244935	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

#20243516

C₂

yes

maximal

C₄₇×#1273

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

—

C₂²

yes

characteristic

#15688993

C₂²

yes

—

#15688886

—

#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

—

#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

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×Q₈

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

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

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#135

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

#476

yes

—

C₄×C₄₇

#479

yes

—

C₄×C₄₇

#481

yes

—

C₄×C₄₇

#485

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

100

C₂×C₄₇

#2094

yes

characteristic

101

C₂×C₄₇

#2097

yes

—

102

C₂×C₄⋊Q₈

—

103

C₂×#106

—

104

C₂×#103

—

105

C₂×C₄×C₈

C₂×D₄₇

yes

—

106

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

107

#384

—

108

#384

—

109

#340

—

110

#340

—

111

C₄₇

#10686

yes

characteristic, complemented, Sylow

112

C₂²×Q₈

—

113

C₂²×C₈

D₁₈₈

yes

—

114

C₂²×C₈

D₁₈₈

yes

—

115

C₄⋊Q₈

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄²

C₂²×D₄₇

yes

characteristic

125

#106

—

188

—

126

#103

—

188

—

127

C₄×C₈

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₂×Q₈

—

131

C₂×Q₈

—

188

—

132

C₂×Q₈

—

188

—

133

C₂×Q₈

—

188

—

134

C₂²×C₄

—

188

—

135

C₂²×C₄

—

188

—

136

C₂²×C₄

—

188

—

137

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

138

C₂²×C₄

C₂×D₁₈₈

yes

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

#10625325

yes

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

188

—

147

C₄⋊C₄

—

188

—

148

C₄⋊C₄

—

188

—

149

C₄⋊C₄

—

188

—

150

C₄⋊C₄

—

188

—

151

C₄⋊C₄

—

188

—

152

C₄²

#10625352

yes

characteristic

153

C₄²

#10625352

yes

—

154

C₄²

Q₈×D₄₇

yes

characteristic

155

C₂³

#13094021

yes

characteristic

156

Q₈

—

376

—

157

Q₈

—

188

—

158

C₂×C₄

—

376

—

159

C₂×C₄

—

376

—

160

C₂×C₄

—

376

—

161

C₂×C₄

—

376

—

162

C₂×C₄

—

376

—

163

C₂×C₄

—

376

—

164

C₂×C₄

—

376

—

165

C₂×C₄

—

188

—

166

C₂×C₄

#13094004

yes

characteristic

167

C₂×C₄

#13094004

yes

—

168

C₂×C₄

#13094008

yes

characteristic

169

C₂×C₄

#13094008

yes

—

170

C₂×C₄

C₂×#10625325

yes

—

171

C₂×C₄

#13094046

yes

—

172

C₂×C₄

#13094052

yes

—

173

C₂×C₄

#13094057

yes

—

174

C₈

—

175

C₈

—

176

C₂²

#15689060

yes

characteristic

177

C₂²

#15689496

yes

characteristic

178

C₂²

#15689507

yes

—

179

C₂²

#15689512

yes

characteristic

180

C₂²

#15689517

yes

—

181

C₄

—

376

—

182

C₄

—

752

—

183

C₄

—

752

—

184

C₄

#15689414

yes

—

185

C₄

#15689431

yes

—

186

C₄

#15689442

yes

—

187

C₄

#15689465

yes

—

188

C₂

#20242989

yes

characteristic

189

C₂

#20243021

yes

characteristic

190

C₂

#20243041

yes

—

191

C₂

#20244920

yes

characteristic

192

C₂

#20244935

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	#10686
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
111. C₄₇	#10686	semidirect

H ≅ G/N

type

111. C₄₇

#10686

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
#20243516	2
C₄₇×#1273	2
#10686	47

subgroup

index

C₄₇×#1273

Subgroups — #26121399

Conjugacy classes of subgroups

Complements

Maximal subgroups