Group #26128155 — subgroups

Subgroups — #26128155

Order 12032.

214 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20245257	C₂	1	1	yes	maximal, characteristic
2	#20242207	C₂	1	1	yes	maximal, characteristic
3	#20242206	C₂	1	1	yes	maximal, characteristic
4	#20241453	C₂	1	1	yes	maximal, characteristic
5	#20241359	C₂	1	1	yes	maximal, characteristic
6	#20241310	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#1334	C₂	1	1	yes	maximal, characteristic
8	C₂×#13093970	C₄	1	1	yes	characteristic
9	Q₈×C₄₇⋊C₈	—	2	1	no	—
10	#15689247	—	2	1	no	—
11	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
12	#15688950	C₄	1	1	yes	characteristic
13	#15688927	C₄	1	1	yes	characteristic
14	#15688926	C₄	1	1	yes	characteristic
15	#15688899	C₂²	1	1	yes	characteristic
16	#15688874	C₂²	1	1	yes	characteristic
17	Q₈⋊(C₄₇⋊C₈)	—	2	1	no	—
18	#15688853	—	2	1	no	—
19	#15688844	C₂²	1	1	yes	characteristic
20	C₄×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₄₇×#346	C₂²	1	1	yes	characteristic
22	C₄₇×#334	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₉₄	—	2	1	no	—
25	#13093970	C₈	1	4	yes	complemented
26	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
28	#13093958	C₂×C₄	1	1	yes	characteristic
29	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
30	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
31	Dic₄₇⋊C₈	—	4	1	no	—
32	#13093949	—	2	1	no	—
33	C₄×C₄₇⋊C₈	—	4	1	no	—
34	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
36	C₈×Dic₄₇	—	4	1	no	—
37	C₂×Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₄₇	—	2	1	no	—
39	C₂×C₁₆×C₄₇	—	2	1	no	—
40	#107×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊C₈×C₄₇	—	2	1	no	—
42	#103×C₄₇	—	2	1	no	—
43	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
44	C₂×Q₈×C₄₇	—	2	1	no	—
45	C₂×Dic₉₄	—	2	1	no	—
46	Q₁₆×C₄₇	—	2	1	no	—
47	Q₁₆×C₄₇	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
51	C₄⋊C₄×C₄₇	—	2	1	no	—
52	C₄⋊C₄×C₄₇	—	4	1	no	—
53	C₄²×C₄₇	—	4	1	no	—
54	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄⋊Dic₄₇	—	2	1	no	—
56	Dic₄₇⋊C₄	—	4	1	no	—
57	C₄×Dic₄₇	—	4	1	no	—
58	C₂×C₄₇⋊C₈	—	2	1	no	—
59	C₂×C₄₇⋊C₈	—	2	1	no	—
60	C₂×C₄₇⋊C₈	—	4	1	no	—
61	Dic₁₈₈	—	2	1	no	—
62	Dic₁₈₈	C₂×C₈	1	2	yes	—
63	C₁₆×C₄₇	—	4	1	no	—
64	C₄₇⋊C₁₆	—	2	1	no	—
65	C₄₇⋊C₁₆	C₂×C₈	1	2	yes	—
66	Q₈×C₄₇	—	2	2	no	—
67	Q₈×C₄₇	—	4	1	no	—
68	C₂×C₄×C₄₇	—	4	1	no	—
69	C₂×C₄×C₄₇	—	4	1	no	—
70	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₂×Dic₄₇	—	4	1	no	—
74	C₂×Dic₄₇	—	4	1	no	—
75	Dic₉₄	—	2	2	no	—
76	Dic₉₄	—	4	1	no	—
77	C₈×C₄₇	—	2	1	no	—
78	C₈×C₄₇	—	8	1	no	—
79	C₈×C₄₇	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₄₇	#131	1	1	yes	characteristic
81	C₄₇⋊C₈	—	4	1	no	—
82	C₄₇⋊C₈	—	4	1	no	—
83	C₄₇⋊C₈	—	8	1	no	—
84	#12066	—	47	1	no	maximal, Sylow
85	C₂²×C₄₇	#442	1	1	yes	characteristic
86	C₄×C₄₇	—	2	1	no	—
87	C₄×C₄₇	—	2	1	no	—
88	C₄×C₄₇	—	4	2	no	—
89	C₄×C₄₇	—	8	1	no	—
90	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
91	C₄×C₄₇	#444	1	1	yes	characteristic
92	Dic₄₇	—	4	2	no	—
93	Dic₄₇	—	8	1	no	—
94	#2176	—	47	1	no	—
95	#1579	—	47	1	no	—
96	#1579	—	47	1	no	—
97	#1368	—	47	1	no	—
98	#1334	—	47	1	no	—
99	#1334	D₄₇	1	1	yes	characteristic
100	#1313	—	47	1	no	—
101	C₂×C₄₇	#1580	1	1	yes	characteristic
102	C₂×C₄₇	#2180	1	1	yes	characteristic
103	C₂×C₄₇	#2180	1	1	yes	characteristic
104	C₂×Q₃₂	—	47	1	no	—
105	C₈×Q₈	—	94	1	no	—
106	C₈×Q₈	—	94	1	no	—
107	C₄×Q₁₆	—	47	1	no	—
108	C₄×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
109	#367	—	47	1	no	—
110	#358	—	47	1	no	—
111	#358	—	47	1	no	—
112	#346	—	47	1	no	—
113	#346	C₂×D₄₇	1	1	yes	characteristic
114	#334	—	47	1	no	—
115	#334	C₂×D₄₇	1	1	yes	characteristic
116	Q₈⋊C₈	—	94	1	no	—
117	Q₈⋊C₈	—	94	1	no	—
118	#322	—	47	1	no	—
119	C₄₇	#12066	1	1	yes	characteristic, complemented, Sylow
120	C₂×Q₁₆	—	47	1	no	—
121	C₂×Q₁₆	C₄×D₄₇	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	94	1	no	—
124	Q₃₂	—	47	4	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	47	1	no	—
127	C₂×C₁₆	—	47	1	no	—
128	#107	—	47	1	no	—
129	#107	C₄×D₄₇	1	1	yes	characteristic
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	94	1	no	—
132	C₄⋊C₈	—	188	1	no	—
133	C₄⋊C₈	—	188	1	no	—
134	#103	—	2	1	no	—
135	#103	—	94	1	no	—
136	C₄×C₈	—	47	1	no	—
137	C₄×C₈	—	47	1	no	—
138	C₄×C₈	—	188	1	no	—
139	C₄×C₈	—	188	1	no	—
140	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	94	1	no	—
143	Q₁₆	—	2	1	no	—
144	Q₁₆	—	47	2	no	—
145	Q₁₆	—	94	1	no	—
146	Q₁₆	C₈×D₄₇	1	2	yes	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	94	1	no	—
149	C₂×C₈	—	94	1	no	—
150	C₂×C₈	—	188	1	no	—
151	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
152	C₂×C₈	#10625352	1	1	yes	characteristic
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	94	1	no	—
156	C₄⋊C₄	—	188	1	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	188	1	no	—
159	C₄²	D₄×D₄₇	1	1	yes	characteristic
160	C₁₆	—	4	1	no	—
161	C₁₆	—	47	2	no	—
162	C₁₆	—	94	1	no	—
163	Q₈	—	2	2	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	94	2	no	—
166	Q₈	—	188	1	no	—
167	C₂×C₄	—	4	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	188	1	no	—
170	C₂×C₄	—	188	1	no	—
171	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
172	C₂×C₄	#13094048	1	1	yes	characteristic
173	C₂×C₄	#13094051	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	8	1	no	—
176	C₈	—	376	1	no	—
177	C₈	—	188	1	no	—
178	C₈	—	188	1	no	—
179	C₈	C₂×C₈×D₄₇	1	1	yes	characteristic
180	C₈	#13094032	1	1	yes	characteristic
181	C₂²	#15689232	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	2	no	—
185	C₄	—	8	1	no	—
186	C₄	—	376	1	no	—
187	C₄	—	188	2	no	—
188	C₄	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
189	C₄	#15689242	1	1	yes	characteristic
190	C₂	#20242209	1	1	yes	characteristic
191	C₂	#20245276	1	1	yes	characteristic
192	C₂	#20245282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245257

C₂

yes

maximal, characteristic

#20242207

C₂

yes

maximal, characteristic

#20242206

C₂

yes

maximal, characteristic

#20241453

C₂

yes

maximal, characteristic

#20241359

C₂

yes

maximal, characteristic

#20241310

C₂

yes

maximal, characteristic

C₄₇×#1334

C₂

yes

maximal, characteristic

C₂×#13093970

C₄

yes

characteristic

Q₈×C₄₇⋊C₈

—

#15689247

—

C₄×Dic₁₈₈

C₂²

yes

characteristic

#15688950

C₄

yes

characteristic

#15688927

C₄

yes

characteristic

#15688926

C₄

yes

characteristic

#15688899

C₂²

yes

characteristic

#15688874

C₂²

yes

characteristic

Q₈⋊(C₄₇⋊C₈)

—

#15688853

—

#15688844

C₂²

yes

characteristic

C₄×Q₁₆×C₄₇

C₂²

yes

characteristic

C₄₇×#346

C₂²

yes

characteristic

C₄₇×#334

C₂²

yes

characteristic

C₂×Dic₁₈₈

C₂×C₄

yes

characteristic

C₄×Dic₉₄

—

#13093970

C₈

yes

complemented

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

#13093958

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₈×Dic₄₇

—

C₂×Q₁₆×C₄₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₁₆×C₄₇

—

#107×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

Q₁₆×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

C₂×C₈

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₂×C₈

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×D₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

#131

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

#444

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

—

—

—

—

D₄₇

yes

characteristic

100

#1313

—

101

C₂×C₄₇

#1580

yes

characteristic

102

C₂×C₄₇

#2180

yes

characteristic

103

C₂×C₄₇

#2180

yes

characteristic

104

C₂×Q₃₂

—

105

C₈×Q₈

—

106

C₈×Q₈

—

107

C₄×Q₁₆

—

108

C₄×Q₁₆

C₂×D₄₇

yes

characteristic

109

#367

—

110

#358

—

111

#358

—

112

#346

—

113

#346

C₂×D₄₇

yes

characteristic

114

#334

—

115

#334

C₂×D₄₇

yes

characteristic

116

Q₈⋊C₈

—

117

Q₈⋊C₈

—

118

#322

—

119

C₄₇

#12066

yes

characteristic, complemented, Sylow

120

C₂×Q₁₆

—

121

C₂×Q₁₆

C₄×D₄₇

yes

characteristic

122

C₄×Q₈

—

123

C₄×Q₈

—

124

Q₃₂

—

125

C₂×C₁₆

—

126

C₂×C₁₆

—

127

C₂×C₁₆

—

128

#107

—

129

#107

C₄×D₄₇

yes

characteristic

130

C₄⋊C₈

—

131

C₄⋊C₈

—

132

C₄⋊C₈

—

188

—

133

C₄⋊C₈

—

188

—

134

#103

—

135

#103

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

188

—

139

C₄×C₈

—

188

—

140

C₄×C₈

C₂²×D₄₇

yes

characteristic

141

C₂×Q₈

—

142

C₂×Q₈

—

143

Q₁₆

—

144

Q₁₆

—

145

Q₁₆

—

146

Q₁₆

C₈×D₄₇

yes

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

188

—

151

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

152

C₂×C₈

#10625352

yes

characteristic

153

C₄⋊C₄

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

188

—

157

C₄²

—

158

C₄²

—

188

—

159

C₄²

D₄×D₄₇

yes

characteristic

160

C₁₆

—

161

C₁₆

—

162

C₁₆

—

163

Q₈

—

164

Q₈

—

165

Q₈

—

166

Q₈

—

188

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

188

—

170

C₂×C₄

—

188

—

171

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

172

C₂×C₄

#13094048

yes

characteristic

173

C₂×C₄

#13094051

yes

characteristic

174

C₈

—

175

C₈

—

176

C₈

—

376

—

177

C₈

—

188

—

178

C₈

—

188

—

179

C₈

C₂×C₈×D₄₇

yes

characteristic

180

C₈

#13094032

yes

characteristic

181

C₂²

#15689232

yes

characteristic

182

C₄

—

183

C₄

—

184

C₄

—

185

C₄

—

186

C₄

—

376

—

187

C₄

—

188

—

188

C₄

(C₂²⋊D₄₇)⋊C₈

yes

characteristic

189

C₄

#15689242

yes

characteristic

190

C₂

#20242209

yes

characteristic

191

C₂

#20245276

yes

characteristic

192

C₂

#20245282

yes

characteristic

	subgroup	G/H
center	C₂²	#15689232
commutator	C₈×C₄₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1334	C₂
socle, prime radical	C₂²×C₄₇	#442
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1334	D₄₇
hypocenter	C₄₇	#12066
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

subgroup

G/H

center

C₂²

#15689232

commutator

C₈×C₄₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1334

C₂

socle, prime radical

C₂²×C₄₇

#442

cosocle

C₄×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1334

N	H ≅ G/N	type
25. #13093970	C₈	semidirect
119. C₄₇	#12066	semidirect

H ≅ G/N

type

25. #13093970

C₈

semidirect

119. C₄₇

#12066

semidirect

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

subgroup	index
#20245257	2
#20242207	2
#20242206	2
#20241453	2
#20241359	2
#20241310	2
C₄₇×#1334	2
#12066	47

subgroup

index

C₄₇×#1334

Subgroups — #26128155

Conjugacy classes of subgroups

Complements

Maximal subgroups