Group #26116115 — subgroups

Subgroups — #26116115

Order 12032.

710 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 191 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	C₂×#15689500	C₂	1	1	yes	maximal, characteristic
2	#20244193	C₂	1	4	yes	maximal, complemented
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₃₇₆	—	2	2	no	—
6	C₂×#13094021	C₂²	1	2	yes	—
7	#15689500	—	2	2	no	—
8	#15689500	C₄	1	4	yes	complemented
9	C₂×#13093944	—	2	8	no	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
11	#15688844	—	2	2	no	—
12	C₂×#106×C₄₇	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	2	yes	—
15	C₂²×D₁₈₈	—	2	4	no	—
16	C₂²×D₁₈₈	—	2	2	no	—
17	C₂×D₃₇₆	—	2	8	no	—
18	C₂×D₃₇₆	—	4	4	no	—
19	C₂×D₃₇₆	—	4	4	no	—
20	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
21	#13094021	—	2	2	no	—
22	#13094021	—	2	4	no	—
23	#13094021	C₂×C₄	1	4	yes	—
24	#13093944	—	4	16	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	#106×C₄₇	—	4	4	no	—
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	C₂³×D₄₇	—	4	2	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
38	C₂×D₁₈₈	—	2	4	no	—
39	C₂×D₁₈₈	—	2	8	no	—
40	C₂×D₁₈₈	—	4	4	no	—
41	C₂×D₁₈₈	—	4	4	no	—
42	C₂×D₁₈₈	—	4	8	no	—
43	C₂×D₁₈₈	—	4	8	no	—
44	C₂×D₁₈₈	—	4	4	no	—
45	C₂×D₁₈₈	—	4	2	no	—
46	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
47	C₂×C₈×C₄₇	D₈	1	4	yes	—
48	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
49	C₄⋊C₄×C₄₇	—	4	4	no	—
50	C₄⋊C₄×C₄₇	—	4	8	no	—
51	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₄₇	#41	1	2	yes	—
54	C₂×C₄₇⋊C₈	—	2	4	no	—
55	C₂×C₄₇⋊C₈	—	2	8	no	—
56	D₃₇₆	—	4	16	no	—
57	C₂³×C₄₇	#127	1	1	yes	characteristic
58	C₂²×D₄₇	—	4	4	no	—
59	C₂²×D₄₇	—	8	2	no	—
60	C₂²×D₄₇	—	8	2	no	—
61	C₂²×D₄₇	—	8	4	no	—
62	C₂²×D₄₇	—	8	4	no	—
63	C₂×C₄×C₄₇	—	8	2	no	—
64	C₂×C₄×C₄₇	—	8	4	no	—
65	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₄₇	#121	1	2	yes	—
68	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
70	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
71	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	8	yes	—
72	D₁₈₈	—	4	8	no	—
73	D₁₈₈	—	4	16	no	—
74	D₁₈₈	—	8	8	no	—
75	D₁₈₈	—	8	4	no	—
76	C₈×C₄₇	#102	1	8	yes	complemented
77	C₄₇⋊C₈	—	4	4	no	—
78	#9837	—	47	1	no	maximal, Sylow
79	C₂²×C₄₇	#390	1	1	yes	characteristic
80	C₂²×C₄₇	#492	1	1	yes	characteristic
81	C₂²×C₄₇	#493	1	1	yes	characteristic
82	C₂²×C₄₇	#496	1	4	yes	—
83	C₂×D₄₇	—	8	4	no	—
84	C₂×D₄₇	—	8	8	no	—
85	C₂×D₄₇	—	16	2	no	—
86	C₂×D₄₇	—	16	2	no	—
87	C₂×D₄₇	—	16	4	no	—
88	C₄×C₄₇	—	16	2	no	—
89	C₄×C₄₇	C₂×#102	1	2	yes	—
90	C₄×C₄₇	#442	1	4	yes	—
91	C₄×C₄₇	#469	1	4	yes	—
92	C₄×C₄₇	#486	1	2	yes	—
93	C₂×#493	—	47	1	no	—
94	#1957	—	47	4	no	—
95	#1844	D₄₇	1	1	yes	characteristic, complemented
96	C₂×#322	—	47	1	no	—
97	C₂×C₄₇	#1715	1	2	yes	—
98	C₂×C₄₇	#1721	1	2	yes	—
99	C₂×C₄₇	#1957	1	1	yes	characteristic
100	C₂×C₄₇	#1972	1	2	yes	—
101	D₄₇	—	16	4	no	—
102	C₂²×D₈	—	94	2	no	—
103	C₂×#127	—	47	2	no	—
104	#493	—	47	4	no	—
105	#493	—	94	2	no	—
106	C₂×#106	—	2	2	no	—
107	C₂×#102	—	94	8	no	—
108	C₂×C₄×C₈	—	47	2	no	—
109	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
110	#384	C₂×D₄₇	1	2	yes	—
111	#322	—	94	2	no	—
112	C₄₇	#9837	1	1	yes	characteristic, complemented, Sylow
113	C₂²×D₄	—	94	4	no	—
114	C₂²×D₄	—	94	2	no	—
115	C₂×D₈	—	94	8	no	—
116	C₂×D₈	—	188	4	no	—
117	C₂×D₈	—	188	4	no	—
118	C₂²×C₈	—	47	4	no	—
119	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
120	#127	—	47	4	no	—
121	#127	—	94	2	no	—
122	#127	—	94	4	no	—
123	C₂×C₄⋊C₄	—	2	4	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	#106	—	4	4	no	—
127	#102	—	188	16	no	—
128	C₄×C₈	—	94	2	no	—
129	C₄×C₈	—	94	2	no	—
130	C₄×C₈	C₄×D₄₇	1	2	yes	—
131	C₄×C₈	D₁₈₈	1	2	yes	—
132	C₂⁴	—	188	2	no	—
133	C₂×D₄	—	94	4	no	—
134	C₂×D₄	—	94	8	no	—
135	C₂×D₄	—	188	4	no	—
136	C₂×D₄	—	188	2	no	—
137	C₂×D₄	—	188	4	no	—
138	C₂×D₄	—	188	4	no	—
139	C₂×D₄	—	188	8	no	—
140	C₂×D₄	—	188	8	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₄×D₄₇	1	2	yes	—
143	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	D₈	—	188	16	no	—
145	C₂×C₈	—	94	4	no	—
146	C₂×C₈	—	94	8	no	—
147	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
148	C₂×C₈	#10625334	1	4	yes	—
149	C₂×C₈	#10625336	1	4	yes	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	8	no	—
152	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
153	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
154	C₄²	#10625354	1	2	yes	—
155	C₂³	—	376	2	no	—
156	C₂³	—	376	2	no	—
157	C₂³	—	376	4	no	—
158	C₂³	—	376	4	no	—
159	C₂³	—	188	4	no	—
160	C₂³	#13094020	1	1	yes	characteristic
161	D₄	—	376	8	no	—
162	D₄	—	376	4	no	—
163	D₄	—	188	16	no	—
164	D₄	—	188	8	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	4	no	—
167	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
168	C₂×C₄	#13093986	1	2	yes	—
169	C₂×C₄	#13093999	1	2	yes	—
170	C₂×C₄	#13094013	1	1	yes	characteristic
171	C₂×C₄	C₂×#10625334	1	2	yes	—
172	C₂×C₄	C₂×#10625336	1	2	yes	—
173	C₂×C₄	#13094051	1	8	yes	—
174	C₈	—	188	4	no	—
175	C₈	#13093944	1	8	yes	complemented
176	C₂²	—	376	8	no	—
177	C₂²	—	376	4	no	—
178	C₂²	—	752	2	no	—
179	C₂²	—	752	2	no	—
180	C₂²	—	752	4	no	—
181	C₂²	#15689024	1	1	yes	characteristic
182	C₂²	#15689497	1	1	yes	characteristic
183	C₂²	#15689499	1	1	yes	characteristic
184	C₂²	#15689509	1	4	yes	—
185	C₄	—	16	2	no	—
186	C₄	C₂×#13093944	1	2	yes	—
187	C₄	#15689234	1	4	yes	—
188	C₄	#15689377	1	4	yes	—
189	C₄	#15689471	1	2	yes	—
190	C₂	—	752	4	no	—
191	C₂	#20243017	1	2	yes	—
192	C₂	#20243052	1	2	yes	—
193	C₂	#20244193	1	1	yes	characteristic
194	C₂	#20244262	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689500

C₂

yes

maximal, characteristic

#20244193

C₂

yes

maximal, complemented

C₂×#15688844

C₂

yes

maximal, characteristic, complemented

C₄₇×#1844

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₇₆

—

C₂×#13094021

C₂²

yes

—

#15689500

—

#15689500

C₄

yes

complemented

C₂×#13093944

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688844

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₂²×D₁₈₈

—

C₂²×D₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂²×C₄₇⋊C₈

D₄

yes

—

#13094021

—

#13094021

—

#13094021

C₂×C₄

yes

—

#13093944

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

#102

yes

complemented

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#102

yes

—

C₄×C₄₇

#442

yes

—

C₄×C₄₇

#469

yes

—

C₄×C₄₇

#486

yes

—

C₂×#493

—

#1957

—

#1844

D₄₇

yes

characteristic, complemented

C₂×#322

—

C₂×C₄₇

#1715

yes

—

C₂×C₄₇

#1721

yes

—

C₂×C₄₇

#1957

yes

characteristic

100

C₂×C₄₇

#1972

yes

—

101

D₄₇

—

102

C₂²×D₈

—

103

C₂×#127

—

104

#493

—

105

#493

—

106

C₂×#106

—

107

C₂×#102

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

110

#384

C₂×D₄₇

yes

—

111

#322

—

112

C₄₇

#9837

yes

characteristic, complemented, Sylow

113

C₂²×D₄

—

114

C₂²×D₄

—

115

C₂×D₈

—

116

C₂×D₈

—

188

—

117

C₂×D₈

—

188

—

118

C₂²×C₈

—

119

C₂²×C₈

C₂²⋊D₄₇

yes

—

120

#127

—

121

#127

—

122

#127

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₄₇

yes

characteristic

126

#106

—

127

#102

—

188

—

128

C₄×C₈

—

129

C₄×C₈

—

130

C₄×C₈

C₄×D₄₇

yes

—

131

C₄×C₈

D₁₈₈

yes

—

132

C₂⁴

—

188

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

188

—

136

C₂×D₄

—

188

—

137

C₂×D₄

—

188

—

138

C₂×D₄

—

188

—

139

C₂×D₄

—

188

—

140

C₂×D₄

—

188

—

141

C₂²×C₄

—

142

C₂²×C₄

D₄×D₄₇

yes

—

143

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

144

D₈

—

188

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

148

C₂×C₈

#10625334

yes

—

149

C₂×C₈

#10625336

yes

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄²

C₂×C₄×D₄₇

yes

characteristic

153

C₄²

C₂×D₁₈₈

yes

characteristic

154

C₄²

#10625354

yes

—

155

C₂³

—

376

—

156

C₂³

—

376

—

157

C₂³

—

376

—

158

C₂³

—

376

—

159

C₂³

—

188

—

160

C₂³

#13094020

yes

characteristic

161

D₄

—

376

—

162

D₄

—

376

—

163

D₄

—

188

—

164

D₄

—

188

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

168

C₂×C₄

#13093986

yes

—

169

C₂×C₄

#13093999

yes

—

170

C₂×C₄

#13094013

yes

characteristic

171

C₂×C₄

C₂×#10625334

yes

—

172

C₂×C₄

C₂×#10625336

yes

—

173

C₂×C₄

#13094051

yes

—

174

C₈

—

188

—

175

C₈

#13093944

yes

complemented

176

C₂²

—

376

—

177

C₂²

—

376

—

178

C₂²

—

752

—

179

C₂²

—

752

—

180

C₂²

—

752

—

181

C₂²

#15689024

yes

characteristic

182

C₂²

#15689497

yes

characteristic

183

C₂²

#15689499

yes

characteristic

184

C₂²

#15689509

yes

—

185

C₄

—

186

C₄

C₂×#13093944

yes

—

187

C₄

#15689234

yes

—

188

C₄

#15689377

yes

—

189

C₄

#15689471

yes

—

190

C₂

—

752

—

191

C₂

#20243017

yes

—

192

C₂

#20243052

yes

—

193

C₂

#20244193

yes

characteristic

194

C₂

#20244262

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#15689500	C₂
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#9837
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1844

C₂

C₂³×C₄₇

C₂

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
2. #20244193	C₂	semidirect
3. C₂×#15688844	C₂	semidirect
4. C₄₇×#1844	C₂	semidirect
8. #15689500	C₄	semidirect
76. C₈×C₄₇	#102	semidirect
95. #1844	D₄₇	semidirect
112. C₄₇	#9837	semidirect
175. C₈	#13093944	semidirect

H ≅ G/N

type

2. #20244193

C₂

semidirect

3. C₂×#15688844

C₂

semidirect

4. C₄₇×#1844

C₂

semidirect

8. #15689500

C₄

semidirect

76. C₈×C₄₇

#102

semidirect

95. #1844

D₄₇

semidirect

112. C₄₇

#9837

semidirect

175. C₈

#13093944

semidirect

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

subgroup	index
C₂×#15689500	2
#20244193	2
C₂×#15688844	2
C₄₇×#1844	2
#9837	47

subgroup

index

C₄₇×#1844

Subgroups — #26116115

Conjugacy classes of subgroups

Complements

Maximal subgroups