Group #24999347 — subgroups

Subgroups — #24999347

Order 11008.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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	#19145741	C₂	1	2	yes	maximal
2	#19142993	C₂	1	1	yes	maximal, characteristic
3	#19135572	C₂	1	1	yes	maximal, characteristic
4	#19133266	C₂	1	2	yes	maximal
5	C₄₃×#1276	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	1	yes	characteristic
7	C₂×#13063325	—	2	2	no	—
8	C₂×#13063325	C₂²	1	2	yes	—
9	#14583297	—	2	2	no	—
10	#14583250	—	2	2	no	—
11	#14583230	—	2	4	no	—
12	#14583185	—	2	4	no	—
13	#14582923	C₂²	1	1	yes	characteristic
14	#14582769	—	2	4	no	—
15	#14582659	—	2	2	no	—
16	#14582644	—	2	4	no	—
17	C₄₃×#431	C₂²	1	2	yes	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₆	—	2	2	no	—
20	C₂×#10621877	—	2	1	no	—
21	#13063325	—	2	2	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	#13063325	—	4	4	no	—
26	#13063325	—	4	4	no	—
27	#13063325	D₄	1	2	yes	—
28	#13063317	—	2	1	no	—
29	#13063304	—	2	2	no	—
30	C₄×Dic₈₆	—	2	2	no	—
31	#13063284	—	2	1	no	—
32	C₂×C₄⋊Dic₄₃	—	2	8	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
35	#13063257	—	2	8	no	—
36	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
37	#13063250	—	2	8	no	—
38	C₂×C₄²×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₃	D₄	1	4	yes	—
42	C₂²⋊C₈×C₄₃	—	2	2	no	—
43	C₄×C₈×C₄₃	—	2	2	no	—
44	C₂²×C₄×C₄₃	—	2	2	no	—
45	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
47	C₂²×Dic₄₃	—	2	4	no	—
48	C₂×Dic₈₆	—	2	2	no	—
49	C₂×Dic₈₆	—	4	2	no	—
50	C₂×Dic₈₆	—	4	2	no	—
51	C₂×Dic₈₆	—	4	4	no	—
52	C₂×Dic₈₆	—	4	4	no	—
53	C₂×C₈×C₄₃	—	2	4	no	—
54	C₄²×C₄₃	—	2	8	no	—
55	C₄²×C₄₃	—	2	2	no	—
56	C₄²×C₄₃	SD₁₆	1	4	yes	—
57	C₄²×C₄₃	C₂×D₄	1	2	yes	—
58	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₃	#41	1	1	yes	characteristic
60	C₂²⋊Dic₄₃	—	4	1	no	—
61	C₄⋊Dic₄₃	—	2	4	no	—
62	C₄⋊Dic₄₃	—	2	2	no	—
63	C₄⋊Dic₄₃	—	2	4	no	—
64	C₄⋊Dic₄₃	—	2	2	no	—
65	C₄⋊Dic₄₃	—	4	16	no	—
66	C₄⋊Dic₄₃	—	4	2	no	—
67	C₄⋊Dic₄₃	—	4	2	no	—
68	Dic₄₃⋊C₄	—	4	2	no	—
69	C₄×Dic₄₃	—	4	1	no	—
70	#10621877	—	4	2	no	—
71	C₂×C₄₃⋊C₈	—	4	1	no	—
72	C₂³×C₄₃	#120	1	1	yes	characteristic
73	C₂×C₄×C₄₃	—	2	8	no	—
74	C₂×C₄×C₄₃	—	2	4	no	—
75	C₂×C₄×C₄₃	—	2	4	no	—
76	C₂×C₄×C₄₃	#120	1	2	yes	—
77	C₂×C₄×C₄₃	#121	1	2	yes	—
78	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
80	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₄₃	—	4	2	no	—
83	C₂×Dic₄₃	—	4	4	no	—
84	C₂×Dic₄₃	—	4	8	no	—
85	Dic₈₆	—	4	4	no	—
86	Dic₈₆	—	8	4	no	—
87	Dic₈₆	—	8	2	no	—
88	Dic₈₆	—	8	2	no	—
89	C₈×C₄₃	—	4	4	no	—
90	C₄₃⋊C₈	—	8	1	no	—
91	#10299	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	—	2	2	no	—
93	C₂²×C₄₃	#392	1	1	yes	characteristic
94	C₂²×C₄₃	#450	1	2	yes	—
95	C₄×C₄₃	—	2	8	no	—
96	C₄×C₄₃	—	2	2	no	—
97	C₄×C₄₃	#456	1	4	yes	—
98	C₄×C₄₃	#460	1	2	yes	—
99	C₄×C₄₃	#492	1	1	yes	characteristic
100	C₄×C₄₃	#496	1	1	yes	characteristic
101	Dic₄₃	—	8	2	no	—
102	Dic₄₃	—	8	4	no	—
103	#3229	—	43	2	no	—
104	#2868	—	43	1	no	—
105	#1994	—	43	1	no	—
106	#1546	—	43	2	no	—
107	#1276	D₄₃	1	1	yes	characteristic
108	C₂×C₄₃	—	2	2	no	—
109	C₂×C₄₃	#2005	1	2	yes	—
110	C₂×C₄₃	#2014	1	1	yes	characteristic
111	#533	—	43	1	no	—
112	C₂×C₄⋊Q₈	—	43	2	no	—
113	C₂×C₄⋊Q₈	—	86	2	no	—
114	#498	—	86	2	no	—
115	#487	—	86	2	no	—
116	#483	—	86	4	no	—
117	#475	—	86	4	no	—
118	#431	C₂×D₄₃	1	2	yes	—
119	#423	—	43	1	no	—
120	#384	—	86	4	no	—
121	C₄³	C₂×D₄₃	1	1	yes	characteristic
122	#339	—	86	2	no	—
123	#332	—	86	4	no	—
124	C₄₃	#10299	1	1	yes	characteristic, complemented, Sylow
125	C₂²×Q₈	—	86	2	no	—
126	C₂×M₁₆	—	86	1	no	—
127	C₄⋊Q₈	—	43	2	no	—
128	C₄⋊Q₈	—	86	2	no	—
129	C₄⋊Q₈	—	86	2	no	—
130	C₄⋊Q₈	—	86	2	no	—
131	C₄⋊Q₈	—	86	2	no	—
132	C₄⋊Q₈	—	172	4	no	—
133	C₄⋊Q₈	—	172	4	no	—
134	#125	—	86	1	no	—
135	C₂²⋊Q₈	—	86	2	no	—
136	C₄×Q₈	—	86	2	no	—
137	#117	—	86	1	no	—
138	C₂×C₄⋊C₄	—	43	4	no	—
139	C₂×C₄⋊C₄	—	86	8	no	—
140	C₂×C₄⋊C₄	—	86	2	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	D₁₇₂	1	2	yes	—
143	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
144	#106	—	86	8	no	—
145	C₄⋊C₈	—	86	2	no	—
146	C₄⋊C₈	D₁₇₂	1	4	yes	—
147	#103	—	86	8	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₂×Q₈	—	86	2	no	—
151	C₂×Q₈	—	172	2	no	—
152	C₂×Q₈	—	172	2	no	—
153	C₂×Q₈	—	172	4	no	—
154	C₂×Q₈	—	172	4	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	86	4	no	—
157	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
158	C₂²×C₄	D₄×D₄₃	1	2	yes	—
159	M₁₆	—	172	2	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	172	1	no	—
162	C₄⋊C₄	—	86	4	no	—
163	C₄⋊C₄	—	86	2	no	—
164	C₄⋊C₄	—	86	4	no	—
165	C₄⋊C₄	—	86	2	no	—
166	C₄⋊C₄	—	172	16	no	—
167	C₄⋊C₄	—	172	2	no	—
168	C₄⋊C₄	—	172	2	no	—
169	C₄⋊C₄	—	172	2	no	—
170	C₂²⋊C₄	—	172	1	no	—
171	C₄²	—	2	8	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	172	1	no	—
174	C₄²	#10621873	1	4	yes	—
175	C₄²	C₂×D₁₇₂	1	2	yes	—
176	C₄²	D₄×D₄₃	1	1	yes	characteristic
177	C₄²	#10621902	1	1	yes	characteristic
178	C₂³	#13063295	1	1	yes	characteristic
179	Q₈	—	344	4	no	—
180	Q₈	—	344	2	no	—
181	Q₈	—	344	2	no	—
182	Q₈	—	172	4	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	172	8	no	—
187	C₂×C₄	—	172	2	no	—
188	C₂×C₄	—	172	4	no	—
189	C₂×C₄	#13063295	1	2	yes	—
190	C₂×C₄	#13063300	1	2	yes	—
191	C₂×C₄	#13063300	1	1	yes	characteristic
192	C₂×C₄	#13063322	1	1	yes	characteristic
193	C₂×C₄	C₂×#10621873	1	2	yes	—
194	C₂×C₄	#13063353	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	344	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	#14582809	1	1	yes	characteristic
199	C₂²	#14583052	1	2	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	344	2	no	—
203	C₄	—	344	4	no	—
204	C₄	#14583083	1	4	yes	—
205	C₄	#14583100	1	2	yes	—
206	C₄	#14583272	1	1	yes	characteristic
207	C₄	#14583288	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	#19135634	1	2	yes	—
210	C₂	#19135685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145741

C₂

yes

maximal

#19142993

C₂

yes

maximal, characteristic

#19135572

C₂

yes

maximal, characteristic

#19133266

C₂

yes

maximal

C₄₃×#1276

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

characteristic

C₂×#13063325

—

C₂×#13063325

C₂²

yes

—

—

—

—

—

C₂²

yes

—

—

—

C₄₃×#431

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

—

—

—

—

—

—

—

D₄

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

#13063284

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

#13063257

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

D₄

yes

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10299

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#392

yes

characteristic

C₂²×C₄₃

#450

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#456

yes

—

C₄×C₄₃

#460

yes

—

C₄×C₄₃

#492

yes

characteristic

100

C₄×C₄₃

#496

yes

characteristic

101

Dic₄₃

—

102

Dic₄₃

—

103

#3229

—

104

#2868

—

105

#1994

—

106

#1546

—

107

#1276

D₄₃

yes

characteristic

108

C₂×C₄₃

—

109

C₂×C₄₃

#2005

yes

—

110

C₂×C₄₃

#2014

yes

characteristic

111

#533

—

112

C₂×C₄⋊Q₈

—

113

C₂×C₄⋊Q₈

—

114

#498

—

115

#487

—

116

#483

—

117

#475

—

118

#431

C₂×D₄₃

yes

—

119

#423

—

120

#384

—

121

C₄³

C₂×D₄₃

yes

characteristic

122

#339

—

123

#332

—

124

C₄₃

#10299

yes

characteristic, complemented, Sylow

125

C₂²×Q₈

—

126

C₂×M₁₆

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₄⋊Q₈

—

132

C₄⋊Q₈

—

172

—

133

C₄⋊Q₈

—

172

—

134

#125

—

135

C₂²⋊Q₈

—

136

C₄×Q₈

—

137

#117

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄⋊C₄

—

141

C₂×C₄²

—

142

C₂×C₄²

D₁₇₂

yes

—

143

C₂×C₄²

C₂²×D₄₃

yes

characteristic

144

#106

—

145

C₄⋊C₈

—

146

C₄⋊C₈

D₁₇₂

yes

—

147

#103

—

148

C₂²⋊C₈

—

149

C₄×C₈

—

150

C₂×Q₈

—

151

C₂×Q₈

—

172

—

152

C₂×Q₈

—

172

—

153

C₂×Q₈

—

172

—

154

C₂×Q₈

—

172

—

155

C₂²×C₄

—

156

C₂²×C₄

—

157

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

158

C₂²×C₄

D₄×D₄₃

yes

—

159

M₁₆

—

172

—

160

C₂×C₈

—

161

C₂×C₈

—

172

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

172

—

167

C₄⋊C₄

—

172

—

168

C₄⋊C₄

—

172

—

169

C₄⋊C₄

—

172

—

170

C₂²⋊C₄

—

172

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

172

—

174

C₄²

#10621873

yes

—

175

C₄²

C₂×D₁₇₂

yes

—

176

C₄²

D₄×D₄₃

yes

characteristic

177

C₄²

#10621902

yes

characteristic

178

C₂³

#13063295

yes

characteristic

179

Q₈

—

344

—

180

Q₈

—

344

—

181

Q₈

—

344

—

182

Q₈

—

172

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

172

—

187

C₂×C₄

—

172

—

188

C₂×C₄

—

172

—

189

C₂×C₄

#13063295

yes

—

190

C₂×C₄

#13063300

yes

—

191

C₂×C₄

#13063300

yes

characteristic

192

C₂×C₄

#13063322

yes

characteristic

193

C₂×C₄

C₂×#10621873

yes

—

194

C₂×C₄

#13063353

yes

—

195

C₈

—

196

C₈

—

344

—

197

C₂²

—

198

C₂²

#14582809

yes

characteristic

199

C₂²

#14583052

yes

—

200

C₄

—

201

C₄

—

202

C₄

—

344

—

203

C₄

—

344

—

204

C₄

#14583083

yes

—

205

C₄

#14583100

yes

—

206

C₄

#14583272

yes

characteristic

207

C₄

#14583288

yes

characteristic

208

C₂

—

209

C₂

#19135634

yes

—

210

C₂

#19135685

yes

characteristic

	subgroup	G/H
center	C₂²	#14582809
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂³×C₄₃	#120
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#10299
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#14582809

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1276

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
124. C₄₃	#10299	semidirect

H ≅ G/N

type

124. C₄₃

#10299

semidirect

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

subgroup	index
#19145741	2
#19142993	2
#19135572	2
#19133266	2
C₄₃×#1276	2
#10299	43

subgroup

index

C₄₃×#1276

Subgroups — #24999347

Conjugacy classes of subgroups

Complements

Maximal subgroups