Group #24999691 — subgroups

Subgroups — #24999691

Order 11008.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19142973	C₂	1	1	yes	maximal, characteristic
2	#19135572	C₂	1	1	yes	maximal, characteristic
3	#19133160	C₂	1	2	yes	maximal
4	#19132454	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	1	no	—
8	#14582923	C₂²	1	1	yes	characteristic
9	#14582805	—	2	4	no	—
10	#14582765	—	2	1	no	—
11	#14582765	C₂²	1	2	yes	—
12	#14582659	—	2	1	no	—
13	#14582659	—	2	1	no	—
14	#14582637	—	2	4	no	—
15	C₄₃×#431	C₂²	1	2	yes	—
16	C₄³×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₆	—	2	2	no	—
18	C₂×#10621877	—	2	1	no	—
19	#13063325	—	4	2	no	—
20	#13063325	—	4	2	no	—
21	#13063325	Q₈	1	2	yes	complemented
22	#13063317	—	2	1	no	—
23	#13063304	—	2	2	no	—
24	C₄×Dic₈₆	—	2	2	no	—
25	#13063284	—	2	1	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	C₄₃⋊#95	—	2	4	no	—
30	C₄₃⋊#95	—	2	2	no	—
31	C₄₃⋊#95	D₄	1	4	yes	—
32	C₂×C₄²×C₄₃	—	2	2	no	—
33	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₄₃	—	2	2	no	—
37	C₂²⋊C₈×C₄₃	—	2	2	no	—
38	C₄×C₈×C₄₃	—	2	2	no	—
39	C₂²×C₄×C₄₃	—	2	1	no	—
40	C₂²×C₄×C₄₃	—	2	1	no	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₂²×Dic₄₃	—	2	4	no	—
44	C₂×Dic₈₆	—	2	2	no	—
45	C₂×Dic₈₆	—	4	4	no	—
46	C₂×Dic₈₆	—	4	8	no	—
47	C₂×C₈×C₄₃	—	4	2	no	—
48	C₄²×C₄₃	—	2	1	no	—
49	C₄²×C₄₃	—	2	1	no	—
50	C₄²×C₄₃	—	4	4	no	—
51	C₄²×C₄₃	SD₁₆	1	2	yes	—
52	C₄²×C₄₃	Q₁₆	1	2	yes	—
53	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₄₃	#41	1	2	yes	—
55	C₄²×C₄₃	#41	1	1	yes	characteristic
56	C₂²⋊Dic₄₃	—	4	1	no	—
57	C₄⋊Dic₄₃	—	2	2	no	—
58	C₄⋊Dic₄₃	—	2	2	no	—
59	C₄⋊Dic₄₃	—	4	4	no	—
60	C₄⋊Dic₄₃	—	4	2	no	—
61	C₄⋊Dic₄₃	—	4	2	no	—
62	Dic₄₃⋊C₄	—	4	2	no	—
63	C₄×Dic₄₃	—	4	1	no	—
64	#10621877	—	4	1	no	—
65	#10621877	—	4	1	no	—
66	C₂×C₄₃⋊C₈	—	4	1	no	—
67	C₂³×C₄₃	#120	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	—	2	2	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	4	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₃	#124	1	2	yes	—
76	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
81	C₂×Dic₄₃	—	4	8	no	—
82	C₂×Dic₄₃	—	4	2	no	—
83	C₂×Dic₄₃	—	4	4	no	—
84	Dic₈₆	—	8	8	no	—
85	Dic₈₆	—	8	2	no	—
86	C₈×C₄₃	—	8	2	no	—
87	C₄₃⋊C₈	—	8	1	no	—
88	#10363	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	—	2	2	no	—
90	C₂²×C₄₃	#393	1	1	yes	characteristic
91	C₂²×C₄₃	#450	1	1	yes	characteristic
92	C₂²×C₄₃	#451	1	1	yes	characteristic
93	C₄×C₄₃	—	2	4	no	—
94	C₄×C₄₃	—	2	2	no	—
95	C₄×C₄₃	—	2	2	no	—
96	C₄×C₄₃	—	4	2	no	—
97	C₄×C₄₃	#475	1	1	yes	characteristic
98	C₄×C₄₃	#477	1	1	yes	characteristic
99	C₄×C₄₃	#487	1	1	yes	characteristic
100	C₄×C₄₃	#489	1	1	yes	characteristic
101	Dic₄₃	—	8	4	no	—
102	Dic₄₃	—	8	2	no	—
103	#2862	—	43	1	no	—
104	#1994	—	43	1	no	—
105	#1518	—	43	2	no	—
106	#1286	—	43	2	no	—
107	#1276	D₄₃	1	1	yes	characteristic
108	C₂×C₄₃	—	2	1	no	—
109	C₂×C₄₃	—	2	1	no	—
110	C₂×C₄₃	#2024	1	1	yes	characteristic
111	C₂×C₄₃	#2031	1	2	yes	—
112	#533	—	43	1	no	—
113	C₂×C₄⋊Q₈	—	86	1	no	—
114	#431	C₂×D₄₃	1	2	yes	—
115	#423	—	43	1	no	—
116	#391	—	86	4	no	—
117	#382	—	43	2	no	—
118	#382	—	86	1	no	—
119	C₄³	C₂×D₄₃	1	1	yes	characteristic
120	#339	—	86	1	no	—
121	#339	—	86	1	no	—
122	#328	—	86	4	no	—
123	C₄₃	#10363	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	86	2	no	—
125	C₂×M₁₆	—	86	1	no	—
126	C₄⋊Q₈	—	43	2	no	—
127	C₄⋊Q₈	—	172	2	no	—
128	C₄⋊Q₈	—	172	2	no	—
129	#125	—	86	1	no	—
130	C₂²⋊Q₈	—	86	2	no	—
131	C₄×Q₈	—	86	2	no	—
132	#117	—	86	1	no	—
133	C₂×C₄⋊C₄	—	86	2	no	—
134	C₂×C₄⋊C₄	—	86	2	no	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
137	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	86	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	#95	—	43	4	no	—
144	#95	—	86	4	no	—
145	#95	—	86	2	no	—
146	C₂×Q₈	—	86	2	no	—
147	C₂×Q₈	—	172	4	no	—
148	C₂×Q₈	—	172	8	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	86	4	no	—
152	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
153	C₂²×C₄	D₄×D₄₃	1	2	yes	—
154	M₁₆	—	172	1	no	—
155	M₁₆	—	172	1	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	172	1	no	—
158	C₄⋊C₄	—	86	2	no	—
159	C₄⋊C₄	—	86	2	no	—
160	C₄⋊C₄	—	172	4	no	—
161	C₄⋊C₄	—	172	2	no	—
162	C₄⋊C₄	—	172	2	no	—
163	C₄⋊C₄	—	172	2	no	—
164	C₂²⋊C₄	—	172	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	4	4	no	—
168	C₄²	—	172	1	no	—
169	C₄²	#10621873	1	2	yes	—
170	C₄²	Dic₁₇₂	1	2	yes	—
171	C₄²	#10621898	1	2	yes	—
172	C₄²	#10621900	1	1	yes	characteristic
173	C₄²	Q₈×D₄₃	1	1	yes	characteristic
174	C₂³	#13063295	1	1	yes	characteristic
175	Q₈	—	344	2	no	—
176	Q₈	—	344	8	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	172	8	no	—
183	C₂×C₄	—	172	4	no	—
184	C₂×C₄	—	172	2	no	—
185	C₂×C₄	#13063303	1	2	yes	—
186	C₂×C₄	#13063305	1	1	yes	characteristic
187	C₂×C₄	#13063311	1	2	yes	—
188	C₂×C₄	#13063313	1	1	yes	characteristic
189	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₁₇₂	1	1	yes	characteristic
191	C₂×C₄	#13063353	1	1	yes	characteristic
192	C₂×C₄	#13063358	1	1	yes	characteristic
193	C₈	—	8	2	no	—
194	C₈	—	344	1	no	—
195	C₂²	—	2	2	no	—
196	C₂²	#14582812	1	1	yes	characteristic
197	C₂²	#14583052	1	1	yes	characteristic
198	C₂²	#14583057	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	344	4	no	—
204	C₄	—	344	2	no	—
205	C₄	#14583185	1	1	yes	characteristic
206	C₄	#14583195	1	1	yes	characteristic
207	C₄	#14583250	1	1	yes	characteristic
208	C₄	#14583259	1	1	yes	characteristic
209	C₂	—	2	1	no	—
210	C₂	—	2	1	no	—
211	C₂	#19135737	1	1	yes	characteristic
212	C₂	#19135768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19142973

C₂

yes

maximal, characteristic

#19135572

C₂

yes

maximal, characteristic

#19133160

C₂

yes

maximal

#19132454

C₂

yes

maximal

C₄₃×#1276

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

characteristic

C₂×#13063325

—

#14582923

C₂²

yes

—

—

C₂²

yes

—

#14582659

—

#14582659

—

#14582637

—

C₄₃×#431

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

—

—

—

Q₈

yes

complemented

#13063317

—

#13063304

—

C₄×Dic₈₆

—

#13063284

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×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×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

—

C₄²×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#124

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10363

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

100

C₄×C₄₃

#489

yes

characteristic

101

Dic₄₃

—

102

Dic₄₃

—

103

#2862

—

104

#1994

—

105

#1518

—

106

#1286

—

107

#1276

D₄₃

yes

characteristic

108

C₂×C₄₃

—

109

C₂×C₄₃

—

110

C₂×C₄₃

#2024

yes

characteristic

111

C₂×C₄₃

#2031

yes

—

112

#533

—

113

C₂×C₄⋊Q₈

—

114

#431

C₂×D₄₃

yes

—

115

#423

—

116

#391

—

117

#382

—

118

#382

—

119

C₄³

C₂×D₄₃

yes

characteristic

120

#339

—

121

#339

—

122

#328

—

123

C₄₃

#10363

yes

characteristic, complemented, Sylow

124

C₂²×Q₈

—

125

C₂×M₁₆

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

172

—

128

C₄⋊Q₈

—

172

—

129

#125

—

130

C₂²⋊Q₈

—

131

C₄×Q₈

—

132

#117

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄²

—

136

C₂×C₄²

D₁₇₂

yes

characteristic

137

C₂×C₄²

D₁₇₂

yes

characteristic

138

C₂×C₄²

C₂²×D₄₃

yes

characteristic

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

#95

—

144

#95

—

145

#95

—

146

C₂×Q₈

—

147

C₂×Q₈

—

172

—

148

C₂×Q₈

—

172

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

153

C₂²×C₄

D₄×D₄₃

yes

—

154

M₁₆

—

172

—

155

M₁₆

—

172

—

156

C₂×C₈

—

157

C₂×C₈

—

172

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

172

—

161

C₄⋊C₄

—

172

—

162

C₄⋊C₄

—

172

—

163

C₄⋊C₄

—

172

—

164

C₂²⋊C₄

—

172

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

172

—

169

C₄²

#10621873

yes

—

170

C₄²

Dic₁₇₂

yes

—

171

C₄²

#10621898

yes

—

172

C₄²

#10621900

yes

characteristic

173

C₄²

Q₈×D₄₃

yes

characteristic

174

C₂³

#13063295

yes

characteristic

175

Q₈

—

344

—

176

Q₈

—

344

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

172

—

183

C₂×C₄

—

172

—

184

C₂×C₄

—

172

—

185

C₂×C₄

#13063303

yes

—

186

C₂×C₄

#13063305

yes

characteristic

187

C₂×C₄

#13063311

yes

—

188

C₂×C₄

#13063313

yes

characteristic

189

C₂×C₄

C₂×#10621873

yes

characteristic

190

C₂×C₄

C₂×Dic₁₇₂

yes

characteristic

191

C₂×C₄

#13063353

yes

characteristic

192

C₂×C₄

#13063358

yes

characteristic

193

C₈

—

194

C₈

—

344

—

195

C₂²

—

196

C₂²

#14582812

yes

characteristic

197

C₂²

#14583052

yes

characteristic

198

C₂²

#14583057

yes

characteristic

199

C₄

—

200

C₄

—

201

C₄

—

202

C₄

—

203

C₄

—

344

—

204

C₄

—

344

—

205

C₄

#14583185

yes

characteristic

206

C₄

#14583195

yes

characteristic

207

C₄

#14583250

yes

characteristic

208

C₄

#14583259

yes

characteristic

209

C₂

—

210

C₂

—

211

C₂

#19135737

yes

characteristic

212

C₂

#19135768

yes

—

	subgroup	G/H
center	C₂²	#14582812
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#393
prime radical	C₂³×C₄₃	#120
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#10363
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#14582812

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
21. #13063325	Q₈	semidirect
123. C₄₃	#10363	semidirect

H ≅ G/N

type

21. #13063325

Q₈

semidirect

123. C₄₃

#10363

semidirect

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

subgroup	index
#19142973	2
#19135572	2
#19133160	2
#19132454	2
C₄₃×#1276	2
#10363	43

subgroup

index

C₄₃×#1276

Subgroups — #24999691

Conjugacy classes of subgroups

Complements

Maximal subgroups