Group #24989588 — subgroups

Subgroups — #24989588

Order 11008.

388 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19143306	C₂	1	1	yes	maximal, characteristic
2	#19133356	C₂	1	2	yes	maximal, complemented
3	#19133229	C₂	1	2	yes	maximal
4	#19132506	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1307	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063333	C₄	1	2	yes	—
7	#14583449	C₂²	1	1	yes	characteristic
8	#14582976	—	2	2	no	—
9	#14582964	—	2	2	no	—
10	#14582956	C₂²	1	2	yes	—
11	#14582867	C₄	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582651	—	2	2	no	—
14	#14582649	—	2	2	no	—
15	#14582638	—	2	4	no	—
16	C₄₃×#432	C₂²	1	2	yes	—
17	C₄₃×#423	C₂²	1	1	yes	characteristic
18	C₂×#10621898	C₂×C₄	1	2	yes	—
19	#13063333	—	4	4	no	—
20	C₂×#10621872	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₄₃	—	2	2	no	—
24	#13063325	Q₈	1	1	yes	characteristic
25	#13063322	Q₈	1	1	yes	characteristic
26	#13063313	D₄	1	2	yes	—
27	#13063299	—	2	2	no	—
28	C₄×D₁₇₂	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
31	(C₂×D₄₃)⋊C₈	—	2	4	no	—
32	#13063236	—	2	2	no	—
33	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
34	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
35	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	—	2	2	no	—
38	C₄⋊C₈×C₄₃	—	2	2	no	—
39	C₂²⋊C₈×C₄₃	—	2	2	no	—
40	#97×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	#41	1	2	yes	—
43	C₂×C₂²⋊D₄₃	—	2	2	no	—
44	#10621898	—	4	2	no	—
45	#10621898	—	4	2	no	—
46	C₂×D₁₇₂	—	2	2	no	—
47	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
48	C₂×C₄×D₄₃	—	2	2	no	—
49	C₂×Dic₈₆	C₄⋊C₄	1	2	yes	—
50	M₁₆×C₄₃	—	4	2	no	—
51	C₂×C₈×C₄₃	—	2	2	no	—
52	C₂×C₈×C₄₃	—	4	2	no	—
53	C₄²×C₄₃	C₂×D₄	1	2	yes	—
54	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	(C₂×D₄₃)⋊C₄	—	2	4	no	—
57	C₄⋊Dic₄₃	—	2	2	no	—
58	C₂×C₄₃⋊C₈	—	2	2	no	—
59	C₂×C₄₃⋊C₈	—	4	1	no	—
60	C₂×C₄₃⋊C₈	—	4	2	no	—
61	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
62	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	#10621872	—	4	4	no	—
65	C₈×D₄₃	—	4	4	no	—
66	C₂³×C₄₃	#117	1	1	yes	characteristic
67	C₂²×D₄₃	—	2	2	no	—
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	4	2	no	—
71	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₃	#121	1	2	yes	—
74	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
75	C₂²⋊D₄₃	—	4	4	no	—
76	C₂×Dic₄₃	—	2	2	no	—
77	D₁₇₂	—	4	2	no	—
78	D₁₇₂	—	4	4	no	—
79	C₄×D₄₃	—	4	2	no	—
80	C₄×D₄₃	—	4	2	no	—
81	Dic₈₆	—	4	2	no	—
82	C₈×C₄₃	—	4	2	no	—
83	C₈×C₄₃	—	8	2	no	—
84	C₄₃⋊C₈	—	2	1	no	—
85	C₄₃⋊C₈	—	2	2	no	—
86	C₄₃⋊C₈	—	8	2	no	—
87	C₄₃⋊C₈	#100	1	2	yes	—
88	#8630	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	—	2	2	no	—
91	C₂²×C₄₃	#380	1	1	yes	characteristic
92	C₂×D₄₃	—	4	2	no	—
93	C₂×D₄₃	—	4	4	no	—
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	—	2	2	no	—
96	C₄×C₄₃	—	2	2	no	—
97	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
98	C₄×C₄₃	#413	1	1	yes	characteristic
99	Dic₄₃	—	4	2	no	—
100	#2931	—	43	1	no	—
101	#1568	—	43	2	no	—
102	#1536	—	43	2	no	—
103	#1307	—	43	1	no	—
104	#1307	D₄₃	1	1	yes	characteristic, complemented
105	C₂×C₄₃	—	4	1	no	—
106	C₂×C₄₃	#1800	1	1	yes	characteristic
107	C₂×C₄₃	#1814	1	2	yes	—
108	D₄₃	—	8	2	no	—
109	C₂×#131	—	43	2	no	—
110	#532	—	43	1	no	—
111	#436	—	86	2	no	—
112	C₈×D₄	—	86	2	no	—
113	#432	—	43	2	no	—
114	#432	C₂×D₄₃	1	2	yes	—
115	#423	C₂×D₄₃	1	1	yes	characteristic
116	#408	—	43	2	no	—
117	C₂×C₄×C₈	—	43	1	no	—
118	#335	—	86	2	no	—
119	#334	—	86	2	no	—
120	#329	—	86	4	no	—
121	C₄₃	#8630	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	—	43	2	no	—
123	#131	—	172	4	no	—
124	C₂×M₁₆	—	86	2	no	—
125	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
126	C₂²×C₈	—	43	1	no	—
127	C₂²×C₈	—	43	1	no	—
128	C₂²×C₈	—	86	2	no	—
129	C₄⋊Q₈	—	43	1	no	—
130	#127	—	43	1	no	—
131	#124	—	43	2	no	—
132	#121	—	86	2	no	—
133	C₄×D₄	—	86	2	no	—
134	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	86	2	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	—	86	4	no	—
140	C₂²⋊C₈	—	86	2	no	—
141	#97	—	2	2	no	—
142	#97	—	86	2	no	—
143	C₄×C₈	—	43	2	no	—
144	C₄×C₈	—	43	2	no	—
145	#41	—	172	2	no	—
146	#41	—	172	2	no	—
147	C₂×Q₈	—	43	2	no	—
148	C₂×D₄	—	43	2	no	—
149	C₂×D₄	—	86	2	no	—
150	C₂×D₄	—	86	2	no	—
151	C₂²×C₄	—	86	2	no	—
152	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
153	C₂²×C₄	#10621898	1	2	yes	—
154	M₁₆	—	4	2	no	—
155	M₁₆	—	172	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	43	1	no	—
159	C₂×C₈	—	43	2	no	—
160	C₂×C₈	—	43	1	no	—
161	C₂×C₈	—	86	2	no	—
162	C₂×C₈	—	172	4	no	—
163	C₂×C₈	—	172	1	no	—
164	C₂×C₈	—	172	2	no	—
165	C₄⋊C₄	—	86	2	no	—
166	C₂²⋊C₄	—	86	4	no	—
167	C₄²	D₄×D₄₃	1	2	yes	—
168	C₄²	D₄×D₄₃	1	1	yes	characteristic
169	C₄²	Q₈×D₄₃	1	1	yes	characteristic
170	C₂³	—	86	2	no	—
171	C₂³	#13063282	1	1	yes	characteristic
172	Q₈	—	172	2	no	—
173	D₄	—	172	2	no	—
174	D₄	—	172	4	no	—
175	D₄	—	172	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	86	2	no	—
180	C₂×C₄	—	172	2	no	—
181	C₂×C₄	—	172	2	no	—
182	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
183	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
184	C₂×C₄	#13063298	1	2	yes	—
185	C₂×C₄	#13063303	1	2	yes	—
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	—	43	2	no	—
189	C₈	—	86	1	no	—
190	C₈	—	86	2	no	—
191	C₈	—	344	2	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	172	2	no	—
195	C₂²	—	172	4	no	—
196	C₂²	#14582758	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	172	2	no	—
201	C₄	#100×D₄₃	1	1	yes	characteristic
202	C₄	#14582882	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	344	2	no	—
205	C₂	#19134586	1	1	yes	characteristic
206	C₂	#19134648	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143306

C₂

yes

maximal, characteristic

#19133356

C₂

yes

maximal, complemented

#19133229

C₂

yes

maximal

#19132506

C₂

yes

maximal, characteristic, complemented

C₄₃×#1307

C₂

yes

maximal, characteristic, complemented

C₂×#13063333

C₄

yes

—

#14583449

C₂²

yes

—

—

C₂²

yes

—

#14582867

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

—

C₄₃×#432

C₂²

yes

—

C₄₃×#423

C₂²

yes

characteristic

C₂×#10621898

C₂×C₄

yes

—

#13063333

—

C₂×#10621872

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈×D₄₃

—

#13063325

Q₈

yes

characteristic

#13063322

Q₈

yes

characteristic

#13063313

D₄

yes

—

#13063299

—

C₄×D₁₇₂

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

(C₂×D₄₃)⋊C₈

—

#13063236

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₂²⋊D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₄⋊C₄

yes

—

C₂×C₄×D₄₃

—

C₂×Dic₈₆

C₄⋊C₄

yes

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

#10621872

—

C₈×D₄₃

—

C₂³×C₄₃

#117

yes

characteristic

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂²⋊D₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#100

yes

—

#8630

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#380

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

100

#2931

—

101

#1568

—

102

#1536

—

103

#1307

—

104

#1307

D₄₃

yes

characteristic, complemented

105

C₂×C₄₃

—

106

C₂×C₄₃

#1800

yes

characteristic

107

C₂×C₄₃

#1814

yes

—

108

D₄₃

—

109

C₂×#131

—

110

#532

—

111

#436

—

112

C₈×D₄

—

113

#432

—

114

#432

C₂×D₄₃

yes

—

115

#423

C₂×D₄₃

yes

characteristic

116

#408

—

117

C₂×C₄×C₈

—

118

#335

—

119

#334

—

120

#329

—

121

C₄₃

#8630

yes

characteristic, complemented, Sylow

122

C₂×#41

—

123

#131

—

172

—

124

C₂×M₁₆

—

125

C₂×M₁₆

C₄×D₄₃

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₄⋊Q₈

—

130

#127

—

131

#124

—

132

#121

—

133

C₄×D₄

—

134

C₂×C₄²

C₂²×D₄₃

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₂²⋊C₈

—

139

C₂²⋊C₈

—

140

C₂²⋊C₈

—

141

#97

—

142

#97

—

143

C₄×C₈

—

144

C₄×C₈

—

145

#41

—

172

—

146

#41

—

172

—

147

C₂×Q₈

—

148

C₂×D₄

—

149

C₂×D₄

—

150

C₂×D₄

—

151

C₂²×C₄

—

152

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

153

C₂²×C₄

#10621898

yes

—

154

M₁₆

—

155

M₁₆

—

172

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

172

—

163

C₂×C₈

—

172

—

164

C₂×C₈

—

172

—

165

C₄⋊C₄

—

166

C₂²⋊C₄

—

167

C₄²

D₄×D₄₃

yes

—

168

C₄²

D₄×D₄₃

yes

characteristic

169

C₄²

Q₈×D₄₃

yes

characteristic

170

C₂³

—

171

C₂³

#13063282

yes

characteristic

172

Q₈

—

172

—

173

D₄

—

172

—

174

D₄

—

172

—

175

D₄

—

172

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

172

—

181

C₂×C₄

—

172

—

182

C₂×C₄

C₂²⋊C₄×D₄₃

yes

characteristic

183

C₂×C₄

C₄⋊C₄×D₄₃

yes

characteristic

184

C₂×C₄

#13063298

yes

—

185

C₂×C₄

#13063303

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

—

190

C₈

—

191

C₈

—

344

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

172

—

195

C₂²

—

172

—

196

C₂²

#14582758

yes

characteristic

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

172

—

201

C₄

#100×D₄₃

yes

characteristic

202

C₄

#14582882

yes

characteristic

203

C₂

—

204

C₂

—

344

—

205

C₂

#19134586

yes

characteristic

206

C₂

#19134648

yes

—

	subgroup	G/H
center	C₂²	#14582758
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1307	C₂
socle	C₂²×C₄₃	#380
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1307	D₄₃
hypocenter	C₄₃	#8630
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

subgroup

G/H

center

C₂²

#14582758

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1307

C₂

C₂²×C₄₃

C₂²

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1307

N	H ≅ G/N	type
2. #19133356	C₂	semidirect
4. #19132506	C₂	semidirect
5. C₄₃×#1307	C₂	semidirect
104. #1307	D₄₃	semidirect
121. C₄₃	#8630	semidirect

H ≅ G/N

type

2. #19133356

C₂

semidirect

4. #19132506

C₂

semidirect

5. C₄₃×#1307

C₂

semidirect

104. #1307

D₄₃

semidirect

121. C₄₃

#8630

semidirect

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

subgroup	index
#19143306	2
#19133356	2
#19133229	2
#19132506	2
C₄₃×#1307	2
#8630	43

subgroup

index

C₄₃×#1307

Subgroups — #24989588

Conjugacy classes of subgroups

Complements

Maximal subgroups