Group #25001701 — subgroups

Subgroups — #25001701

Order 11008.

526 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 153 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	#19135401	C₂	1	1	yes	maximal, characteristic
2	#19135400	C₂	1	2	yes	maximal
3	#19134751	C₂	1	2	yes	maximal, complemented
4	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	C₂×#13063256	—	2	4	no	—
8	C₂×#13063247	—	2	2	no	—
9	C₂×#13063245	—	2	4	no	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
11	#14582769	C₂²	1	1	yes	characteristic
12	C₂×#106×C₄₃	—	2	2	no	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#384	C₂²	1	2	yes	—
15	C₂²×D₁₇₂	—	2	2	no	—
16	C₂²×D₁₇₂	—	2	1	no	—
17	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
18	#13063322	—	2	2	no	—
19	#13063322	—	2	1	no	—
20	#13063322	Q₈	1	2	yes	—
21	C₂×C₄⋊Dic₄₃	—	2	1	no	—
22	C₂×C₄⋊Dic₄₃	—	2	2	no	—
23	#13063256	—	4	8	no	—
24	#13063247	—	4	4	no	—
25	#13063245	—	4	8	no	—
26	C₄×C₄₃⋊C₈	—	2	2	no	—
27	C₄×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₈×C₄₃	D₄	1	2	yes	—
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
32	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
33	#106×C₄₃	—	4	4	no	—
34	C₄×C₈×C₄₃	—	2	2	no	—
35	C₂³×D₄₃	—	4	1	no	—
36	C₂²×C₄×C₄₃	—	4	2	no	—
37	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₄₃	—	4	1	no	—
40	C₂×D₁₇₂	—	2	2	no	—
41	C₂×D₁₇₂	—	2	4	no	—
42	C₂×D₁₇₂	—	4	1	no	—
43	C₂×D₁₇₂	—	4	2	no	—
44	C₂×D₁₇₂	—	4	4	no	—
45	C₂×D₁₇₂	—	4	2	no	—
46	C₂×D₁₇₂	—	4	2	no	—
47	C₂×D₁₇₂	—	4	4	no	—
48	C₂×C₈×C₄₃	—	2	2	no	—
49	C₂×C₈×C₄₃	—	2	2	no	—
50	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
51	C₄⋊C₄×C₄₃	—	4	4	no	—
52	C₄⋊C₄×C₄₃	—	4	4	no	—
53	C₄⋊C₄×C₄₃	—	4	4	no	—
54	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₄₃	#41	1	2	yes	—
56	C₄²×C₄₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₄₃	—	4	2	no	—
58	C₄⋊Dic₄₃	—	4	4	no	—
59	C₂×C₄₃⋊C₈	—	2	4	no	—
60	C₂×C₄₃⋊C₈	—	2	4	no	—
61	C₂×C₄₃⋊C₈	SD₁₆	1	8	yes	—
62	C₂³×C₄₃	#127	1	1	yes	characteristic
63	C₂²×D₄₃	—	4	2	no	—
64	C₂²×D₄₃	—	8	2	no	—
65	C₂²×D₄₃	—	8	1	no	—
66	C₂²×D₄₃	—	8	2	no	—
67	C₂²×D₄₃	—	8	1	no	—
68	C₂×C₄×C₄₃	—	8	2	no	—
69	C₂×C₄×C₄₃	—	8	2	no	—
70	C₂×C₄×C₄₃	—	8	2	no	—
71	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#123	1	2	yes	—
75	C₂×C₄×C₄₃	C₂×SD₁₆	1	4	yes	—
76	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
79	C₂×Dic₄₃	—	8	1	no	—
80	C₂×Dic₄₃	—	8	2	no	—
81	D₁₇₂	—	4	4	no	—
82	D₁₇₂	—	4	8	no	—
83	D₁₇₂	—	8	2	no	—
84	D₁₇₂	—	8	4	no	—
85	C₈×C₄₃	—	2	4	no	—
86	C₄₃⋊C₈	—	2	8	no	—
87	#10683	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	#399	1	1	yes	characteristic
89	C₂²×C₄₃	#492	1	2	yes	—
90	C₂²×C₄₃	#492	1	1	yes	characteristic
91	C₂²×C₄₃	#496	1	2	yes	—
92	C₂²×C₄₃	#496	1	1	yes	characteristic
93	C₂×D₄₃	—	8	4	no	—
94	C₂×D₄₃	—	8	2	no	—
95	C₂×D₄₃	—	16	1	no	—
96	C₂×D₄₃	—	16	2	no	—
97	C₂×D₄₃	—	16	1	no	—
98	C₄×C₄₃	—	16	2	no	—
99	C₄×C₄₃	#476	1	2	yes	—
100	C₄×C₄₃	#476	1	4	yes	—
101	C₄×C₄₃	#481	1	2	yes	—
102	C₄×C₄₃	#481	1	4	yes	—
103	Dic₄₃	—	16	1	no	—
104	#1957	—	43	1	no	—
105	#1957	—	43	2	no	—
106	#1844	—	43	2	no	—
107	#1844	D₄₃	1	1	yes	characteristic, complemented
108	#1273	—	43	1	no	—
109	C₂×C₄₃	#1705	1	1	yes	characteristic
110	C₂×C₄₃	#1715	1	2	yes	—
111	C₂×C₄₃	#1715	1	1	yes	characteristic
112	C₂×C₄₃	#2094	1	2	yes	—
113	C₂×C₄₃	#2094	1	1	yes	characteristic
114	D₄₃	—	16	2	no	—
115	C₂×#127	—	43	1	no	—
116	C₂×#106	—	2	2	no	—
117	C₂×#106	—	86	4	no	—
118	C₂×#102	—	86	4	no	—
119	C₂×#102	—	86	2	no	—
120	C₂×C₄×C₈	—	43	2	no	—
121	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
122	#384	—	43	1	no	—
123	#384	C₂×D₄₃	1	2	yes	—
124	C₄₃	#10683	1	1	yes	characteristic, complemented, Sylow
125	C₂²×D₄	—	86	2	no	—
126	C₂²×D₄	—	86	1	no	—
127	C₂²×C₈	—	43	4	no	—
128	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
129	#127	—	43	2	no	—
130	#127	—	86	2	no	—
131	#127	—	86	1	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	86	2	no	—
136	C₂×C₄⋊C₄	—	86	1	no	—
137	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
138	#106	—	4	4	no	—
139	#106	—	172	8	no	—
140	#102	—	172	8	no	—
141	#102	—	172	4	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	86	2	no	—
144	C₄×C₈	—	86	2	no	—
145	C₂⁴	—	172	1	no	—
146	C₂×D₄	—	86	2	no	—
147	C₂×D₄	—	86	4	no	—
148	C₂×D₄	—	172	1	no	—
149	C₂×D₄	—	172	2	no	—
150	C₂×D₄	—	172	4	no	—
151	C₂×D₄	—	172	2	no	—
152	C₂×D₄	—	172	2	no	—
153	C₂×D₄	—	172	4	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	172	1	no	—
156	C₂²×C₄	D₄×D₄₃	1	2	yes	—
157	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	43	8	no	—
161	C₂×C₈	—	86	4	no	—
162	C₂×C₈	—	86	4	no	—
163	C₂×C₈	#10621884	1	4	yes	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	4	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	172	2	no	—
168	C₄⋊C₄	—	172	4	no	—
169	C₄²	#10621898	1	2	yes	—
170	C₄²	Q₈×D₄₃	1	1	yes	characteristic
171	C₄²	#10621902	1	1	yes	characteristic
172	C₂³	—	344	2	no	—
173	C₂³	—	344	1	no	—
174	C₂³	—	344	2	no	—
175	C₂³	—	344	1	no	—
176	C₂³	—	172	2	no	—
177	C₂³	#13063321	1	1	yes	characteristic
178	D₄	—	344	2	no	—
179	D₄	—	344	4	no	—
180	D₄	—	172	4	no	—
181	D₄	—	172	8	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	—	8	2	no	—
185	C₂×C₄	—	344	1	no	—
186	C₂×C₄	—	344	2	no	—
187	C₂×C₄	#13063301	1	1	yes	characteristic
188	C₂×C₄	#13063303	1	2	yes	—
189	C₂×C₄	#13063308	1	1	yes	characteristic
190	C₂×C₄	#13063310	1	2	yes	—
191	C₂×C₄	SD₁₆×D₄₃	1	4	yes	—
192	C₂×C₄	C₂×#10621884	1	2	yes	—
193	C₂×C₄	#13063351	1	2	yes	—
194	C₂×C₄	#13063352	1	4	yes	—
195	C₈	—	2	4	no	—
196	C₈	—	86	8	no	—
197	C₂²	—	344	4	no	—
198	C₂²	—	344	2	no	—
199	C₂²	—	688	2	no	—
200	C₂²	—	688	1	no	—
201	C₂²	—	688	1	no	—
202	C₂²	#14582834	1	1	yes	characteristic
203	C₂²	#14583270	1	2	yes	—
204	C₂²	#14583273	1	1	yes	characteristic
205	C₂²	#14583285	1	2	yes	—
206	C₂²	#14583286	1	1	yes	characteristic
207	C₄	—	16	2	no	—
208	C₄	—	688	1	no	—
209	C₄	#14583186	1	2	yes	—
210	C₄	#14583188	1	4	yes	—
211	C₄	#14583217	1	2	yes	—
212	C₄	#14583219	1	4	yes	—
213	C₂	—	688	2	no	—
214	C₂	#19134183	1	1	yes	characteristic
215	C₂	#19134222	1	2	yes	—
216	C₂	#19134224	1	1	yes	characteristic
217	C₂	#19136125	1	2	yes	—
218	C₂	#19136129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19135401

C₂

yes

maximal, characteristic

#19135400

C₂

yes

maximal

#19134751

C₂

yes

maximal, complemented

C₄₃⋊#1273

C₂

yes

maximal, characteristic, complemented

C₄₃×#1844

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

—

—

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

characteristic

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

C₂²

yes

—

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

C₂²×C₄₃⋊C₈

D₄

yes

—

#13063322

—

#13063322

—

#13063322

Q₈

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063256

—

#13063247

—

#13063245

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#106×C₄₃

—

C₄×C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

SD₁₆

yes

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#492

yes

characteristic

C₂²×C₄₃

#496

yes

—

C₂²×C₄₃

#496

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#476

yes

—

100

C₄×C₄₃

#476

yes

—

101

C₄×C₄₃

#481

yes

—

102

C₄×C₄₃

#481

yes

—

103

Dic₄₃

—

104

#1957

—

105

#1957

—

106

#1844

—

107

#1844

D₄₃

yes

characteristic, complemented

108

#1273

—

109

C₂×C₄₃

#1705

yes

characteristic

110

C₂×C₄₃

#1715

yes

—

111

C₂×C₄₃

#1715

yes

characteristic

112

C₂×C₄₃

#2094

yes

—

113

C₂×C₄₃

#2094

yes

characteristic

114

D₄₃

—

115

C₂×#127

—

116

C₂×#106

—

117

C₂×#106

—

118

C₂×#102

—

119

C₂×#102

—

120

C₂×C₄×C₈

—

121

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

122

#384

—

123

#384

C₂×D₄₃

yes

—

124

C₄₃

#10683

yes

characteristic, complemented, Sylow

125

C₂²×D₄

—

126

C₂²×D₄

—

127

C₂²×C₈

—

128

C₂²×C₈

C₂²⋊D₄₃

yes

—

129

#127

—

130

#127

—

131

#127

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄²

C₂²×D₄₃

yes

characteristic

138

#106

—

139

#106

—

172

—

140

#102

—

172

—

141

#102

—

172

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₂⁴

—

172

—

146

C₂×D₄

—

147

C₂×D₄

—

148

C₂×D₄

—

172

—

149

C₂×D₄

—

172

—

150

C₂×D₄

—

172

—

151

C₂×D₄

—

172

—

152

C₂×D₄

—

172

—

153

C₂×D₄

—

172

—

154

C₂²×C₄

—

155

C₂²×C₄

—

172

—

156

C₂²×C₄

D₄×D₄₃

yes

—

157

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

#10621884

yes

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

172

—

168

C₄⋊C₄

—

172

—

169

C₄²

#10621898

yes

—

170

C₄²

Q₈×D₄₃

yes

characteristic

171

C₄²

#10621902

yes

characteristic

172

C₂³

—

344

—

173

C₂³

—

344

—

174

C₂³

—

344

—

175

C₂³

—

344

—

176

C₂³

—

172

—

177

C₂³

#13063321

yes

characteristic

178

D₄

—

344

—

179

D₄

—

344

—

180

D₄

—

172

—

181

D₄

—

172

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

344

—

186

C₂×C₄

—

344

—

187

C₂×C₄

#13063301

yes

characteristic

188

C₂×C₄

#13063303

yes

—

189

C₂×C₄

#13063308

yes

characteristic

190

C₂×C₄

#13063310

yes

—

191

C₂×C₄

SD₁₆×D₄₃

yes

—

192

C₂×C₄

C₂×#10621884

yes

—

193

C₂×C₄

#13063351

yes

—

194

C₂×C₄

#13063352

yes

—

195

C₈

—

196

C₈

—

197

C₂²

—

344

—

198

C₂²

—

344

—

199

C₂²

—

688

—

200

C₂²

—

688

—

201

C₂²

—

688

—

202

C₂²

#14582834

yes

characteristic

203

C₂²

#14583270

yes

—

204

C₂²

#14583273

yes

characteristic

205

C₂²

#14583285

yes

—

206

C₂²

#14583286

yes

characteristic

207

C₄

—

208

C₄

—

688

—

209

C₄

#14583186

yes

—

210

C₄

#14583188

yes

—

211

C₄

#14583217

yes

—

212

C₄

#14583219

yes

—

213

C₂

—

688

—

214

C₂

#19134183

yes

characteristic

215

C₂

#19134222

yes

—

216

C₂

#19134224

yes

characteristic

217

C₂

#19136125

yes

—

218

C₂

#19136129

yes

characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1844	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	#1844	D₄₃
hypocenter	C₄₃	#10683
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13063321

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1844

C₂

C₂³×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
3. #19134751	C₂	semidirect
4. C₄₃⋊#1273	C₂	semidirect
5. C₄₃×#1844	C₂	semidirect
107. #1844	D₄₃	semidirect
124. C₄₃	#10683	semidirect

H ≅ G/N

type

3. #19134751

C₂

semidirect

4. C₄₃⋊#1273

C₂

semidirect

5. C₄₃×#1844

C₂

semidirect

107. #1844

D₄₃

semidirect

124. C₄₃

#10683

semidirect

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

subgroup	index
#19135401	2
#19135400	2
#19134751	2
C₄₃⋊#1273	2
C₄₃×#1844	2
#10683	43

subgroup

index

#19135401

#19135400

#19134751

C₄₃⋊#1273

C₄₃×#1844

#10683

Subgroups — #25001701

Conjugacy classes of subgroups

Complements

Maximal subgroups