Group #25001649 — subgroups

Subgroups — #25001649

Order 11008.

442 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 155 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	#19135557	C₂	1	2	yes	maximal
2	#19134755	C₂	1	1	yes	maximal, characteristic
3	#19134754	C₂	1	2	yes	maximal
4	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1956	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	—	2	2	no	—
7	C₂×#13063258	—	2	4	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#14582769	C₂²	1	2	yes	—
10	#14582661	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×#103×C₄₃	—	2	2	no	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×#384	C₂²	1	1	yes	characteristic
15	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
16	C₂×C₄⋊Dic₄₃	—	2	2	no	—
17	C₂×C₄⋊Dic₄₃	—	2	2	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	#13063259	—	4	4	no	—
20	#13063258	—	4	8	no	—
21	C₄×C₄₃⋊C₈	—	2	2	no	—
22	C₄×C₄₃⋊C₈	—	2	2	no	—
23	C₂²×Q₈×C₄₃	—	2	1	no	—
24	C₂²×C₈×C₄₃	D₄	1	2	yes	—
25	C₄⋊Q₈×C₄₃	—	2	2	no	—
26	C₄⋊Q₈×C₄₃	—	2	1	no	—
27	C₄⋊Q₈×C₄₃	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
31	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
32	#103×C₄₃	—	4	4	no	—
33	C₄×C₈×C₄₃	—	2	2	no	—
34	C₂×Q₈×C₄₃	—	2	2	no	—
35	C₂×Q₈×C₄₃	—	4	1	no	—
36	C₂×Q₈×C₄₃	—	4	2	no	—
37	C₂×Q₈×C₄₃	—	4	2	no	—
38	C₂²×C₄×C₄₃	—	4	1	no	—
39	C₂²×C₄×C₄₃	—	4	1	no	—
40	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₄₃	—	4	2	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	2	2	no	—
45	C₂×C₈×C₄₃	D₈	1	4	yes	—
46	C₄⋊C₄×C₄₃	—	2	4	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊C₄×C₄₃	—	4	2	no	—
49	C₄⋊C₄×C₄₃	—	4	4	no	—
50	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₃	#41	1	1	yes	characteristic
52	C₄²×C₄₃	#41	1	2	yes	—
53	C₄⋊Dic₄₃	—	4	4	no	—
54	C₄⋊Dic₄₃	—	4	4	no	—
55	C₄⋊Dic₄₃	—	4	4	no	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	—	2	4	no	—
58	C₂×C₄₃⋊C₈	Q₁₆	1	8	yes	—
59	C₂³×C₄₃	#127	1	1	yes	characteristic
60	Q₈×C₄₃	—	4	4	no	—
61	Q₈×C₄₃	—	8	2	no	—
62	C₂×C₄×C₄₃	—	4	2	no	—
63	C₂×C₄×C₄₃	—	8	1	no	—
64	C₂×C₄×C₄₃	—	8	1	no	—
65	C₂×C₄×C₄₃	—	8	1	no	—
66	C₂×C₄×C₄₃	—	8	2	no	—
67	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
70	C₂×C₄×C₄₃	#123	1	2	yes	—
71	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
72	C₂×C₄×C₄₃	C₂×Q₁₆	1	4	yes	—
73	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×C₄×C₄₃	#137	1	2	yes	—
75	C₂×Dic₄₃	—	8	2	no	—
76	C₂×Dic₄₃	—	8	2	no	—
77	C₂×Dic₄₃	—	8	2	no	—
78	C₈×C₄₃	—	2	4	no	—
79	C₄₃⋊C₈	—	2	8	no	—
80	#10674	—	43	1	no	maximal, Sylow
81	C₂²×C₄₃	#399	1	1	yes	characteristic
82	C₂²×C₄₃	#493	1	1	yes	characteristic
83	C₂²×C₄₃	#494	1	2	yes	—
84	C₂²×C₄₃	#496	1	2	yes	—
85	C₂²×C₄₃	#497	1	1	yes	characteristic
86	C₄×C₄₃	—	8	2	no	—
87	C₄×C₄₃	—	16	1	no	—
88	C₄×C₄₃	#474	1	2	yes	—
89	C₄×C₄₃	#477	1	4	yes	—
90	C₄×C₄₃	#480	1	2	yes	—
91	C₄×C₄₃	#484	1	4	yes	—
92	Dic₄₃	—	16	2	no	—
93	#1990	—	43	2	no	—
94	#1956	D₄₃	1	1	yes	characteristic
95	#1845	—	43	1	no	—
96	#1845	—	43	2	no	—
97	#1273	—	43	1	no	—
98	C₂×C₄₃	#1713	1	1	yes	characteristic
99	C₂×C₄₃	#1721	1	1	yes	characteristic
100	C₂×C₄₃	#1723	1	2	yes	—
101	C₂×C₄₃	#2093	1	1	yes	characteristic
102	C₂×C₄₃	#2095	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
104	C₂×#107	—	86	4	no	—
105	C₂×#107	—	86	2	no	—
106	C₂×#103	—	2	2	no	—
107	C₂×C₄×C₈	—	43	2	no	—
108	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
109	#384	—	43	2	no	—
110	#384	C₂×D₄₃	1	1	yes	characteristic
111	#340	—	86	4	no	—
112	C₄₃	#10674	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	—	2	1	no	—
114	C₂²×C₈	—	43	4	no	—
115	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	2	2	no	—
118	C₄⋊Q₈	Dic₈₆	1	2	yes	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	1	no	—
122	C₂×C₄⋊C₄	—	86	2	no	—
123	C₂×C₄⋊C₄	—	86	2	no	—
124	C₂×C₄⋊C₄	—	86	2	no	—
125	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
126	#107	—	172	8	no	—
127	#107	—	172	4	no	—
128	#103	—	4	4	no	—
129	C₄×C₈	—	2	2	no	—
130	C₄×C₈	—	86	2	no	—
131	C₄×C₈	—	86	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	1	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	4	2	no	—
136	C₂²×C₄	—	4	1	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	—	172	2	no	—
139	C₂²×C₄	D₄×D₄₃	1	2	yes	—
140	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	43	8	no	—
144	C₂×C₈	—	86	4	no	—
145	C₂×C₈	—	86	4	no	—
146	C₂×C₈	#10621882	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	172	4	no	—
152	C₄⋊C₄	—	172	4	no	—
153	C₄⋊C₄	—	172	4	no	—
154	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
155	C₄²	#10621898	1	1	yes	characteristic
156	C₄²	#10621900	1	2	yes	—
157	C₂³	#13063321	1	1	yes	characteristic
158	Q₈	—	4	4	no	—
159	Q₈	—	8	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	8	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	—	344	2	no	—
166	C₂×C₄	—	344	2	no	—
167	C₂×C₄	—	344	2	no	—
168	C₂×C₄	#13063302	1	2	yes	—
169	C₂×C₄	#13063304	1	1	yes	characteristic
170	C₂×C₄	#13063306	1	1	yes	characteristic
171	C₂×C₄	#13063307	1	2	yes	—
172	C₂×C₄	C₂×#10621882	1	2	yes	—
173	C₂×C₄	Q₁₆×D₄₃	1	4	yes	—
174	C₂×C₄	#13063352	1	4	yes	—
175	C₂×C₄	#13063355	1	2	yes	—
176	C₈	—	2	4	no	—
177	C₈	—	86	8	no	—
178	C₂²	#14582835	1	1	yes	characteristic
179	C₂²	#14583275	1	1	yes	characteristic
180	C₂²	#14583277	1	2	yes	—
181	C₂²	#14583285	1	2	yes	—
182	C₂²	#14583294	1	1	yes	characteristic
183	C₄	—	8	2	no	—
184	C₄	—	16	1	no	—
185	C₄	—	688	2	no	—
186	C₄	#14583179	1	2	yes	—
187	C₄	#14583192	1	4	yes	—
188	C₄	#14583210	1	2	yes	—
189	C₄	#14583233	1	4	yes	—
190	C₂	#19134211	1	1	yes	characteristic
191	C₂	#19134259	1	1	yes	characteristic
192	C₂	#19134275	1	2	yes	—
193	C₂	#19136123	1	1	yes	characteristic
194	C₂	#19136131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135557

C₂

yes

maximal

#19134755

C₂

yes

maximal, characteristic

#19134754

C₂

yes

maximal

C₄₃⋊#1273

C₂

yes

maximal, characteristic

C₄₃×#1956

C₂

yes

maximal, characteristic

C₂×#13063259

—

C₂×#13063258

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582769

C₂²

yes

—

#14582661

—

C₂×C₄⋊Q₈×C₄₃

C₂²

yes

characteristic

C₂×#103×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#384

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063258

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×Q₈×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

—

C₄⋊Q₈×C₄₃

Q₈

yes

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#103×C₄₃

—

C₄×C₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

C₄²×C₄₃

yes

C₄²×C₄₃

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

Q₁₆

yes

—

C₂³×C₄₃

#127

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

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

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#496

yes

—

C₂²×C₄₃

#497

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#474

yes

—

C₄×C₄₃

#477

yes

—

C₄×C₄₃

#480

yes

—

C₄×C₄₃

#484

yes

—

Dic₄₃

—

#1990

—

#1956

D₄₃

yes

—

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

100

C₂×C₄₃

#1723

yes

—

101

C₂×C₄₃

#2093

yes

characteristic

102

C₂×C₄₃

#2095

yes

—

103

C₂×C₄⋊Q₈

C₂×D₄₃

yes

characteristic

104

C₂×#107

—

105

C₂×#107

—

106

C₂×#103

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

109

#384

—

110

#384

C₂×D₄₃

yes

characteristic

111

#340

—

112

C₄₃

#10674

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

—

114

C₂²×C₈

—

115

C₂²×C₈

C₂²⋊D₄₃

yes

—

116

C₄⋊Q₈

—

117

C₄⋊Q₈

—

118

C₄⋊Q₈

Dic₈₆

yes

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

—

125

C₂×C₄²

C₂²×D₄₃

yes

characteristic

126

#107

—

172

—

127

#107

—

172

—

128

#103

—

129

C₄×C₈

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₂×Q₈

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

—

172

—

139

C₂²×C₄

D₄×D₄₃

yes

—

140

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

#10621882

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

172

—

152

C₄⋊C₄

—

172

—

153

C₄⋊C₄

—

172

—

154

C₄²

C₂×Dic₈₆

yes

characteristic

155

C₄²

#10621898

yes

characteristic

156

C₄²

#10621900

yes

—

157

C₂³

#13063321

yes

characteristic

158

Q₈

—

159

Q₈

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

344

—

166

C₂×C₄

—

344

—

167

C₂×C₄

—

344

—

168

C₂×C₄

#13063302

yes

—

169

C₂×C₄

#13063304

yes

characteristic

170

C₂×C₄

#13063306

yes

characteristic

171

C₂×C₄

#13063307

yes

—

172

C₂×C₄

C₂×#10621882

yes

—

173

C₂×C₄

Q₁₆×D₄₃

yes

—

174

C₂×C₄

#13063352

yes

—

175

C₂×C₄

#13063355

yes

—

176

C₈

—

177

C₈

—

178

C₂²

#14582835

yes

characteristic

179

C₂²

#14583275

yes

characteristic

180

C₂²

#14583277

yes

—

181

C₂²

#14583285

yes

—

182

C₂²

#14583294

yes

characteristic

183

C₄

—

184

C₄

—

185

C₄

—

688

—

186

C₄

#14583179

yes

—

187

C₄

#14583192

yes

—

188

C₄

#14583210

yes

—

189

C₄

#14583233

yes

—

190

C₂

#19134211

yes

characteristic

191

C₂

#19134259

yes

characteristic

192

C₂

#19134275

yes

—

193

C₂

#19136123

yes

characteristic

194

C₂

#19136131

yes

—

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1956	C₂
socle, prime radical	C₂³×C₄₃	#127
hypercenter	#1956	D₄₃
hypocenter	C₄₃	#10674
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

subgroup

G/H

center

C₂³

#13063321

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1956

C₂

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1956

N	H ≅ G/N	type
112. C₄₃	#10674	semidirect

H ≅ G/N

type

112. C₄₃

#10674

semidirect

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

subgroup	index
#19135557	2
#19134755	2
#19134754	2
C₄₃⋊#1273	2
C₄₃×#1956	2
#10674	43

subgroup

index

#19135557

#19134755

#19134754

C₄₃⋊#1273

C₄₃×#1956

#10674

Subgroups — #25001649

Conjugacy classes of subgroups

Complements

Maximal subgroups