Group #25001050 — subgroups

Subgroups — #25001050

Order 11008.

406 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 129 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	#19139837	C₂	1	1	yes	maximal, characteristic
2	#19135359	C₂	1	2	yes	maximal
3	#19134754	C₂	1	1	yes	maximal, characteristic
4	#19132434	C₂	1	2	yes	maximal
5	C₄₃×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063258	—	2	2	no	—
7	C₂×#13063257	—	2	2	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
9	#14582838	—	2	2	no	—
10	#14582798	—	2	2	no	—
11	#14582769	C₂²	1	1	yes	characteristic
12	#14582764	—	2	2	no	—
13	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
14	#14582662	—	2	2	no	—
15	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#384	C₂²	1	1	yes	characteristic
17	C₂²×C₄₃⋊C₈	—	2	1	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	1	no	—
20	C₂×C₄⋊Dic₄₃	—	2	2	no	—
21	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
22	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₄₃	—	2	2	no	—
24	#13063258	—	4	4	no	—
25	#13063257	—	4	4	no	—
26	C₄×C₄₃⋊C₈	—	2	1	no	—
27	C₄×C₄₃⋊C₈	Q₈	1	2	yes	—
28	C₄₃⋊#95	—	2	4	no	—
29	C₄₃⋊#95	—	2	2	no	—
30	C₄₃⋊#95	—	2	2	no	—
31	C₂²×C₈×C₄₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₃	D₄	1	2	yes	—
34	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₄₃	—	2	2	no	—
36	C₄⋊C₈×C₄₃	Q₈	1	4	yes	—
37	C₂²×C₄×C₄₃	—	2	2	no	—
38	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
40	C₂²×Dic₄₃	—	2	2	no	—
41	C₂²×Dic₄₃	—	2	2	no	—
42	C₂²×Dic₄₃	—	4	1	no	—
43	C₂×C₈×C₄₃	—	2	4	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₂×C₈×C₄₃	—	4	2	no	—
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄⋊C₄×C₄₃	—	4	2	no	—
49	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
50	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₄₃	#41	1	1	yes	characteristic
52	C₄⋊Dic₄₃	—	2	2	no	—
53	C₄⋊Dic₄₃	—	2	2	no	—
54	C₄⋊Dic₄₃	—	2	2	no	—
55	C₄⋊Dic₄₃	—	2	2	no	—
56	C₄⋊Dic₄₃	—	4	2	no	—
57	C₄⋊Dic₄₃	—	4	2	no	—
58	C₄⋊Dic₄₃	—	4	2	no	—
59	C₄⋊Dic₄₃	—	8	2	no	—
60	C₄×Dic₄₃	—	4	2	no	—
61	C₄×Dic₄₃	—	4	2	no	—
62	C₂×C₄₃⋊C₈	—	2	2	no	—
63	C₂×C₄₃⋊C₈	—	2	2	no	—
64	C₂×C₄₃⋊C₈	—	2	2	no	—
65	C₂³×C₄₃	#120	1	1	yes	characteristic
66	C₂×C₄×C₄₃	—	2	1	no	—
67	C₂×C₄×C₄₃	—	2	1	no	—
68	C₂×C₄×C₄₃	—	4	4	no	—
69	C₂×C₄×C₄₃	—	4	2	no	—
70	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₄₃	#135	1	2	yes	—
75	C₂×C₄×C₄₃	#135	1	2	yes	—
76	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄×C₄₃	#137	1	2	yes	—
78	C₂×Dic₄₃	—	4	2	no	—
79	C₂×Dic₄₃	—	4	2	no	—
80	C₂×Dic₄₃	—	4	2	no	—
81	C₂×Dic₄₃	—	4	2	no	—
82	C₂×Dic₄₃	—	4	2	no	—
83	C₂×Dic₄₃	—	4	2	no	—
84	C₂×Dic₄₃	—	8	2	no	—
85	C₂×Dic₄₃	—	8	1	no	—
86	C₈×C₄₃	—	4	4	no	—
87	C₄₃⋊C₈	—	2	4	no	—
88	#10576	—	43	1	no	maximal, Sylow
89	C₂²×C₄₃	#395	1	1	yes	characteristic
90	C₂²×C₄₃	#449	1	2	yes	—
91	C₂²×C₄₃	#452	1	2	yes	—
92	C₂²×C₄₃	#453	1	1	yes	characteristic
93	C₂²×C₄₃	#456	1	1	yes	characteristic
94	C₄×C₄₃	—	2	1	no	—
95	C₄×C₄₃	—	2	1	no	—
96	C₄×C₄₃	—	8	2	no	—
97	C₄×C₄₃	#478	1	2	yes	—
98	C₄×C₄₃	#479	1	2	yes	—
99	C₄×C₄₃	#499	1	2	yes	—
100	C₄×C₄₃	#501	1	2	yes	—
101	Dic₄₃	—	8	2	no	—
102	Dic₄₃	—	8	2	no	—
103	Dic₄₃	—	16	1	no	—
104	#2571	—	43	1	no	—
105	#1948	—	43	2	no	—
106	#1845	—	43	1	no	—
107	#1296	D₄₃	1	1	yes	characteristic
108	#1279	—	43	2	no	—
109	C₂×C₄₃	#1620	1	1	yes	characteristic
110	C₂×C₄₃	#1634	1	1	yes	characteristic
111	C₂×C₄₃	#1650	1	2	yes	—
112	C₂×C₄₃	#2059	1	2	yes	—
113	C₂×C₄₃	#2060	1	1	yes	characteristic
114	C₂×#107	—	86	2	no	—
115	C₂×#106	—	86	2	no	—
116	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
117	C₂×C₄×C₈	—	43	1	no	—
118	#400	—	86	2	no	—
119	#389	—	86	2	no	—
120	#384	—	43	1	no	—
121	#384	C₂×D₄₃	1	1	yes	characteristic
122	#382	—	86	2	no	—
123	C₄×C₄⋊C₄	—	43	2	no	—
124	#340	—	86	2	no	—
125	C₄₃	#10576	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	2	2	no	—
127	C₂²×C₈	—	86	1	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	43	2	no	—
130	C₂×C₄⋊C₄	—	43	2	no	—
131	C₂×C₄⋊C₄	—	86	2	no	—
132	C₂×C₄⋊C₄	—	86	1	no	—
133	C₂×C₄⋊C₄	—	86	2	no	—
134	C₂×C₄⋊C₄	D₁₇₂	1	2	yes	—
135	C₂×C₄²	—	86	2	no	—
136	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
137	#107	—	172	4	no	—
138	#106	—	172	4	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	Dic₈₆	1	4	yes	—
141	C₄×C₈	—	43	2	no	—
142	C₄×C₈	—	86	1	no	—
143	#95	—	86	2	no	—
144	#95	—	86	4	no	—
145	#95	—	86	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	86	2	no	—
148	C₂²×C₄	—	86	2	no	—
149	C₂²×C₄	—	172	1	no	—
150	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
151	C₂²×C₄	D₄×D₄₃	1	2	yes	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	86	2	no	—
156	C₂×C₈	—	86	2	no	—
157	C₂×C₈	—	86	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	86	2	no	—
162	C₄⋊C₄	—	86	2	no	—
163	C₄⋊C₄	—	86	2	no	—
164	C₄⋊C₄	—	86	2	no	—
165	C₄⋊C₄	—	344	2	no	—
166	C₄⋊C₄	—	172	2	no	—
167	C₄⋊C₄	—	172	2	no	—
168	C₄⋊C₄	—	172	2	no	—
169	C₄²	—	172	2	no	—
170	C₄²	—	172	2	no	—
171	C₄²	C₂×Dic₈₆	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	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	344	2	no	—
180	C₂×C₄	—	344	1	no	—
181	C₂×C₄	—	172	2	no	—
182	C₂×C₄	—	172	2	no	—
183	C₂×C₄	—	172	2	no	—
184	C₂×C₄	—	172	2	no	—
185	C₂×C₄	—	172	2	no	—
186	C₂×C₄	—	172	2	no	—
187	C₂×C₄	#13063302	1	2	yes	—
188	C₂×C₄	#13063305	1	1	yes	characteristic
189	C₂×C₄	#13063324	1	2	yes	—
190	C₂×C₄	#13063325	1	1	yes	characteristic
191	C₂×C₄	#13063346	1	2	yes	—
192	C₂×C₄	#13063346	1	2	yes	—
193	C₂×C₄	#13063349	1	2	yes	—
194	C₂×C₄	#13063357	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	86	4	no	—
197	C₂²	#14582820	1	1	yes	characteristic
198	C₂²	#14583045	1	2	yes	—
199	C₂²	#14583064	1	2	yes	—
200	C₂²	#14583069	1	1	yes	characteristic
201	C₂²	#14583084	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	8	2	no	—
205	C₄	—	344	2	no	—
206	C₄	—	344	2	no	—
207	C₄	—	688	1	no	—
208	C₄	#14583201	1	2	yes	—
209	C₄	#14583204	1	2	yes	—
210	C₄	#14583299	1	2	yes	—
211	C₄	#14583307	1	2	yes	—
212	C₂	#19133655	1	1	yes	characteristic
213	C₂	#19133741	1	1	yes	characteristic
214	C₂	#19133852	1	2	yes	—
215	C₂	#19135932	1	2	yes	—
216	C₂	#19135941	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19139837

C₂

yes

maximal, characteristic

#19135359

C₂

yes

maximal

#19134754

C₂

yes

maximal, characteristic

#19132434

C₂

yes

maximal

C₄₃×#1296

C₂

yes

maximal, characteristic

C₂×#13063258

—

C₂×#13063257

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

C₂²

yes

characteristic

#14582764

—

C₄×C₄⋊Dic₄₃

C₂²

yes

—

#14582662

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₄₃×#384

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

C₂×C₄×Dic₄₃

—

#13063258

—

#13063257

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

Q₈

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

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

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#452

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#478

yes

—

C₄×C₄₃

#479

yes

—

C₄×C₄₃

#499

yes

—

100

C₄×C₄₃

#501

yes

—

101

Dic₄₃

—

102

Dic₄₃

—

103

Dic₄₃

—

104

#2571

—

105

#1948

—

106

#1845

—

107

#1296

D₄₃

yes

characteristic

108

#1279

—

109

C₂×C₄₃

#1620

yes

characteristic

110

C₂×C₄₃

#1634

yes

characteristic

111

C₂×C₄₃

#1650

yes

—

112

C₂×C₄₃

#2059

yes

—

113

C₂×C₄₃

#2060

yes

characteristic

114

C₂×#107

—

115

C₂×#106

—

116

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

117

C₂×C₄×C₈

—

118

#400

—

119

#389

—

120

#384

—

121

#384

C₂×D₄₃

yes

characteristic

122

#382

—

123

C₄×C₄⋊C₄

—

124

#340

—

125

C₄₃

#10576

yes

characteristic, complemented, Sylow

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄⋊C₄

—

134

C₂×C₄⋊C₄

D₁₇₂

yes

—

135

C₂×C₄²

—

136

C₂×C₄²

C₂²×D₄₃

yes

characteristic

137

#107

—

172

—

138

#106

—

172

—

139

C₄⋊C₈

—

140

C₄⋊C₈

Dic₈₆

yes

—

141

C₄×C₈

—

142

C₄×C₈

—

143

#95

—

144

#95

—

145

#95

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

172

—

150

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

151

C₂²×C₄

D₄×D₄₃

yes

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

344

—

166

C₄⋊C₄

—

172

—

167

C₄⋊C₄

—

172

—

168

C₄⋊C₄

—

172

—

169

C₄²

—

172

—

170

C₄²

—

172

—

171

C₄²

C₂×Dic₈₆

yes

—

172

C₄²

#10621900

yes

characteristic

173

C₄²

Q₈×D₄₃

yes

characteristic

174

C₂³

#13063295

yes

characteristic

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

344

—

180

C₂×C₄

—

344

—

181

C₂×C₄

—

172

—

182

C₂×C₄

—

172

—

183

C₂×C₄

—

172

—

184

C₂×C₄

—

172

—

185

C₂×C₄

—

172

—

186

C₂×C₄

—

172

—

187

C₂×C₄

#13063302

yes

—

188

C₂×C₄

#13063305

yes

characteristic

189

C₂×C₄

#13063324

yes

—

190

C₂×C₄

#13063325

yes

characteristic

191

C₂×C₄

#13063346

yes

—

192

C₂×C₄

#13063346

yes

—

193

C₂×C₄

#13063349

yes

—

194

C₂×C₄

#13063357

yes

—

195

C₈

—

196

C₈

—

197

C₂²

#14582820

yes

characteristic

198

C₂²

#14583045

yes

—

199

C₂²

#14583064

yes

—

200

C₂²

#14583069

yes

characteristic

201

C₂²

#14583084

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

344

—

206

C₄

—

344

—

207

C₄

—

688

—

208

C₄

#14583201

yes

—

209

C₄

#14583204

yes

—

210

C₄

#14583299

yes

—

211

C₄

#14583307

yes

—

212

C₂

#19133655

yes

characteristic

213

C₂

#19133741

yes

characteristic

214

C₂

#19133852

yes

—

215

C₂

#19135932

yes

—

216

C₂

#19135941

yes

characteristic

	subgroup	G/H
center	C₂³	#13063295
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1296	C₂
socle, prime radical	C₂³×C₄₃	#120
hypercenter	#1296	D₄₃
hypocenter	C₄₃	#10576
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

subgroup

G/H

center

C₂³

#13063295

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1296

C₂

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

N	H ≅ G/N	type
125. C₄₃	#10576	semidirect

H ≅ G/N

type

125. C₄₃

#10576

semidirect

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

subgroup	index
#19139837	2
#19135359	2
#19134754	2
#19132434	2
C₄₃×#1296	2
#10576	43

subgroup

index

C₄₃×#1296

Subgroups — #25001050

Conjugacy classes of subgroups

Complements

Maximal subgroups