Group #24991562 — subgroups

Subgroups — #24991562

Order 11008.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#19143701	C₂	1	1	yes	maximal, characteristic, complemented
2	#19133260	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133064	C₂	1	2	yes	maximal
4	#19132494	C₂	1	2	yes	maximal, complemented
5	C₄₃×#3135	C₂	1	1	yes	maximal, characteristic
6	#14583030	—	2	4	no	—
7	#14582960	—	2	2	no	—
8	#14582956	C₂²	1	2	yes	—
9	#14582952	C₂²	1	2	yes	—
10	#14582645	—	2	2	no	—
11	D₄⋊(C₄₃⋊C₈)	—	2	4	no	—
12	C₄₃×#533	C₂²	1	1	yes	characteristic
13	C₄₃×#532	C₂²	1	1	yes	characteristic
14	C₄₃×#491	C₄	1	2	yes	—
15	C₄₃×#490	C₄	1	2	yes	—
16	C₄₃×#489	—	2	2	no	—
17	C₄₃×#416	—	2	2	no	—
18	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
19	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
22	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
24	#13063236	—	2	2	no	—
25	#13063236	—	2	2	no	—
26	C₄×C₄₃⋊C₈	—	2	2	no	—
27	C₄×C₄₃⋊C₈	—	2	2	no	—
28	C₂×#41×C₄₃	—	2	1	no	—
29	C₂²×C₈×C₄₃	—	2	1	no	—
30	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
32	#127×C₄₃	C₂×C₄	1	1	yes	characteristic
33	#125×C₄₃	C₂×C₄	1	2	yes	—
34	#124×C₄₃	—	2	1	no	—
35	C₂²⋊Q₈×C₄₃	—	2	2	no	—
36	#121×C₄₃	—	2	2	no	—
37	C₄×Q₈×C₄₃	—	2	2	no	—
38	C₄×D₄×C₄₃	—	2	2	no	—
39	#117×C₄₃	D₄	1	2	yes	—
40	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
41	#103×C₄₃	—	2	4	no	—
42	#102×C₄₃	—	2	4	no	—
43	#97×C₄₃	—	2	1	no	—
44	#97×C₄₃	C₂×C₄	1	2	yes	—
45	#41×C₄₃	—	4	4	no	—
46	C₂×Q₈×C₄₃	—	2	2	no	—
47	C₂×Q₈×C₄₃	—	2	1	no	—
48	C₂×D₄×C₄₃	—	2	1	no	—
49	C₂×D₄×C₄₃	—	2	2	no	—
50	C₂×D₄×C₄₃	—	4	1	no	—
51	C₂²×C₄×C₄₃	—	4	1	no	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	#41	1	2	yes	—
54	C₂×C₈×C₄₃	—	2	2	no	—
55	C₂×C₈×C₄₃	—	4	2	no	—
56	C₄⋊C₄×C₄₃	—	2	4	no	—
57	C₄⋊C₄×C₄₃	—	2	2	no	—
58	C₄⋊C₄×C₄₃	—	2	2	no	—
59	C₄⋊C₄×C₄₃	C₂²⋊C₄	1	4	yes	—
60	C₂²⋊C₄×C₄₃	—	2	2	no	—
61	C₂²⋊C₄×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₄₃	C₂×D₄	1	2	yes	—
65	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄₃⋊C₈	—	2	4	no	—
67	C₂×C₄₃⋊C₈	—	4	2	no	—
68	C₂³×C₄₃	—	4	1	no	—
69	C₂³×C₄₃	#123	1	1	yes	characteristic
70	Q₈×C₄₃	—	4	2	no	—
71	Q₈×C₄₃	—	8	2	no	—
72	D₄×C₄₃	—	4	4	no	—
73	D₄×C₄₃	—	4	2	no	—
74	D₄×C₄₃	—	8	2	no	—
75	C₂×C₄×C₄₃	—	2	2	no	—
76	C₂×C₄×C₄₃	—	2	4	no	—
77	C₂×C₄×C₄₃	—	4	2	no	—
78	C₂×C₄×C₄₃	—	4	1	no	—
79	C₂×C₄×C₄₃	—	8	2	no	—
80	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
82	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
83	C₂×C₄×C₄₃	#121	1	2	yes	—
84	C₈×C₄₃	—	4	2	no	—
85	C₄₃⋊C₈	—	4	4	no	—
86	C₄₃⋊C₈	—	8	2	no	—
87	#9008	—	43	1	no	maximal, Sylow
88	C₂²×C₄₃	—	2	2	no	—
89	C₂²×C₄₃	—	2	1	no	—
90	C₂²×C₄₃	—	8	2	no	—
91	C₂²×C₄₃	—	8	1	no	—
92	C₂²×C₄₃	#386	1	1	yes	characteristic
93	C₄×C₄₃	—	2	2	no	—
94	C₄×C₄₃	—	2	2	no	—
95	C₄×C₄₃	—	2	1	no	—
96	C₄×C₄₃	—	4	4	no	—
97	C₄×C₄₃	—	8	1	no	—
98	C₄×C₄₃	#454	1	2	yes	—
99	#3135	D₄₃	1	1	yes	characteristic
100	#2987	—	43	1	no	—
101	#1544	—	43	1	no	—
102	#1496	—	43	2	no	—
103	#1302	—	43	2	no	—
104	C₂×C₄₃	—	4	1	no	—
105	C₂×C₄₃	—	16	1	no	—
106	C₂×C₄₃	#1881	1	2	yes	—
107	C₂×C₄₃	#1888	1	1	yes	characteristic
108	#533	C₂×D₄₃	1	1	yes	characteristic
109	#532	C₂×D₄₃	1	1	yes	characteristic
110	#491	Dic₄₃	1	2	yes	—
111	#490	Dic₄₃	1	2	yes	—
112	#489	—	2	2	no	—
113	#446	—	86	4	no	—
114	#433	—	86	2	no	—
115	#432	—	43	2	no	—
116	#431	—	43	2	no	—
117	#416	—	2	2	no	—
118	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
119	#332	—	86	2	no	—
120	D₄⋊C₈	—	86	4	no	—
121	C₄₃	#9008	1	1	yes	characteristic, complemented, Sylow
122	C₂×#41	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic
125	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
126	#127	C₂×Dic₄₃	1	1	yes	characteristic
127	#125	C₂×Dic₄₃	1	2	yes	—
128	#124	—	2	1	no	—
129	C₂²⋊Q₈	—	2	2	no	—
130	#121	—	2	2	no	—
131	C₄×Q₈	—	2	2	no	—
132	C₄×D₄	—	2	2	no	—
133	#117	C₂²⋊D₄₃	1	2	yes	—
134	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
135	C₄⋊C₈	—	43	4	no	—
136	C₄⋊C₈	—	86	4	no	—
137	C₄⋊C₈	—	86	2	no	—
138	#103	—	2	4	no	—
139	#102	—	2	4	no	—
140	C₂²⋊C₈	—	86	2	no	—
141	C₂²⋊C₈	—	86	2	no	—
142	#97	—	2	1	no	—
143	#97	—	86	2	no	—
144	#97	—	86	2	no	—
145	#97	C₂×Dic₄₃	1	2	yes	—
146	C₄×C₈	—	86	2	no	—
147	C₄×C₈	—	86	2	no	—
148	#41	—	4	4	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	—	2	1	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	1	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	#10621900	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	86	4	no	—
160	C₂×C₈	—	172	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	C₂²⋊Dic₄₃	1	4	yes	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	D₄×D₄₃	1	2	yes	—
169	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
170	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
171	C₂³	—	4	1	no	—
172	C₂³	#13063306	1	1	yes	characteristic
173	Q₈	—	4	2	no	—
174	Q₈	—	8	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
185	C₂×C₄	#13063293	1	1	yes	characteristic
186	C₂×C₄	#13063297	1	2	yes	—
187	C₈	—	4	2	no	—
188	C₈	—	344	2	no	—
189	C₈	—	172	4	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	8	1	no	—
193	C₂²	—	8	2	no	—
194	C₂²	#14582779	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	4	no	—
199	C₄	—	8	1	no	—
200	C₄	#14583071	1	2	yes	—
201	C₂	—	4	1	no	—
202	C₂	—	16	1	no	—
203	C₂	#19134959	1	2	yes	—
204	C₂	#19134999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19143701

C₂

yes

maximal, characteristic, complemented

#19133260

C₂

yes

maximal, characteristic, complemented

#19133064

C₂

yes

maximal

#19132494

C₂

yes

maximal, complemented

C₄₃×#3135

C₂

yes

maximal, characteristic

#14583030

—

#14582960

—

#14582956

C₂²

yes

—

#14582952

C₂²

yes

—

#14582645

—

D₄⋊(C₄₃⋊C₈)

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#532

C₂²

yes

characteristic

C₄₃×#491

C₄

yes

—

C₄₃×#490

C₄

yes

—

C₄₃×#489

—

C₄₃×#416

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

C₂²⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂×#41×C₄₃

—

C₂²×C₈×C₄₃

—

C₄⋊Q₈×C₄₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₄₃

D₄

yes

—

#127×C₄₃

C₂×C₄

yes

characteristic

#125×C₄₃

C₂×C₄

yes

—

#124×C₄₃

—

C₂²⋊Q₈×C₄₃

—

#121×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×D₄×C₄₃

—

#117×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#103×C₄₃

—

#102×C₄₃

—

#97×C₄₃

—

#97×C₄₃

C₂×C₄

yes

—

#41×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

#123

yes

characteristic

Q₈×C₄₃

—

Q₈×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9008

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#386

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#454

yes

—

#3135

D₄₃

yes

characteristic

100

#2987

—

101

#1544

—

102

#1496

—

103

#1302

—

104

C₂×C₄₃

—

105

C₂×C₄₃

—

106

C₂×C₄₃

#1881

yes

—

107

C₂×C₄₃

#1888

yes

characteristic

108

#533

C₂×D₄₃

yes

characteristic

109

#532

C₂×D₄₃

yes

characteristic

110

#491

Dic₄₃

yes

—

111

#490

Dic₄₃

yes

—

112

#489

—

113

#446

—

114

#433

—

115

#432

—

116

#431

—

117

#416

—

118

C₂×#97

C₂×D₄₃

yes

characteristic

119

#332

—

120

D₄⋊C₈

—

121

C₄₃

#9008

yes

characteristic, complemented, Sylow

122

C₂×#41

—

123

C₂²×C₈

—

124

C₄⋊Q₈

C₂×Dic₄₃

yes

characteristic

125

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

126

#127

C₂×Dic₄₃

yes

characteristic

127

#125

C₂×Dic₄₃

yes

—

128

#124

—

129

C₂²⋊Q₈

—

130

#121

—

131

C₄×Q₈

—

132

C₄×D₄

—

133

#117

C₂²⋊D₄₃

yes

—

134

C₂×C₄²

C₂²×D₄₃

yes

characteristic

135

C₄⋊C₈

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

#103

—

139

#102

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

142

#97

—

143

#97

—

144

#97

—

145

#97

C₂×Dic₄₃

yes

—

146

C₄×C₈

—

147

C₄×C₈

—

148

#41

—

149

C₂×Q₈

—

150

C₂×Q₈

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂²×C₄

—

155

C₂²×C₄

#10621900

yes

—

156

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

172

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

C₂²⋊Dic₄₃

yes

—

165

C₂²⋊C₄

—

166

C₂²⋊C₄

—

167

C₄²

—

168

C₄²

D₄×D₄₃

yes

—

169

C₄²

C₂²×Dic₄₃

yes

characteristic

170

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

171

C₂³

—

172

C₂³

#13063306

yes

characteristic

173

Q₈

—

174

Q₈

—

175

D₄

—

176

D₄

—

177

D₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

C₂×C₂²⋊Dic₄₃

yes

characteristic

184

C₂×C₄

D₄×Dic₄₃

yes

—

185

C₂×C₄

#13063293

yes

characteristic

186

C₂×C₄

#13063297

yes

—

187

C₈

—

188

C₈

—

344

—

189

C₈

—

172

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

#14582779

yes

characteristic

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

#14583071

yes

—

201

C₂

—

202

C₂

—

203

C₂

#19134959

yes

—

204

C₂

#19134999

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063293
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3135	C₂
socle	C₂²×C₄₃	#386
prime radical	C₄₃×#532	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#3135	D₄₃
hypocenter	C₄₃	#9008
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

subgroup

G/H

center

C₂×C₄

#13063293

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#3135

C₂

socle

C₂²×C₄₃

#386

prime radical

C₄₃×#532

C₂²

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3135

N	H ≅ G/N	type
1. #19143701	C₂	semidirect
2. #19133260	C₂	semidirect
4. #19132494	C₂	semidirect
121. C₄₃	#9008	semidirect

H ≅ G/N

type

1. #19143701

C₂

semidirect

2. #19133260

C₂

semidirect

4. #19132494

C₂

semidirect

121. C₄₃

#9008

semidirect

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

subgroup	index
#19143701	2
#19133260	2
#19133064	2
#19132494	2
C₄₃×#3135	2
#9008	43

subgroup

index

C₄₃×#3135

Subgroups — #24991562

Conjugacy classes of subgroups

Complements

Maximal subgroups