Group #24999270 — subgroups

Subgroups — #24999270

Order 11008.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19145517	C₂	1	2	yes	maximal
2	#19142973	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133277	C₂	1	2	yes	maximal, complemented
4	#19132428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1962	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	2	yes	—
7	#14583271	—	2	2	no	—
8	#14583251	—	2	2	no	—
9	#14583104	—	2	4	no	—
10	#14583037	—	2	4	no	—
11	#14582952	C₂²	1	2	yes	—
12	#14582805	—	2	4	no	—
13	#14582765	—	2	2	no	—
14	#14582647	—	2	4	no	—
15	C₄₃×#532	C₂²	1	1	yes	characteristic
16	C₄₃×#423	C₂²	1	1	yes	characteristic
17	C₂×#104×C₄₃	—	2	2	no	—
18	C₄³×C₄₃	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₆	D₄	1	4	yes	—
20	C₂×#10621883	—	2	8	no	—
21	#13063325	—	2	2	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	2	2	no	—
25	C₂×C₄⋊Dic₄₃	—	2	2	no	—
26	C₂×C₄⋊Dic₄₃	—	2	2	no	—
27	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
28	#13063248	—	2	8	no	—
29	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₄×C₄₃⋊C₈	—	2	2	no	—
31	C₄₃⋊#95	—	2	8	no	—
32	C₂×#41×C₄₃	—	2	1	no	—
33	C₂×M₁₆×C₄₃	—	2	1	no	—
34	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
35	#127×C₄₃	D₄	1	1	yes	characteristic
36	#124×C₄₃	—	2	1	no	—
37	#121×C₄₃	—	2	2	no	—
38	C₄×D₄×C₄₃	—	2	2	no	—
39	C₂×C₄²×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	D₄	1	2	yes	—
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₃	—	2	2	no	—
43	#104×C₄₃	—	4	4	no	—
44	#41×C₄₃	—	4	2	no	—
45	#41×C₄₃	—	4	2	no	—
46	C₂×Q₈×C₄₃	—	2	1	no	—
47	C₂×D₄×C₄₃	—	2	2	no	—
48	C₂×D₄×C₄₃	—	2	1	no	—
49	C₂×D₄×C₄₃	—	4	1	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	4	1	no	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₄₃	—	2	4	no	—
55	C₂×Dic₈₆	—	2	4	no	—
56	C₂×Dic₈₆	—	2	4	no	—
57	C₂×Dic₈₆	—	4	8	no	—
58	M₁₆×C₄₃	—	4	2	no	—
59	C₂×C₈×C₄₃	—	4	1	no	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₂²⋊C₄×C₄₃	—	4	2	no	—
62	C₄²×C₄₃	—	2	2	no	—
63	C₄²×C₄₃	—	4	4	no	—
64	C₄²×C₄₃	SD₁₆	1	4	yes	complemented
65	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₄₃	C₂×D₄	1	2	yes	—
67	C₄²×C₄₃	#41	1	1	yes	characteristic
68	#10621883	—	8	8	no	—
69	C₄⋊Dic₄₃	—	4	2	no	—
70	C₄⋊Dic₄₃	—	4	2	no	—
71	C₄⋊Dic₄₃	—	4	2	no	—
72	C₄⋊Dic₄₃	—	4	2	no	—
73	C₂×C₄₃⋊C₈	—	2	4	no	—
74	C₂³×C₄₃	—	4	1	no	—
75	C₂³×C₄₃	#120	1	1	yes	characteristic
76	Q₈×C₄₃	—	4	2	no	—
77	D₄×C₄₃	—	4	4	no	—
78	D₄×C₄₃	—	4	2	no	—
79	D₄×C₄₃	—	8	2	no	—
80	C₂×C₄×C₄₃	—	2	4	no	—
81	C₂×C₄×C₄₃	—	2	4	no	—
82	C₂×C₄×C₄₃	—	4	4	no	—
83	C₂×C₄×C₄₃	—	4	1	no	—
84	C₂×C₄×C₄₃	—	8	1	no	—
85	C₂×C₄×C₄₃	—	8	1	no	—
86	C₂×C₄×C₄₃	#120	1	2	yes	—
87	C₂×C₄×C₄₃	#121	1	2	yes	—
88	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₄₃	#137	1	2	yes	—
92	C₂×Dic₄₃	—	4	8	no	—
93	C₂×Dic₄₃	—	4	4	no	—
94	Dic₈₆	—	4	8	no	—
95	Dic₈₆	—	8	4	no	—
96	C₈×C₄₃	—	8	1	no	—
97	C₄₃⋊C₈	—	4	4	no	—
98	#10287	—	43	1	no	maximal, Sylow
99	C₂²×C₄₃	—	2	2	no	—
100	C₂²×C₄₃	—	8	2	no	—
101	C₂²×C₄₃	—	8	1	no	—
102	C₂²×C₄₃	#392	1	1	yes	characteristic
103	C₂²×C₄₃	#448	1	2	yes	—
104	C₄×C₄₃	—	2	2	no	—
105	C₄×C₄₃	—	4	4	no	—
106	C₄×C₄₃	—	8	1	no	—
107	C₄×C₄₃	#455	1	4	yes	—
108	C₄×C₄₃	#461	1	2	yes	—
109	C₄×C₄₃	#492	1	1	yes	characteristic
110	C₄×C₄₃	#497	1	1	yes	characteristic
111	Dic₄₃	—	8	4	no	—
112	#3206	—	43	2	no	—
113	#2862	—	43	1	no	—
114	#1962	D₄₃	1	1	yes	characteristic
115	#1548	—	43	2	no	—
116	#1276	—	43	1	no	—
117	C₂×C₄₃	—	2	2	no	—
118	C₂×C₄₃	—	16	1	no	—
119	C₂×C₄₃	#2004	1	2	yes	—
120	C₂×C₄₃	#2013	1	1	yes	characteristic
121	#532	C₂×D₄₃	1	1	yes	characteristic
122	C₂×C₄⋊Q₈	—	43	2	no	—
123	#492	—	86	2	no	—
124	#487	—	86	2	no	—
125	#461	—	86	4	no	—
126	#448	—	86	4	no	—
127	#431	—	43	2	no	—
128	#423	C₂×D₄₃	1	1	yes	characteristic
129	C₂×#104	—	2	2	no	—
130	#391	—	86	4	no	—
131	#382	—	86	2	no	—
132	C₄³	C₂×D₄₃	1	1	yes	characteristic
133	#333	—	86	4	no	—
134	C₄₃	#10287	1	1	yes	characteristic, complemented, Sylow
135	C₂×#41	—	2	1	no	—
136	C₂²×Q₈	—	43	4	no	—
137	C₂×SD₁₆	—	86	8	no	—
138	C₂×M₁₆	—	2	1	no	—
139	C₄⋊Q₈	—	86	2	no	—
140	C₄⋊Q₈	—	86	2	no	—
141	C₄⋊Q₈	—	86	2	no	—
142	C₄⋊Q₈	—	86	2	no	—
143	C₄⋊Q₈	D₁₇₂	1	1	yes	characteristic
144	#127	D₁₇₂	1	1	yes	characteristic
145	#124	—	2	1	no	—
146	#121	—	2	2	no	—
147	C₄×D₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	86	2	no	—
149	C₂×C₄⋊C₄	—	86	2	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
152	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	43	4	no	—
155	#104	—	4	4	no	—
156	#103	—	86	8	no	—
157	C₂²⋊C₈	—	86	2	no	—
158	C₄×C₈	—	86	2	no	—
159	#95	—	86	8	no	—
160	#41	—	4	2	no	—
161	#41	—	4	2	no	—
162	C₂×Q₈	—	2	1	no	—
163	C₂×Q₈	—	86	4	no	—
164	C₂×Q₈	—	86	4	no	—
165	C₂×Q₈	—	172	8	no	—
166	C₂×D₄	—	2	2	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	4	1	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	—	86	4	no	—
172	C₂²×C₄	D₄×D₄₃	1	2	yes	—
173	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
174	SD₁₆	—	344	8	no	—
175	M₁₆	—	4	2	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	86	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	172	2	no	—
180	C₄⋊C₄	—	172	2	no	—
181	C₄⋊C₄	—	172	2	no	—
182	C₄⋊C₄	—	172	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	4	4	no	—
186	C₄²	#10621883	1	4	yes	complemented
187	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
188	C₄²	#10621898	1	1	yes	characteristic
189	C₄²	D₄×D₄₃	1	2	yes	—
190	C₂³	—	4	1	no	—
191	C₂³	#13063293	1	1	yes	characteristic
192	Q₈	—	4	2	no	—
193	Q₈	—	344	4	no	—
194	Q₈	—	172	8	no	—
195	D₄	—	4	4	no	—
196	D₄	—	4	2	no	—
197	D₄	—	8	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	172	8	no	—
205	C₂×C₄	—	172	4	no	—
206	C₂×C₄	#13063295	1	2	yes	—
207	C₂×C₄	#13063298	1	2	yes	—
208	C₂×C₄	#13063299	1	1	yes	characteristic
209	C₂×C₄	#13063321	1	1	yes	characteristic
210	C₂×C₄	C₂×#10621883	1	2	yes	—
211	C₂×C₄	#13063355	1	2	yes	—
212	C₈	—	8	1	no	—
213	C₈	—	172	4	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	1	no	—
217	C₂²	#14582808	1	1	yes	characteristic
218	C₂²	#14583037	1	2	yes	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	—	8	1	no	—
222	C₄	—	344	4	no	—
223	C₄	#14583078	1	4	yes	—
224	C₄	#14583104	1	2	yes	—
225	C₄	#14583271	1	1	yes	characteristic
226	C₄	#14583294	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	—	16	1	no	—
229	C₂	#19135623	1	2	yes	—
230	C₂	#19135677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145517

C₂

yes

maximal

#19142973

C₂

yes

maximal, characteristic, complemented

#19133277

C₂

yes

maximal, complemented

#19132428

C₂

yes

maximal, characteristic, complemented

C₄₃×#1962

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#14582805

—

#14582765

—

#14582647

—

C₄₃×#532

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

characteristic

C₂×#104×C₄₃

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

D₄

yes

—

—

—

—

—

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063248

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₄₃⋊#95

—

C₂×#41×C₄₃

—

C₂×M₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

D₄

yes

characteristic

#127×C₄₃

D₄

yes

characteristic

#124×C₄₃

—

#121×C₄₃

—

C₄×D₄×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

#104×C₄₃

—

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

SD₁₆

yes

complemented

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

#10621883

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

#120

yes

characteristic

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×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10287

—

maximal, Sylow

C₂²×C₄₃

—

100

C₂²×C₄₃

—

101

C₂²×C₄₃

—

102

C₂²×C₄₃

#392

yes

characteristic

103

C₂²×C₄₃

#448

yes

—

104

C₄×C₄₃

—

105

C₄×C₄₃

—

106

C₄×C₄₃

—

107

C₄×C₄₃

#455

yes

—

108

C₄×C₄₃

#461

yes

—

109

C₄×C₄₃

#492

yes

characteristic

110

C₄×C₄₃

#497

yes

characteristic

111

Dic₄₃

—

112

#3206

—

113

#2862

—

114

#1962

D₄₃

yes

characteristic

115

#1548

—

116

#1276

—

117

C₂×C₄₃

—

118

C₂×C₄₃

—

119

C₂×C₄₃

#2004

yes

—

120

C₂×C₄₃

#2013

yes

characteristic

121

#532

C₂×D₄₃

yes

characteristic

122

C₂×C₄⋊Q₈

—

123

#492

—

124

#487

—

125

#461

—

126

#448

—

127

#431

—

128

#423

C₂×D₄₃

yes

characteristic

129

C₂×#104

—

130

#391

—

131

#382

—

132

C₄³

C₂×D₄₃

yes

characteristic

133

#333

—

134

C₄₃

#10287

yes

characteristic, complemented, Sylow

135

C₂×#41

—

136

C₂²×Q₈

—

137

C₂×SD₁₆

—

138

C₂×M₁₆

—

139

C₄⋊Q₈

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

D₁₇₂

yes

characteristic

144

#127

D₁₇₂

yes

characteristic

145

#124

—

146

#121

—

147

C₄×D₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄²

—

151

C₂×C₄²

C₂²⋊D₄₃

yes

—

152

C₂×C₄²

C₂²×D₄₃

yes

characteristic

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

#104

—

156

#103

—

157

C₂²⋊C₈

—

158

C₄×C₈

—

159

#95

—

160

#41

—

161

#41

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

172

—

166

C₂×D₄

—

167

C₂×D₄

—

168

C₂×D₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

—

172

C₂²×C₄

D₄×D₄₃

yes

—

173

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

174

SD₁₆

—

344

—

175

M₁₆

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₄⋊C₄

—

179

C₄⋊C₄

—

172

—

180

C₄⋊C₄

—

172

—

181

C₄⋊C₄

—

172

—

182

C₄⋊C₄

—

172

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

#10621883

yes

complemented

187

C₄²

C₂×D₁₇₂

yes

characteristic

188

C₄²

#10621898

yes

characteristic

189

C₄²

D₄×D₄₃

yes

—

190

C₂³

—

191

C₂³

#13063293

yes

characteristic

192

Q₈

—

193

Q₈

—

344

—

194

Q₈

—

172

—

195

D₄

—

196

D₄

—

197

D₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

172

—

205

C₂×C₄

—

172

—

206

C₂×C₄

#13063295

yes

—

207

C₂×C₄

#13063298

yes

—

208

C₂×C₄

#13063299

yes

characteristic

209

C₂×C₄

#13063321

yes

characteristic

210

C₂×C₄

C₂×#10621883

yes

—

211

C₂×C₄

#13063355

yes

—

212

C₈

—

213

C₈

—

172

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

#14582808

yes

characteristic

218

C₂²

#14583037

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

344

—

223

C₄

#14583078

yes

—

224

C₄

#14583104

yes

—

225

C₄

#14583271

yes

characteristic

226

C₄

#14583294

yes

characteristic

227

C₂

—

228

C₂

—

229

C₂

#19135623

yes

—

230

C₂

#19135677

yes

characteristic

	subgroup	G/H
center	C₂²	#14582808
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1962	C₂
socle	C₂²×C₄₃	#392
prime radical	C₄₃×#532	C₂²
hypercenter	#1962	D₄₃
hypocenter	C₄₃	#10287
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

subgroup

G/H

center

C₂²

#14582808

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1962

C₂

socle

C₂²×C₄₃

#392

prime radical

C₄₃×#532

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1962

N	H ≅ G/N	type
2. #19142973	C₂	semidirect
3. #19133277	C₂	semidirect
4. #19132428	C₂	semidirect
64. C₄²×C₄₃	SD₁₆	semidirect
134. C₄₃	#10287	semidirect
186. C₄²	#10621883	semidirect

H ≅ G/N

type

2. #19142973

C₂

semidirect

3. #19133277

C₂

semidirect

4. #19132428

C₂

semidirect

64. C₄²×C₄₃

SD₁₆

semidirect

134. C₄₃

#10287

semidirect

186. C₄²

#10621883

semidirect

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

subgroup	index
#19145517	2
#19142973	2
#19133277	2
#19132428	2
C₄₃×#1962	2
#10287	43

subgroup

index

C₄₃×#1962

Subgroups — #24999270

Conjugacy classes of subgroups

Complements

Maximal subgroups