Group #24998844 — subgroups

Subgroups — #24998844

Order 11008.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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	#19145533	C₂	1	2	yes	maximal
2	#19135419	C₂	1	1	yes	maximal, characteristic
3	#19133159	C₂	1	2	yes	maximal, complemented
4	#19132440	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063322	C₂²	1	1	yes	characteristic
7	#14583273	—	2	2	no	—
8	#14583247	—	2	2	no	—
9	#14583099	—	2	4	no	—
10	#14583049	—	2	4	no	—
11	#14582952	C₂²	1	2	yes	—
12	C₂×#13063247	—	2	2	no	—
13	#14582765	C₂²	1	1	yes	characteristic
14	#14582636	—	2	4	no	—
15	C₄₃×#533	C₂²	1	2	yes	—
16	C₄₃×#501	—	2	2	no	—
17	C₄₃×#501	—	2	2	no	—
18	C₄₃×#427	—	2	2	no	—
19	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×D₁₇₂	—	2	1	no	—
21	C₂²×D₁₇₂	—	2	2	no	—
22	C₂×#10621884	—	2	8	no	—
23	#13063322	—	2	2	no	—
24	#13063322	—	2	1	no	—
25	#13063322	D₄	1	2	yes	—
26	C₂×C₄⋊Dic₄₃	—	2	1	no	—
27	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
28	#13063247	—	4	4	no	—
29	#13063245	—	2	8	no	—
30	C₂²⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
31	C₄×C₄₃⋊C₈	—	2	2	no	—
32	C₄₃⋊#95	—	2	2	no	—
33	C₂²×C₈×C₄₃	D₄	1	2	yes	—
34	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
35	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
36	#125×C₄₃	—	2	2	no	—
37	C₂²⋊Q₈×C₄₃	—	2	2	no	—
38	C₂²⋊Q₈×C₄₃	—	2	2	no	—
39	C₄×Q₈×C₄₃	—	2	2	no	—
40	C₄×Q₈×C₄₃	—	2	2	no	—
41	#117×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
43	#107×C₄₃	—	2	4	no	—
44	#106×C₄₃	—	2	2	no	—
45	#106×C₄₃	—	2	2	no	—
46	#97×C₄₃	—	2	2	no	—
47	C₂³×D₄₃	—	4	1	no	—
48	C₂×Q₈×C₄₃	—	2	2	no	—
49	C₂×Q₈×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₄₃	—	4	1	no	—
53	C₂×D₁₇₂	—	2	4	no	—
54	C₂×D₁₇₂	—	2	2	no	—
55	C₂×D₁₇₂	—	4	1	no	—
56	C₂×D₁₇₂	—	4	2	no	—
57	C₂×D₁₇₂	—	4	2	no	—
58	C₂×D₁₇₂	—	4	2	no	—
59	C₂×D₁₇₂	—	8	4	no	—
60	C₂×C₈×C₄₃	—	2	2	no	—
61	C₂×C₈×C₄₃	—	2	2	no	—
62	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₄⋊C₄×C₄₃	—	2	2	no	—
65	C₄⋊C₄×C₄₃	—	2	2	no	—
66	C₄⋊C₄×C₄₃	—	2	2	no	—
67	C₄⋊C₄×C₄₃	—	4	2	no	—
68	C₄⋊C₄×C₄₃	—	4	2	no	—
69	C₂²⋊C₄×C₄₃	—	4	2	no	—
70	C₄²×C₄₃	—	4	2	no	—
71	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₄₃	C₂×D₄	1	2	yes	—
73	C₄²×C₄₃	#41	1	1	yes	characteristic
74	#10621884	—	8	8	no	—
75	C₄⋊Dic₄₃	—	4	2	no	—
76	C₂×C₄₃⋊C₈	—	2	4	no	—
77	C₂³×C₄₃	#127	1	1	yes	characteristic
78	C₂²×D₄₃	—	4	2	no	—
79	C₂²×D₄₃	—	8	4	no	—
80	C₂²×D₄₃	—	8	1	no	—
81	C₂²×D₄₃	—	8	1	no	—
82	Q₈×C₄₃	—	4	4	no	—
83	Q₈×C₄₃	—	8	2	no	—
84	C₂×C₄×C₄₃	—	4	2	no	—
85	C₂×C₄×C₄₃	—	4	2	no	—
86	C₂×C₄×C₄₃	—	4	2	no	—
87	C₂×C₄×C₄₃	#120	1	2	yes	—
88	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
89	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₄₃	#121	1	2	yes	—
91	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₄₃	#137	1	2	yes	—
93	C₂×Dic₄₃	—	8	2	no	—
94	C₂×Dic₄₃	—	8	1	no	—
95	D₁₇₂	—	4	8	no	—
96	D₁₇₂	—	4	4	no	—
97	D₁₇₂	—	8	4	no	—
98	D₁₇₂	—	8	2	no	—
99	C₈×C₄₃	—	2	4	no	—
100	C₄₃⋊C₈	—	4	4	no	—
101	#10220	—	43	1	no	maximal, Sylow
102	C₂²×C₄₃	—	2	2	no	—
103	C₂²×C₄₃	#392	1	1	yes	characteristic
104	C₂²×C₄₃	#492	1	1	yes	characteristic
105	C₂²×C₄₃	#497	1	1	yes	characteristic
106	C₂×D₄₃	—	8	4	no	—
107	C₂×D₄₃	—	8	2	no	—
108	C₂×D₄₃	—	16	2	no	—
109	C₂×D₄₃	—	16	2	no	—
110	C₄×C₄₃	—	2	2	no	—
111	C₄×C₄₃	—	2	2	no	—
112	C₄×C₄₃	—	8	2	no	—
113	C₄×C₄₃	—	8	2	no	—
114	C₄×C₄₃	#448	1	2	yes	—
115	C₄×C₄₃	#461	1	2	yes	—
116	Dic₄₃	—	16	1	no	—
117	#3208	—	43	2	no	—
118	#3162	D₄₃	1	1	yes	characteristic, complemented
119	#1961	—	43	1	no	—
120	#1518	—	43	2	no	—
121	#1281	—	43	1	no	—
122	C₂×C₄₃	—	2	1	no	—
123	C₂×C₄₃	—	2	1	no	—
124	C₂×C₄₃	#2008	1	2	yes	—
125	C₂×C₄₃	#2013	1	1	yes	characteristic
126	D₄₃	—	16	2	no	—
127	#533	C₂×D₄₃	1	2	yes	—
128	C₂×#127	—	43	1	no	—
129	#501	—	2	2	no	—
130	#501	—	2	2	no	—
131	#492	—	86	2	no	—
132	#486	—	86	2	no	—
133	#460	—	86	4	no	—
134	#450	—	86	4	no	—
135	#431	—	43	2	no	—
136	#427	—	2	2	no	—
137	C₂×#102	—	86	2	no	—
138	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
139	#382	—	43	1	no	—
140	#328	—	86	4	no	—
141	C₄₃	#10220	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	86	2	no	—
143	C₂²×D₄	—	86	1	no	—
144	C₂×SD₁₆	—	86	8	no	—
145	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
146	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
147	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
148	#127	—	43	2	no	—
149	#127	—	86	1	no	—
150	#127	—	86	2	no	—
151	#125	—	2	2	no	—
152	C₂²⋊Q₈	—	2	2	no	—
153	C₂²⋊Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₄×Q₈	—	2	2	no	—
156	#117	—	2	2	no	—
157	C₂×C₄⋊C₄	—	86	1	no	—
158	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
159	#107	—	2	4	no	—
160	#106	—	2	2	no	—
161	#106	—	2	2	no	—
162	C₄⋊C₈	—	86	2	no	—
163	#102	—	86	8	no	—
164	#102	—	172	4	no	—
165	C₂²⋊C₈	—	43	4	no	—
166	#97	—	2	2	no	—
167	C₄×C₈	—	86	2	no	—
168	#95	—	86	2	no	—
169	C₂⁴	—	172	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×D₄	—	86	4	no	—
173	C₂×D₄	—	86	2	no	—
174	C₂×D₄	—	344	4	no	—
175	C₂×D₄	—	172	1	no	—
176	C₂×D₄	—	172	2	no	—
177	C₂×D₄	—	172	2	no	—
178	C₂×D₄	—	172	2	no	—
179	C₂²×C₄	—	172	1	no	—
180	C₂²×C₄	D₄×D₄₃	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
182	SD₁₆	—	344	8	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	86	4	no	—
186	C₂×C₈	#10621884	1	4	yes	complemented
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	172	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	D₄×D₄₃	1	1	yes	characteristic
197	C₄²	#10621900	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
199	C₂³	—	344	4	no	—
200	C₂³	—	344	1	no	—
201	C₂³	—	344	1	no	—
202	C₂³	—	172	2	no	—
203	C₂³	#13063321	1	1	yes	characteristic
204	Q₈	—	4	4	no	—
205	Q₈	—	8	2	no	—
206	D₄	—	344	4	no	—
207	D₄	—	344	2	no	—
208	D₄	—	172	8	no	—
209	D₄	—	172	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	344	2	no	—
214	C₂×C₄	—	344	1	no	—
215	C₂×C₄	#13063293	1	2	yes	—
216	C₂×C₄	#13063294	1	1	yes	characteristic
217	C₂×C₄	#13063296	1	1	yes	characteristic
218	C₂×C₄	#13063297	1	2	yes	—
219	C₂×C₄	C₂×#10621884	1	2	yes	—
220	C₂×C₄	#13063356	1	2	yes	—
221	C₈	—	2	4	no	—
222	C₈	—	172	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	344	4	no	—
225	C₂²	—	344	2	no	—
226	C₂²	—	688	2	no	—
227	C₂²	—	688	2	no	—
228	C₂²	#14582807	1	1	yes	characteristic
229	C₂²	#14583273	1	1	yes	characteristic
230	C₂²	#14583291	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	688	1	no	—
236	C₄	#14583038	1	2	yes	—
237	C₄	#14583101	1	2	yes	—
238	C₂	—	2	1	no	—
239	C₂	—	2	1	no	—
240	C₂	—	688	2	no	—
241	C₂	#19135645	1	2	yes	—
242	C₂	#19135678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145533

C₂

yes

maximal

#19135419

C₂

yes

maximal, characteristic

#19133159

C₂

yes

maximal, complemented

#19132440

C₂

yes

maximal, characteristic, complemented

C₄₃×#3162

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#13063247

—

#14582765

C₂²

yes

characteristic

#14582636

—

C₄₃×#533

C₂²

yes

—

C₄₃×#501

—

C₄₃×#501

—

C₄₃×#427

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×D₁₇₂

—

C₂²×D₁₇₂

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

#13063247

—

#13063245

—

C₂²⋊(C₄₃⋊C₈)

D₄

yes

—

C₄×C₄₃⋊C₈

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

#106×C₄₃

—

#106×C₄₃

—

#97×C₄₃

—

C₂³×D₄₃

—

C₂×Q₈×C₄₃

—

C₂×Q₈×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×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

complemented

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

#10621884

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#127

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

100

C₄₃⋊C₈

—

101

#10220

—

maximal, Sylow

102

C₂²×C₄₃

—

103

C₂²×C₄₃

#392

yes

characteristic

104

C₂²×C₄₃

#492

yes

characteristic

105

C₂²×C₄₃

#497

yes

characteristic

106

C₂×D₄₃

—

107

C₂×D₄₃

—

108

C₂×D₄₃

—

109

C₂×D₄₃

—

110

C₄×C₄₃

—

111

C₄×C₄₃

—

112

C₄×C₄₃

—

113

C₄×C₄₃

—

114

C₄×C₄₃

#448

yes

—

115

C₄×C₄₃

#461

yes

—

116

Dic₄₃

—

117

#3208

—

118

#3162

D₄₃

yes

characteristic, complemented

119

#1961

—

120

#1518

—

121

#1281

—

122

C₂×C₄₃

—

123

C₂×C₄₃

—

124

C₂×C₄₃

#2008

yes

—

125

C₂×C₄₃

#2013

yes

characteristic

126

D₄₃

—

127

#533

C₂×D₄₃

yes

—

128

C₂×#127

—

129

#501

—

130

#501

—

131

#492

—

132

#486

—

133

#460

—

134

#450

—

135

#431

—

136

#427

—

137

C₂×#102

—

138

C₂×#97

C₂×D₄₃

yes

characteristic

139

#382

—

140

#328

—

141

C₄₃

#10220

yes

characteristic, complemented, Sylow

142

C₂²×D₄

—

143

C₂²×D₄

—

144

C₂×SD₁₆

—

145

C₂²×C₈

C₂²⋊D₄₃

yes

—

146

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

147

C₄⋊Q₈

C₂²⋊D₄₃

yes

—

148

#127

—

149

#127

—

150

#127

—

151

#125

—

152

C₂²⋊Q₈

—

153

C₂²⋊Q₈

—

154

C₄×Q₈

—

155

C₄×Q₈

—

156

#117

—

157

C₂×C₄⋊C₄

—

158

C₂×C₄²

C₂²×D₄₃

yes

characteristic

159

#107

—

160

#106

—

161

#106

—

162

C₄⋊C₈

—

163

#102

—

164

#102

—

172

—

165

C₂²⋊C₈

—

166

#97

—

167

C₄×C₈

—

168

#95

—

169

C₂⁴

—

172

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×D₄

—

173

C₂×D₄

—

174

C₂×D₄

—

344

—

175

C₂×D₄

—

172

—

176

C₂×D₄

—

172

—

177

C₂×D₄

—

172

—

178

C₂×D₄

—

172

—

179

C₂²×C₄

—

172

—

180

C₂²×C₄

D₄×D₄₃

yes

—

181

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

182

SD₁₆

—

344

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

#10621884

yes

complemented

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

172

—

194

C₂²⋊C₄

—

195

C₄²

—

196

C₄²

D₄×D₄₃

yes

characteristic

197

C₄²

#10621900

yes

characteristic

198

C₄²

C₂×C₂²⋊D₄₃

yes

—

199

C₂³

—

344

—

200

C₂³

—

344

—

201

C₂³

—

344

—

202

C₂³

—

172

—

203

C₂³

#13063321

yes

characteristic

204

Q₈

—

205

Q₈

—

206

D₄

—

344

—

207

D₄

—

344

—

208

D₄

—

172

—

209

D₄

—

172

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

344

—

214

C₂×C₄

—

344

—

215

C₂×C₄

#13063293

yes

—

216

C₂×C₄

#13063294

yes

characteristic

217

C₂×C₄

#13063296

yes

characteristic

218

C₂×C₄

#13063297

yes

—

219

C₂×C₄

C₂×#10621884

yes

—

220

C₂×C₄

#13063356

yes

—

221

C₈

—

222

C₈

—

172

—

223

C₂²

—

224

C₂²

—

344

—

225

C₂²

—

344

—

226

C₂²

—

688

—

227

C₂²

—

688

—

228

C₂²

#14582807

yes

characteristic

229

C₂²

#14583273

yes

characteristic

230

C₂²

#14583291

yes

characteristic

231

C₄

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

688

—

236

C₄

#14583038

yes

—

237

C₄

#14583101

yes

—

238

C₂

—

239

C₂

—

240

C₂

—

688

—

241

C₂

#19135645

yes

—

242

C₂

#19135678

yes

characteristic

	subgroup	G/H
center	C₂²	#14582807
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3162	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂×#13063322	C₂²
hypercenter	#3162	D₄₃
hypocenter	C₄₃	#10220
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3162

subgroup

G/H

center

C₂²

#14582807

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#3162

C₂

C₂²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3162

N	H ≅ G/N	type
3. #19133159	C₂	semidirect
4. #19132440	C₂	semidirect
5. C₄₃×#3162	C₂	semidirect
62. C₂×C₈×C₄₃	SD₁₆	semidirect
118. #3162	D₄₃	semidirect
141. C₄₃	#10220	semidirect
186. C₂×C₈	#10621884	semidirect

H ≅ G/N

type

3. #19133159

C₂

semidirect

4. #19132440

C₂

semidirect

5. C₄₃×#3162

C₂

semidirect

62. C₂×C₈×C₄₃

SD₁₆

semidirect

118. #3162

D₄₃

semidirect

141. C₄₃

#10220

semidirect

186. C₂×C₈

#10621884

semidirect

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

subgroup	index
#19145533	2
#19135419	2
#19133159	2
#19132440	2
C₄₃×#3162	2
#10220	43

subgroup

index

C₄₃×#3162

Subgroups — #24998844

Conjugacy classes of subgroups

Complements

Maximal subgroups