Group #24999282 — subgroups

Subgroups — #24999282

Order 11008.

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

normal subgroups only

This group has 614 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	#19145713	C₂	1	2	yes	maximal
2	#19142993	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133268	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	—	2	2	no	—
7	C₂×#13063325	C₂²	1	2	yes	—
8	#14583302	—	2	2	no	—
9	#14583245	—	2	2	no	—
10	#14583210	—	2	4	no	—
11	#14583179	—	2	4	no	—
12	#14582952	C₂²	1	2	yes	—
13	#14582769	—	2	4	no	—
14	#14582646	—	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	—	2	2	no	—
20	#13063325	—	2	2	no	—
21	#13063325	—	2	2	no	—
22	#13063325	—	2	2	no	—
23	#13063325	—	2	2	no	—
24	#13063325	—	4	4	no	—
25	#13063325	—	4	4	no	—
26	C₂×C₄⋊Dic₄₃	—	2	8	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
29	#13063258	—	2	8	no	—
30	C₄⋊(C₄₃⋊C₈)	D₄	1	4	yes	—
31	#13063246	—	2	8	no	—
32	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
33	C₄×C₄₃⋊C₈	—	2	2	no	—
34	C₂×#41×C₄₃	—	2	1	no	—
35	C₂×M₁₆×C₄₃	—	2	1	no	—
36	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
37	#127×C₄₃	D₄	1	1	yes	characteristic
38	#124×C₄₃	—	2	1	no	—
39	#121×C₄₃	—	2	2	no	—
40	C₄×D₄×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	—	2	2	no	—
42	C₂×C₄²×C₄₃	D₄	1	2	yes	—
43	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₄₃	—	2	2	no	—
45	#104×C₄₃	—	4	4	no	—
46	#41×C₄₃	—	4	2	no	—
47	#41×C₄₃	—	4	2	no	—
48	C₂×Q₈×C₄₃	—	2	1	no	—
49	C₂×D₄×C₄₃	—	2	2	no	—
50	C₂×D₄×C₄₃	—	2	1	no	—
51	C₂×D₄×C₄₃	—	4	1	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	4	1	no	—
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₄₃	—	2	4	no	—
57	C₂×Dic₈₆	—	4	2	no	—
58	C₂×Dic₈₆	—	4	2	no	—
59	C₂×Dic₈₆	—	4	4	no	—
60	C₂×Dic₈₆	—	4	4	no	—
61	M₁₆×C₄₃	—	4	2	no	—
62	C₂×C₈×C₄₃	—	4	1	no	—
63	C₄⋊C₄×C₄₃	—	2	2	no	—
64	C₂²⋊C₄×C₄₃	—	4	2	no	—
65	C₄²×C₄₃	—	2	8	no	—
66	C₄²×C₄₃	—	2	2	no	—
67	C₄²×C₄₃	D₈	1	4	yes	—
68	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₄₃	C₂×D₄	1	2	yes	—
70	C₄²×C₄₃	#41	1	1	yes	characteristic
71	C₄⋊Dic₄₃	—	2	4	no	—
72	C₄⋊Dic₄₃	—	2	4	no	—
73	C₄⋊Dic₄₃	—	4	16	no	—
74	C₄⋊Dic₄₃	—	4	2	no	—
75	C₄⋊Dic₄₃	—	4	2	no	—
76	C₂×C₄₃⋊C₈	—	2	4	no	—
77	C₂³×C₄₃	—	4	1	no	—
78	C₂³×C₄₃	#120	1	1	yes	characteristic
79	Q₈×C₄₃	—	4	2	no	—
80	D₄×C₄₃	—	4	4	no	—
81	D₄×C₄₃	—	4	2	no	—
82	D₄×C₄₃	—	8	2	no	—
83	C₂×C₄×C₄₃	—	2	8	no	—
84	C₂×C₄×C₄₃	—	2	4	no	—
85	C₂×C₄×C₄₃	—	2	4	no	—
86	C₂×C₄×C₄₃	—	4	1	no	—
87	C₂×C₄×C₄₃	—	8	1	no	—
88	C₂×C₄×C₄₃	—	8	1	no	—
89	C₂×C₄×C₄₃	#120	1	2	yes	—
90	C₂×C₄×C₄₃	#121	1	2	yes	—
91	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×Dic₄₃	—	4	4	no	—
96	C₂×Dic₄₃	—	4	8	no	—
97	Dic₈₆	—	8	4	no	—
98	Dic₈₆	—	8	2	no	—
99	Dic₈₆	—	8	2	no	—
100	C₈×C₄₃	—	8	1	no	—
101	C₄₃⋊C₈	—	4	4	no	—
102	#10289	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	—	2	2	no	—
104	C₂²×C₄₃	—	8	2	no	—
105	C₂²×C₄₃	—	8	1	no	—
106	C₂²×C₄₃	#392	1	1	yes	characteristic
107	C₂²×C₄₃	#447	1	2	yes	—
108	C₄×C₄₃	—	2	8	no	—
109	C₄×C₄₃	—	2	2	no	—
110	C₄×C₄₃	—	8	1	no	—
111	C₄×C₄₃	#456	1	4	yes	—
112	C₄×C₄₃	#459	1	2	yes	—
113	C₄×C₄₃	#493	1	1	yes	characteristic
114	C₄×C₄₃	#496	1	1	yes	characteristic
115	Dic₄₃	—	8	4	no	—
116	#3226	—	43	2	no	—
117	#2868	—	43	1	no	—
118	#1962	D₄₃	1	1	yes	characteristic
119	#1546	—	43	2	no	—
120	#1276	—	43	1	no	—
121	C₂×C₄₃	—	2	2	no	—
122	C₂×C₄₃	—	16	1	no	—
123	C₂×C₄₃	#2005	1	2	yes	—
124	C₂×C₄₃	#2011	1	1	yes	characteristic
125	#532	C₂×D₄₃	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	43	2	no	—
127	C₂×C₄⋊Q₈	—	86	2	no	—
128	#500	—	86	2	no	—
129	#486	—	86	2	no	—
130	#480	—	86	4	no	—
131	#474	—	86	4	no	—
132	#431	—	43	2	no	—
133	#423	C₂×D₄₃	1	1	yes	characteristic
134	C₂×#104	—	2	2	no	—
135	#384	—	86	4	no	—
136	C₄³	C₂×D₄₃	1	1	yes	characteristic
137	#332	—	86	4	no	—
138	C₄₃	#10289	1	1	yes	characteristic, complemented, Sylow
139	C₂×#41	—	2	1	no	—
140	C₂²×Q₈	—	86	2	no	—
141	C₂×M₁₆	—	2	1	no	—
142	C₄⋊Q₈	—	86	2	no	—
143	C₄⋊Q₈	—	86	2	no	—
144	C₄⋊Q₈	—	86	2	no	—
145	C₄⋊Q₈	—	86	2	no	—
146	C₄⋊Q₈	—	172	4	no	—
147	C₄⋊Q₈	—	172	4	no	—
148	C₄⋊Q₈	D₁₇₂	1	1	yes	characteristic
149	#127	D₁₇₂	1	1	yes	characteristic
150	#124	—	2	1	no	—
151	#121	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	43	4	no	—
154	C₂×C₄⋊C₄	—	86	8	no	—
155	C₂×C₄⋊C₄	—	86	2	no	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
158	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
159	#107	—	86	8	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	43	4	no	—
162	#104	—	4	4	no	—
163	#102	—	86	8	no	—
164	C₂²⋊C₈	—	86	2	no	—
165	C₄×C₈	—	86	2	no	—
166	#41	—	4	2	no	—
167	#41	—	4	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	172	2	no	—
170	C₂×Q₈	—	172	2	no	—
171	C₂×Q₈	—	172	4	no	—
172	C₂×Q₈	—	172	4	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	—	86	4	no	—
179	C₂²×C₄	D₄×D₄₃	1	2	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	86	4	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	86	4	no	—
186	C₄⋊C₄	—	86	4	no	—
187	C₄⋊C₄	—	172	16	no	—
188	C₄⋊C₄	—	172	2	no	—
189	C₄⋊C₄	—	172	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	—	2	8	no	—
192	C₄²	—	2	2	no	—
193	C₄²	#10621882	1	4	yes	—
194	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
195	C₄²	#10621898	1	1	yes	characteristic
196	C₄²	D₄×D₄₃	1	2	yes	—
197	C₂³	—	4	1	no	—
198	C₂³	#13063293	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	344	4	no	—
201	Q₈	—	344	2	no	—
202	Q₈	—	344	2	no	—
203	D₄	—	4	4	no	—
204	D₄	—	4	2	no	—
205	D₄	—	8	2	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	8	1	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	172	4	no	—
213	C₂×C₄	—	172	8	no	—
214	C₂×C₄	#13063295	1	2	yes	—
215	C₂×C₄	#13063298	1	2	yes	—
216	C₂×C₄	#13063299	1	1	yes	characteristic
217	C₂×C₄	#13063321	1	1	yes	characteristic
218	C₂×C₄	C₂×#10621882	1	2	yes	—
219	C₂×C₄	#13063350	1	2	yes	—
220	C₈	—	8	1	no	—
221	C₈	—	172	4	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	1	no	—
225	C₂²	#14582808	1	1	yes	characteristic
226	C₂²	#14583032	1	2	yes	—
227	C₄	—	2	8	no	—
228	C₄	—	2	2	no	—
229	C₄	—	8	1	no	—
230	C₄	—	344	4	no	—
231	C₄	#14583083	1	4	yes	—
232	C₄	#14583094	1	2	yes	—
233	C₄	#14583275	1	1	yes	characteristic
234	C₄	#14583289	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	16	1	no	—
237	C₂	#19135630	1	2	yes	—
238	C₂	#19135665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19145713

C₂

yes

maximal

#19142993

C₂

yes

maximal, characteristic, complemented

#19133268

C₂

yes

maximal, complemented

#19132428

C₂

yes

maximal, characteristic, complemented

C₄₃×#1962

C₂

yes

maximal, characteristic

C₂×#13063325

—

C₂×#13063325

C₂²

yes

—

—

—

—

—

C₂²

yes

—

#14582769

—

#14582646

—

C₄₃×#532

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

characteristic

C₂×#104×C₄₃

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

—

—

—

—

—

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

#13063258

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

—

#13063246

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

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

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

D₈

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

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×D₈

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

100

C₈×C₄₃

—

101

C₄₃⋊C₈

—

102

#10289

—

maximal, Sylow

103

C₂²×C₄₃

—

104

C₂²×C₄₃

—

105

C₂²×C₄₃

—

106

C₂²×C₄₃

#392

yes

characteristic

107

C₂²×C₄₃

#447

yes

—

108

C₄×C₄₃

—

109

C₄×C₄₃

—

110

C₄×C₄₃

—

111

C₄×C₄₃

#456

yes

—

112

C₄×C₄₃

#459

yes

—

113

C₄×C₄₃

#493

yes

characteristic

114

C₄×C₄₃

#496

yes

characteristic

115

Dic₄₃

—

116

#3226

—

117

#2868

—

118

#1962

D₄₃

yes

characteristic

119

#1546

—

120

#1276

—

121

C₂×C₄₃

—

122

C₂×C₄₃

—

123

C₂×C₄₃

#2005

yes

—

124

C₂×C₄₃

#2011

yes

characteristic

125

#532

C₂×D₄₃

yes

characteristic

126

C₂×C₄⋊Q₈

—

127

C₂×C₄⋊Q₈

—

128

#500

—

129

#486

—

130

#480

—

131

#474

—

132

#431

—

133

#423

C₂×D₄₃

yes

characteristic

134

C₂×#104

—

135

#384

—

136

C₄³

C₂×D₄₃

yes

characteristic

137

#332

—

138

C₄₃

#10289

yes

characteristic, complemented, Sylow

139

C₂×#41

—

140

C₂²×Q₈

—

141

C₂×M₁₆

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

—

172

—

147

C₄⋊Q₈

—

172

—

148

C₄⋊Q₈

D₁₇₂

yes

characteristic

149

#127

D₁₇₂

yes

characteristic

150

#124

—

151

#121

—

152

C₄×D₄

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

—

155

C₂×C₄⋊C₄

—

156

C₂×C₄²

—

157

C₂×C₄²

C₂²⋊D₄₃

yes

—

158

C₂×C₄²

C₂²×D₄₃

yes

characteristic

159

#107

—

160

C₄⋊C₈

—

161

C₄⋊C₈

—

162

#104

—

163

#102

—

164

C₂²⋊C₈

—

165

C₄×C₈

—

166

#41

—

167

#41

—

168

C₂×Q₈

—

169

C₂×Q₈

—

172

—

170

C₂×Q₈

—

172

—

171

C₂×Q₈

—

172

—

172

C₂×Q₈

—

172

—

173

C₂×D₄

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

—

179

C₂²×C₄

D₄×D₄₃

yes

—

180

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

181

M₁₆

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

172

—

188

C₄⋊C₄

—

172

—

189

C₄⋊C₄

—

172

—

190

C₂²⋊C₄

—

191

C₄²

—

192

C₄²

—

193

C₄²

#10621882

yes

—

194

C₄²

C₂×D₁₇₂

yes

characteristic

195

C₄²

#10621898

yes

characteristic

196

C₄²

D₄×D₄₃

yes

—

197

C₂³

—

198

C₂³

#13063293

yes

characteristic

199

Q₈

—

200

Q₈

—

344

—

201

Q₈

—

344

—

202

Q₈

—

344

—

203

D₄

—

204

D₄

—

205

D₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

172

—

213

C₂×C₄

—

172

—

214

C₂×C₄

#13063295

yes

—

215

C₂×C₄

#13063298

yes

—

216

C₂×C₄

#13063299

yes

characteristic

217

C₂×C₄

#13063321

yes

characteristic

218

C₂×C₄

C₂×#10621882

yes

—

219

C₂×C₄

#13063350

yes

—

220

C₈

—

221

C₈

—

172

—

222

C₂²

—

223

C₂²

—

224

C₂²

—

225

C₂²

#14582808

yes

characteristic

226

C₂²

#14583032

yes

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

344

—

231

C₄

#14583083

yes

—

232

C₄

#14583094

yes

—

233

C₄

#14583275

yes

characteristic

234

C₄

#14583289

yes

characteristic

235

C₂

—

236

C₂

—

237

C₂

#19135630

yes

—

238

C₂

#19135665

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	#10289
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. #19142993	C₂	semidirect
3. #19133268	C₂	semidirect
4. #19132428	C₂	semidirect
138. C₄₃	#10289	semidirect

H ≅ G/N

type

2. #19142993

C₂

semidirect

3. #19133268

C₂

semidirect

4. #19132428

C₂

semidirect

138. C₄₃

#10289

semidirect

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

subgroup	index
#19145713	2
#19142993	2
#19133268	2
#19132428	2
C₄₃×#1962	2
#10289	43

subgroup

index

C₄₃×#1962

Subgroups — #24999282

Conjugacy classes of subgroups

Complements

Maximal subgroups