Group #24997466 — subgroups

Subgroups — #24997466

Order 11008.

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

normal subgroups only

This group has 528 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	#19144630	C₂	1	2	yes	maximal
2	#19137888	C₂	1	1	yes	maximal, characteristic
3	#19133023	C₂	1	2	yes	maximal
4	#19132424	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1995	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₈₆	C₂²	1	2	yes	—
7	#14583232	—	2	2	no	—
8	#14583191	—	2	2	no	—
9	#14583106	—	2	2	no	—
10	#14583054	—	2	2	no	—
11	C₄×#10621885	—	2	4	no	—
12	#14582952	C₂²	1	2	yes	—
13	#14582831	—	2	2	no	—
14	#14582820	—	2	2	no	—
15	C₄×C₄⋊Dic₄₃	—	2	2	no	—
16	#14582629	—	2	4	no	—
17	C₄₃×#533	C₂²	1	1	yes	characteristic
18	C₄₃×#423	C₂²	1	1	yes	characteristic
19	C₄₃×#376	C₂²	1	1	yes	characteristic
20	C₄₃×#339	—	2	2	no	—
21	C₂²×Dic₈₆	D₄	1	2	yes	—
22	C₂×#10621885	—	2	4	no	—
23	C₄×Dic₈₆	—	2	2	no	—
24	C₄×Dic₈₆	—	2	2	no	—
25	C₄×Dic₈₆	—	2	2	no	—
26	C₄×Dic₈₆	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	4	no	—
28	C₂×C₄⋊Dic₄₃	D₄	1	2	yes	—
29	C₂×Dic₄₃⋊C₄	—	2	2	no	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	#13063258	—	2	4	no	—
32	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
33	#13063249	—	2	4	no	—
34	#13063248	—	2	4	no	—
35	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
36	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
37	C₄₃⋊#95	—	2	2	no	—
38	C₄₃⋊#95	—	2	2	no	—
39	C₂×M₁₆×C₄₃	—	2	1	no	—
40	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
41	#125×C₄₃	—	2	1	no	—
42	C₂²⋊Q₈×C₄₃	—	2	2	no	—
43	C₄×Q₈×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	1	no	—
45	C₂×C₄²×C₄₃	D₄	1	2	yes	—
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₄₃	—	2	2	no	—
48	#95×C₄₃	—	2	2	no	—
49	C₂×Q₈×C₄₃	—	2	2	no	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₄₃	—	2	2	no	—
54	C₂²×Dic₄₃	—	2	2	no	—
55	C₂×Dic₈₆	—	2	2	no	—
56	C₂×Dic₈₆	—	2	2	no	—
57	C₂×Dic₈₆	—	4	4	no	—
58	M₁₆×C₄₃	—	4	2	no	—
59	C₂×C₈×C₄₃	—	4	1	no	—
60	C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₄⋊C₄×C₄₃	—	4	2	no	—
63	C₂²⋊C₄×C₄₃	—	4	1	no	—
64	C₄²×C₄₃	—	2	2	no	—
65	C₄²×C₄₃	—	4	1	no	—
66	C₄²×C₄₃	Q₁₆	1	4	yes	complemented
67	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₄₃	C₂×D₄	1	2	yes	—
69	C₄²×C₄₃	#41	1	1	yes	characteristic
70	#10621885	—	4	8	no	—
71	C₄⋊Dic₄₃	—	2	2	no	—
72	C₄⋊Dic₄₃	—	2	2	no	—
73	C₄⋊Dic₄₃	—	4	8	no	—
74	Dic₄₃⋊C₄	—	4	2	no	—
75	Dic₄₃⋊C₄	—	4	2	no	—
76	C₄×Dic₄₃	—	4	2	no	—
77	C₄×Dic₄₃	—	4	2	no	—
78	C₂×C₄₃⋊C₈	—	2	4	no	—
79	C₂³×C₄₃	#120	1	1	yes	characteristic
80	Q₈×C₄₃	—	4	4	no	—
81	C₂×C₄×C₄₃	—	2	4	no	—
82	C₂×C₄×C₄₃	—	4	2	no	—
83	C₂×C₄×C₄₃	—	4	4	no	—
84	C₂×C₄×C₄₃	—	4	2	no	—
85	C₂×C₄×C₄₃	#120	1	2	yes	—
86	C₂×C₄×C₄₃	#121	1	2	yes	—
87	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₄₃	#137	1	2	yes	—
91	C₂×Dic₄₃	—	2	4	no	—
92	C₂×Dic₄₃	—	4	2	no	—
93	C₂×Dic₄₃	—	4	2	no	—
94	C₂×Dic₄₃	—	4	4	no	—
95	C₂×Dic₄₃	—	4	2	no	—
96	Dic₈₆	—	2	4	no	—
97	Dic₈₆	—	2	4	no	—
98	Dic₈₆	—	4	2	no	—
99	Dic₈₆	—	4	2	no	—
100	C₈×C₄₃	—	8	1	no	—
101	C₄₃⋊C₈	—	4	4	no	—
102	#10017	—	43	1	no	maximal, Sylow
103	C₂²×C₄₃	—	2	2	no	—
104	C₂²×C₄₃	#392	1	1	yes	characteristic
105	C₂²×C₄₃	#451	1	2	yes	—
106	C₄×C₄₃	—	2	2	no	—
107	C₄×C₄₃	—	2	2	no	—
108	C₄×C₄₃	—	4	4	no	—
109	C₄×C₄₃	—	8	2	no	—
110	C₄×C₄₃	#462	1	2	yes	—
111	C₄×C₄₃	#494	1	1	yes	characteristic
112	C₄×C₄₃	#497	1	1	yes	characteristic
113	Dic₄₃	—	4	4	no	—
114	Dic₄₃	—	8	2	no	—
115	#3104	—	43	2	no	—
116	#2399	—	43	1	no	—
117	#1995	D₄₃	1	1	yes	characteristic
118	#1487	—	43	2	no	—
119	#1274	—	43	1	no	—
120	C₂×C₄₃	—	2	2	no	—
121	C₂×C₄₃	#2010	1	2	yes	—
122	C₂×C₄₃	#2016	1	1	yes	characteristic
123	#533	C₂×D₄₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	—	43	2	no	—
125	#484	—	86	2	no	—
126	#477	—	86	2	no	—
127	#462	—	86	2	no	—
128	#451	—	86	2	no	—
129	C₄×Q₁₆	—	86	4	no	—
130	#431	—	43	2	no	—
131	#423	C₂×D₄₃	1	1	yes	characteristic
132	#398	—	86	2	no	—
133	#395	—	86	2	no	—
134	C₄×C₄⋊C₄	—	86	2	no	—
135	#376	C₂×D₄₃	1	1	yes	characteristic
136	#339	—	2	2	no	—
137	Q₈⋊C₈	—	86	4	no	—
138	C₄₃	#10017	1	1	yes	characteristic, complemented, Sylow
139	C₂²×Q₈	—	43	2	no	—
140	C₂×Q₁₆	—	86	4	no	—
141	C₂×M₁₆	—	2	1	no	—
142	C₄⋊Q₈	D₁₇₂	1	2	yes	—
143	#125	—	2	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	86	2	no	—
147	C₄×Q₈	—	86	2	no	—
148	C₄×Q₈	—	86	2	no	—
149	C₄×Q₈	—	86	2	no	—
150	#117	—	2	1	no	—
151	C₂×C₄⋊C₄	—	43	2	no	—
152	C₂×C₄⋊C₄	—	86	4	no	—
153	C₂×C₄⋊C₄	—	86	2	no	—
154	C₂×C₄²	—	86	2	no	—
155	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
156	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
157	#107	—	86	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	86	2	no	—
160	#103	—	86	4	no	—
161	#103	—	86	4	no	—
162	C₂²⋊C₈	—	86	2	no	—
163	C₄×C₈	—	43	4	no	—
164	#95	—	2	2	no	—
165	#95	—	86	2	no	—
166	#95	—	86	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	86	2	no	—
169	C₂×Q₈	—	86	2	no	—
170	C₂×Q₈	—	172	4	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	86	2	no	—
173	C₂²×C₄	—	86	2	no	—
174	C₂²×C₄	D₄×D₄₃	1	2	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
176	Q₁₆	—	172	8	no	—
177	M₁₆	—	4	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	86	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	86	2	no	—
184	C₄⋊C₄	—	86	2	no	—
185	C₄⋊C₄	—	172	8	no	—
186	C₄⋊C₄	—	172	2	no	—
187	C₄⋊C₄	—	172	2	no	—
188	C₂²⋊C₄	—	4	1	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	172	2	no	—
192	C₄²	—	172	2	no	—
193	C₄²	#10621885	1	4	yes	complemented
194	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
195	C₄²	#10621898	1	1	yes	characteristic
196	C₄²	D₄×D₄₃	1	2	yes	—
197	C₂³	#13063293	1	1	yes	characteristic
198	Q₈	—	4	4	no	—
199	Q₈	—	86	4	no	—
200	Q₈	—	86	4	no	—
201	Q₈	—	172	2	no	—
202	Q₈	—	172	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	86	4	no	—
208	C₂×C₄	—	172	4	no	—
209	C₂×C₄	—	172	2	no	—
210	C₂×C₄	—	172	2	no	—
211	C₂×C₄	—	172	2	no	—
212	C₂×C₄	#13063295	1	2	yes	—
213	C₂×C₄	#13063298	1	2	yes	—
214	C₂×C₄	#13063299	1	1	yes	characteristic
215	C₂×C₄	#13063321	1	1	yes	characteristic
216	C₂×C₄	C₂×#10621885	1	2	yes	—
217	C₂×C₄	#13063355	1	2	yes	—
218	C₈	—	8	1	no	—
219	C₈	—	172	4	no	—
220	C₂²	—	2	2	no	—
221	C₂²	#14582808	1	1	yes	characteristic
222	C₂²	#14583054	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	4	no	—
226	C₄	—	8	2	no	—
227	C₄	—	344	2	no	—
228	C₄	—	172	4	no	—
229	C₄	#14583109	1	2	yes	—
230	C₄	#14583278	1	1	yes	characteristic
231	C₄	#14583294	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	#19135661	1	2	yes	—
234	C₂	#19135694	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19144630

C₂

yes

maximal

#19137888

C₂

yes

maximal, characteristic

#19133023

C₂

yes

maximal

#19132424

C₂

yes

maximal, characteristic

C₄₃×#1995

C₂

yes

maximal, characteristic

C₂×C₄×Dic₈₆

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

#14582831

—

#14582820

—

C₄×C₄⋊Dic₄₃

—

#14582629

—

C₄₃×#533

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

characteristic

C₄₃×#376

C₂²

yes

characteristic

C₄₃×#339

—

C₂²×Dic₈₆

D₄

yes

—

C₂×#10621885

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₄×Dic₈₆

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

#13063258

—

C₄⋊(C₄₃⋊C₈)

—

#13063249

—

#13063248

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

D₄

yes

—

#125×C₄₃

—

C₂²⋊Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

#117×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

#95×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

complemented

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

characteristic

#10621885

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

Q₈×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×Q₁₆

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

100

C₈×C₄₃

—

101

C₄₃⋊C₈

—

102

#10017

—

maximal, Sylow

103

C₂²×C₄₃

—

104

C₂²×C₄₃

#392

yes

characteristic

105

C₂²×C₄₃

#451

yes

—

106

C₄×C₄₃

—

107

C₄×C₄₃

—

108

C₄×C₄₃

—

109

C₄×C₄₃

—

110

C₄×C₄₃

#462

yes

—

111

C₄×C₄₃

#494

yes

characteristic

112

C₄×C₄₃

#497

yes

characteristic

113

Dic₄₃

—

114

Dic₄₃

—

115

#3104

—

116

#2399

—

117

#1995

D₄₃

yes

characteristic

118

#1487

—

119

#1274

—

120

C₂×C₄₃

—

121

C₂×C₄₃

#2010

yes

—

122

C₂×C₄₃

#2016

yes

characteristic

123

#533

C₂×D₄₃

yes

characteristic

124

C₂×C₄×Q₈

—

125

#484

—

126

#477

—

127

#462

—

128

#451

—

129

C₄×Q₁₆

—

130

#431

—

131

#423

C₂×D₄₃

yes

characteristic

132

#398

—

133

#395

—

134

C₄×C₄⋊C₄

—

135

#376

C₂×D₄₃

yes

characteristic

136

#339

—

137

Q₈⋊C₈

—

138

C₄₃

#10017

yes

characteristic, complemented, Sylow

139

C₂²×Q₈

—

140

C₂×Q₁₆

—

141

C₂×M₁₆

—

142

C₄⋊Q₈

D₁₇₂

yes

—

143

#125

—

144

C₂²⋊Q₈

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

C₄×Q₈

—

150

#117

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄²

—

155

C₂×C₄²

C₂²⋊D₄₃

yes

—

156

C₂×C₄²

C₂²×D₄₃

yes

characteristic

157

#107

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

#103

—

161

#103

—

162

C₂²⋊C₈

—

163

C₄×C₈

—

164

#95

—

165

#95

—

166

#95

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

172

—

171

C₂²×C₄

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

D₄×D₄₃

yes

—

175

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

176

Q₁₆

—

172

—

177

M₁₆

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

172

—

186

C₄⋊C₄

—

172

—

187

C₄⋊C₄

—

172

—

188

C₂²⋊C₄

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

172

—

192

C₄²

—

172

—

193

C₄²

#10621885

yes

complemented

194

C₄²

C₂×D₁₇₂

yes

characteristic

195

C₄²

#10621898

yes

characteristic

196

C₄²

D₄×D₄₃

yes

—

197

C₂³

#13063293

yes

characteristic

198

Q₈

—

199

Q₈

—

200

Q₈

—

201

Q₈

—

172

—

202

Q₈

—

172

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

172

—

209

C₂×C₄

—

172

—

210

C₂×C₄

—

172

—

211

C₂×C₄

—

172

—

212

C₂×C₄

#13063295

yes

—

213

C₂×C₄

#13063298

yes

—

214

C₂×C₄

#13063299

yes

characteristic

215

C₂×C₄

#13063321

yes

characteristic

216

C₂×C₄

C₂×#10621885

yes

—

217

C₂×C₄

#13063355

yes

—

218

C₈

—

219

C₈

—

172

—

220

C₂²

—

221

C₂²

#14582808

yes

characteristic

222

C₂²

#14583054

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

344

—

228

C₄

—

172

—

229

C₄

#14583109

yes

—

230

C₄

#14583278

yes

characteristic

231

C₄

#14583294

yes

characteristic

232

C₂

—

233

C₂

#19135661

yes

—

234

C₂

#19135694

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1995	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂³×C₄₃	#120
hypercenter	#1995	D₄₃
hypocenter	C₄₃	#10017
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#13063321

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1995

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
66. C₄²×C₄₃	Q₁₆	semidirect
138. C₄₃	#10017	semidirect
193. C₄²	#10621885	semidirect

H ≅ G/N

type

66. C₄²×C₄₃

Q₁₆

semidirect

138. C₄₃

#10017

semidirect

193. C₄²

#10621885

semidirect

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

subgroup	index
#19144630	2
#19137888	2
#19133023	2
#19132424	2
C₄₃×#1995	2
#10017	43

subgroup

index

C₄₃×#1995

Subgroups — #24997466

Conjugacy classes of subgroups

Complements

Maximal subgroups