Group #24989621 — subgroups

Subgroups — #24989621

Order 11008.

404 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 109 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	#19143307	C₂	1	1	yes	maximal, characteristic
2	#19133353	C₂	1	2	yes	maximal, complemented
3	#19133330	C₂	1	1	yes	maximal, characteristic
4	C₄₃⋊#1309	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1560	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063332	C₄	1	2	yes	—
7	#14582977	—	2	2	no	—
8	D₄×C₄₃⋊C₈	—	2	2	no	—
9	#14582923	C₂²	1	2	yes	—
10	#14582923	C₂²	1	1	yes	characteristic
11	#14582866	C₄	1	2	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582651	—	2	2	no	—
14	#14582651	—	2	2	no	—
15	#14582648	—	2	1	no	—
16	#14582643	—	2	1	no	—
17	C₄₃×#532	C₂²	1	1	yes	characteristic
18	C₄₃×#423	C₂²	1	2	yes	—
19	C₄₃×#333	—	2	1	no	—
20	C₄₃×#331	—	2	1	no	—
21	#13063332	—	4	4	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	C₂×C₄	1	2	yes	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
30	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
32	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
33	C₄×C₄₃⋊C₈	D₄	1	2	yes	—
34	C₂×#41×C₄₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₄₃	—	2	2	no	—
36	C₄⋊Q₈×C₄₃	D₄	1	1	yes	characteristic
37	#127×C₄₃	D₄	1	1	yes	characteristic
38	#124×C₄₃	Q₈	1	2	yes	—
39	#121×C₄₃	—	2	2	no	—
40	C₄×D₄×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₄₃	—	2	2	no	—
43	C₄⋊C₈×C₄₃	—	2	2	no	—
44	#41×C₄₃	—	4	2	no	—
45	#41×C₄₃	—	4	2	no	—
46	C₂×Q₈×C₄₃	C₄⋊C₄	1	2	yes	—
47	C₂×D₄×C₄₃	—	2	2	no	—
48	C₂×D₄×C₄₃	—	2	2	no	—
49	C₂×D₄×C₄₃	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	#41	1	2	yes	—
53	M₁₆×C₄₃	—	8	2	no	—
54	C₂×C₈×C₄₃	—	4	2	no	—
55	C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₂²⋊C₄×C₄₃	—	2	4	no	—
57	C₄²×C₄₃	C₂×D₄	1	2	yes	—
58	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
60	#10621877	—	4	4	no	—
61	#10621877	—	4	2	no	—
62	#10621877	—	8	2	no	—
63	C₂×C₄₃⋊C₈	—	2	2	no	—
64	C₂×C₄₃⋊C₈	—	2	2	no	—
65	C₂×C₄₃⋊C₈	—	4	1	no	—
66	C₂×C₄₃⋊C₈	—	4	4	no	—
67	C₂×C₄₃⋊C₈	—	4	2	no	—
68	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂³×C₄₃	—	2	2	no	—
72	C₂³×C₄₃	#117	1	1	yes	characteristic
73	Q₈×C₄₃	—	4	2	no	—
74	D₄×C₄₃	—	4	2	no	—
75	D₄×C₄₃	—	4	4	no	—
76	D₄×C₄₃	—	4	2	no	—
77	D₄×C₄₃	—	4	2	no	—
78	C₂×C₄×C₄₃	—	2	1	no	—
79	C₂×C₄×C₄₃	—	2	1	no	—
80	C₂×C₄×C₄₃	—	2	2	no	—
81	C₂×C₄×C₄₃	—	2	2	no	—
82	C₂×C₄×C₄₃	—	2	2	no	—
83	C₂×C₄×C₄₃	—	4	2	no	—
84	C₂×C₄×C₄₃	—	4	2	no	—
85	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₄₃	#121	1	2	yes	—
88	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
89	C₈×C₄₃	—	8	2	no	—
90	C₄₃⋊C₈	—	2	1	no	—
91	C₄₃⋊C₈	—	2	2	no	—
92	C₄₃⋊C₈	—	4	2	no	—
93	C₄₃⋊C₈	—	8	2	no	—
94	C₄₃⋊C₈	#100	1	2	yes	—
95	#8638	—	43	1	no	maximal, Sylow
96	C₂²×C₄₃	—	2	1	no	—
97	C₂²×C₄₃	—	2	2	no	—
98	C₂²×C₄₃	—	4	2	no	—
99	C₂²×C₄₃	—	4	4	no	—
100	C₂²×C₄₃	#380	1	1	yes	characteristic
101	C₄×C₄₃	—	2	1	no	—
102	C₄×C₄₃	—	2	2	no	—
103	C₄×C₄₃	—	4	2	no	—
104	C₄×C₄₃	C₂×#100	1	1	yes	characteristic
105	C₄×C₄₃	#413	1	1	yes	characteristic
106	C₄×C₄₃	#472	1	2	yes	—
107	C₄×C₄₃	#473	1	2	yes	—
108	#2931	—	43	1	no	—
109	#1567	—	43	2	no	—
110	#1560	—	43	1	no	—
111	#1560	D₄₃	1	1	yes	characteristic
112	#1309	—	43	2	no	—
113	C₂×C₄₃	—	4	1	no	—
114	C₂×C₄₃	—	8	2	no	—
115	C₂×C₄₃	#1800	1	1	yes	characteristic
116	C₂×C₄₃	#1816	1	2	yes	—
117	C₂×#131	—	43	2	no	—
118	#532	C₂×D₄₃	1	1	yes	characteristic
119	#436	—	86	2	no	—
120	C₈×D₄	—	86	2	no	—
121	#423	—	43	1	no	—
122	#423	—	43	2	no	—
123	#423	C₂×D₄₃	1	2	yes	—
124	#408	—	43	2	no	—
125	C₂×C₄×C₈	—	43	1	no	—
126	#335	—	86	2	no	—
127	#335	—	86	2	no	—
128	#333	—	2	1	no	—
129	#333	—	86	1	no	—
130	#331	—	2	1	no	—
131	#331	—	86	1	no	—
132	C₄₃	#8638	1	1	yes	characteristic, complemented, Sylow
133	C₂×#41	C₄×D₄₃	1	2	yes	—
134	#131	—	172	4	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	—	43	2	no	—
137	C₂×M₁₆	—	86	2	no	—
138	C₂×M₁₆	—	86	2	no	—
139	C₂²×C₈	—	43	1	no	—
140	C₂²×C₈	—	43	1	no	—
141	C₂²×C₈	—	86	2	no	—
142	C₄⋊Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
143	#127	C₂²⋊D₄₃	1	1	yes	characteristic
144	#124	Dic₈₆	1	2	yes	—
145	#121	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	86	2	no	—
151	C₄⋊C₈	—	86	2	no	—
152	C₄⋊C₈	—	86	2	no	—
153	C₂²⋊C₈	—	86	4	no	—
154	C₄×C₈	—	43	2	no	—
155	C₄×C₈	—	43	2	no	—
156	#41	—	4	2	no	—
157	#41	—	4	2	no	—
158	C₂×Q₈	Dic₄₃⋊C₄	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	Dic₄₃⋊C₄	1	2	yes	—
162	C₂²×C₄	—	2	2	no	—
163	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
164	C₂²×C₄	#10621900	1	2	yes	—
165	M₁₆	—	8	2	no	—
166	M₁₆	—	344	2	no	—
167	M₁₆	—	172	4	no	—
168	M₁₆	—	172	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	43	1	no	—
171	C₂×C₈	—	43	2	no	—
172	C₂×C₈	—	43	1	no	—
173	C₂×C₈	—	86	2	no	—
174	C₂×C₈	—	86	2	no	—
175	C₂×C₈	—	172	4	no	—
176	C₂×C₈	—	172	1	no	—
177	C₂×C₈	—	172	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
181	C₄²	D₄×D₄₃	1	2	yes	—
182	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
183	C₂³	—	2	2	no	—
184	C₂³	#13063281	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	D₄	—	4	2	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	2	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#13063297	1	2	yes	—
200	C₂×C₄	#13063302	1	2	yes	—
201	C₈	—	8	2	no	—
202	C₈	—	43	2	no	—
203	C₈	—	86	1	no	—
204	C₈	—	86	2	no	—
205	C₈	—	344	2	no	—
206	C₈	—	172	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	4	no	—
211	C₂²	#14582757	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	4	2	no	—
215	C₄	#100×D₄₃	1	1	yes	characteristic
216	C₄	#14582880	1	1	yes	characteristic
217	C₄	#14583167	1	2	yes	—
218	C₄	#14583172	1	2	yes	—
219	C₂	—	4	1	no	—
220	C₂	—	8	2	no	—
221	C₂	#19134585	1	1	yes	characteristic
222	C₂	#19134657	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143307

C₂

yes

maximal, characteristic

#19133353

C₂

yes

maximal, complemented

#19133330

C₂

yes

maximal, characteristic

C₄₃⋊#1309

C₂

yes

maximal, complemented

C₄₃×#1560

C₂

yes

maximal, characteristic

C₂×#13063332

C₄

yes

—

#14582977

—

D₄×C₄₃⋊C₈

—

#14582923

C₂²

yes

—

#14582923

C₂²

yes

characteristic

#14582866

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

—

—

—

C₄₃×#532

C₂²

yes

characteristic

C₄₃×#423

C₂²

yes

—

C₄₃×#333

—

C₄₃×#331

—

—

—

—

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×#41×C₄₃

C₂×C₄

yes

—

C₂×M₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

D₄

yes

characteristic

#127×C₄₃

D₄

yes

characteristic

#124×C₄₃

Q₈

yes

—

#121×C₄₃

—

C₄×D₄×C₄₃

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#41×C₄₃

—

#41×C₄₃

—

C₂×Q₈×C₄₃

C₄⋊C₄

yes

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

C₄⋊C₄

yes

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₂³×C₄₃

—

C₂³×C₄₃

#117

yes

characteristic

Q₈×C₄₃

—

D₄×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

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#100

yes

—

#8638

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

100

C₂²×C₄₃

#380

yes

characteristic

101

C₄×C₄₃

—

102

C₄×C₄₃

—

103

C₄×C₄₃

—

104

C₄×C₄₃

C₂×#100

yes

characteristic

105

C₄×C₄₃

#413

yes

characteristic

106

C₄×C₄₃

#472

yes

—

107

C₄×C₄₃

#473

yes

—

108

#2931

—

109

#1567

—

110

#1560

—

111

#1560

D₄₃

yes

characteristic

112

#1309

—

113

C₂×C₄₃

—

114

C₂×C₄₃

—

115

C₂×C₄₃

#1800

yes

characteristic

116

C₂×C₄₃

#1816

yes

—

117

C₂×#131

—

118

#532

C₂×D₄₃

yes

characteristic

119

#436

—

120

C₈×D₄

—

121

#423

—

122

#423

—

123

#423

C₂×D₄₃

yes

—

124

#408

—

125

C₂×C₄×C₈

—

126

#335

—

127

#335

—

128

#333

—

129

#333

—

130

#331

—

131

#331

—

132

C₄₃

#8638

yes

characteristic, complemented, Sylow

133

C₂×#41

C₄×D₄₃

yes

—

134

#131

—

172

—

135

C₂×M₁₆

—

136

C₂×M₁₆

—

137

C₂×M₁₆

—

138

C₂×M₁₆

—

139

C₂²×C₈

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

C₂²⋊D₄₃

yes

characteristic

143

#127

C₂²⋊D₄₃

yes

characteristic

144

#124

Dic₈₆

yes

—

145

#121

—

146

C₄×D₄

—

147

C₂×C₄²

C₂²×D₄₃

yes

characteristic

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

C₂²⋊C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

#41

—

157

#41

—

158

C₂×Q₈

Dic₄₃⋊C₄

yes

—

159

C₂×D₄

—

160

C₂×D₄

—

161

C₂×D₄

Dic₄₃⋊C₄

yes

—

162

C₂²×C₄

—

163

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

164

C₂²×C₄

#10621900

yes

—

165

M₁₆

—

166

M₁₆

—

344

—

167

M₁₆

—

172

—

168

M₁₆

—

172

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

172

—

176

C₂×C₈

—

172

—

177

C₂×C₈

—

172

—

178

C₄⋊C₄

—

179

C₂²⋊C₄

—

180

C₄²

C₂×Dic₈₆

yes

characteristic

181

C₄²

D₄×D₄₃

yes

—

182

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

183

C₂³

—

184

C₂³

#13063281

yes

characteristic

185

Q₈

—

186

D₄

—

187

D₄

—

188

D₄

—

189

D₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

C₂²⋊C₄×D₄₃

yes

characteristic

198

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

199

C₂×C₄

#13063297

yes

—

200

C₂×C₄

#13063302

yes

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

344

—

206

C₈

—

172

—

207

C₂²

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

#14582757

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

#100×D₄₃

yes

characteristic

216

C₄

#14582880

yes

characteristic

217

C₄

#14583167

yes

—

218

C₄

#14583172

yes

—

219

C₂

—

220

C₂

—

221

C₂

#19134585

yes

characteristic

222

C₂

#19134657

yes

—

	subgroup	G/H
center	C₂²	#14582757
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1560	C₂
socle	C₂²×C₄₃	#380
prime radical	C₄₃×#532	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1560	D₄₃
hypocenter	C₄₃	#8638
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1560

subgroup

G/H

center

C₂²

#14582757

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1560

C₂

socle

C₂²×C₄₃

#380

prime radical

C₄₃×#532

C₂²

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1560

N	H ≅ G/N	type
2. #19133353	C₂	semidirect
4. C₄₃⋊#1309	C₂	semidirect
132. C₄₃	#8638	semidirect

H ≅ G/N

type

2. #19133353

C₂

semidirect

4. C₄₃⋊#1309

C₂

semidirect

132. C₄₃

#8638

semidirect

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

subgroup	index
#19143307	2
#19133353	2
#19133330	2
C₄₃⋊#1309	2
C₄₃×#1560	2
#8638	43

subgroup

index

#19143307

#19133353

#19133330

C₄₃⋊#1309

C₄₃×#1560

#8638

Subgroups — #24989621

Conjugacy classes of subgroups

Complements

Maximal subgroups