Group #24987522 — subgroups

Subgroups — #24987522

Order 11008.

444 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 129 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	#19137150	C₂	1	1	yes	maximal, characteristic
2	#19134343	C₂	1	2	yes	maximal
3	C₂×#14582640	C₂	1	2	yes	maximal
4	#19132476	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1295	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063317	C₂²	1	1	yes	characteristic
7	C₂×#13063236	C₂²	1	2	yes	—
8	#14582795	C₄	1	2	yes	—
9	#14582781	—	2	2	no	—
10	C₄⋊C₄×Dic₄₃	C₄	1	2	yes	—
11	#14582750	—	2	2	no	—
12	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
13	#14582653	—	2	2	no	—
14	#14582653	—	2	2	no	—
15	#14582640	C₄	1	8	yes	complemented
16	C₂×#97×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#382	C₂²	1	1	yes	characteristic
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	#13063317	—	2	1	no	—
20	#13063317	—	2	1	no	—
21	#13063317	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
23	C₂×Dic₄₃⋊C₄	—	2	2	no	—
24	C₂×Dic₄₃⋊C₄	—	2	2	no	—
25	C₂×C₄×Dic₄₃	—	2	2	no	—
26	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
30	#13063236	—	2	2	no	—
31	#13063236	C₂×C₄	1	4	yes	—
32	C₄₃⋊#95	—	2	2	no	—
33	C₄₃⋊#95	—	2	2	no	—
34	C₂²×C₈×C₄₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	#97×C₄₃	—	2	2	no	—
38	#97×C₄₃	C₂×C₄	1	4	yes	—
39	#95×C₄₃	—	2	2	no	—
40	C₂²×C₄×C₄₃	—	2	2	no	—
41	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
43	C₂²×Dic₄₃	—	2	2	no	—
44	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
46	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₄₃	—	2	4	no	—
48	C₂×C₈×C₄₃	—	4	2	no	—
49	C₂×C₈×C₄₃	—	4	2	no	—
50	C₄⋊C₄×C₄₃	—	4	2	no	—
51	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
52	C₄²×C₄₃	#41	1	1	yes	characteristic
53	C₄²×C₄₃	#41	1	1	yes	characteristic
54	C₄⋊Dic₄₃	—	2	2	no	—
55	C₄⋊Dic₄₃	C₄²	1	4	yes	—
56	Dic₄₃⋊C₄	—	2	4	no	—
57	Dic₄₃⋊C₄	—	2	4	no	—
58	Dic₄₃⋊C₄	—	4	4	no	—
59	C₄×Dic₄₃	—	2	2	no	—
60	C₄×Dic₄₃	—	4	1	no	—
61	C₄×Dic₄₃	—	4	2	no	—
62	C₄×Dic₄₃	—	4	2	no	—
63	C₄×Dic₄₃	—	4	4	no	—
64	C₂×C₄₃⋊C₈	—	2	4	no	—
65	C₂×C₄₃⋊C₈	—	4	2	no	—
66	C₂×C₄₃⋊C₈	—	4	2	no	—
67	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₄₃	—	2	1	no	—
69	C₂×C₄×C₄₃	—	2	1	no	—
70	C₂×C₄×C₄₃	—	2	2	no	—
71	C₂×C₄×C₄₃	—	2	2	no	—
72	C₂×C₄×C₄₃	—	4	2	no	—
73	C₂×C₄×C₄₃	—	4	2	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
79	C₂×Dic₄₃	—	2	2	no	—
80	C₂×Dic₄₃	—	2	4	no	—
81	C₂×Dic₄₃	—	2	1	no	—
82	C₂×Dic₄₃	—	2	2	no	—
83	C₂×Dic₄₃	—	2	2	no	—
84	C₂×Dic₄₃	—	2	2	no	—
85	C₂×Dic₄₃	—	2	2	no	—
86	C₂×Dic₄₃	—	4	2	no	—
87	C₂×Dic₄₃	—	4	2	no	—
88	C₂×Dic₄₃	#101	1	2	yes	—
89	C₈×C₄₃	—	4	4	no	—
90	C₄₃⋊C₈	—	4	4	no	—
91	#8225	—	43	1	no	maximal, Sylow
92	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
94	C₂²×C₄₃	#413	1	1	yes	characteristic
95	C₂²×C₄₃	#421	1	2	yes	—
96	C₂²×C₄₃	#421	1	2	yes	—
97	C₄×C₄₃	—	2	1	no	—
98	C₄×C₄₃	—	2	1	no	—
99	C₄×C₄₃	—	2	2	no	—
100	C₄×C₄₃	—	2	2	no	—
101	C₄×C₄₃	—	8	2	no	—
102	Dic₄₃	—	2	2	no	—
103	Dic₄₃	—	4	1	no	—
104	Dic₄₃	—	4	4	no	—
105	Dic₄₃	—	4	2	no	—
106	C₄×#125	—	43	1	no	—
107	#1745	—	43	2	no	—
108	C₂×#330	—	43	2	no	—
109	#1295	—	43	1	no	—
110	#1295	D₄₃	1	1	yes	characteristic
111	C₂×C₄₃	#1534	1	1	yes	characteristic
112	C₂×C₄₃	#1535	1	2	yes	—
113	C₂×C₄₃	#1537	1	1	yes	characteristic
114	C₂×C₄₃	C₄×#101	1	1	yes	characteristic
115	C₂×C₄₃	#1759	1	2	yes	—
116	C₂×#125	—	43	1	no	—
117	C₂×#97	—	43	2	no	—
118	C₂×#97	C₂×D₄₃	1	2	yes	—
119	#389	—	43	2	no	—
120	#387	—	86	2	no	—
121	#382	C₂×D₄₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	43	2	no	—
123	C₄×C₄⋊C₄	—	86	2	no	—
124	C₄³	—	43	1	no	—
125	#336	—	86	2	no	—
126	#336	—	86	2	no	—
127	#330	—	43	8	no	—
128	C₄₃	#8225	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	86	2	no	—
131	#125	—	43	4	no	—
132	#125	—	86	1	no	—
133	#125	—	86	1	no	—
134	C₂×C₄⋊C₄	—	43	2	no	—
135	C₂×C₄⋊C₄	—	86	2	no	—
136	C₂×C₄⋊C₄	—	86	2	no	—
137	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
138	C₂×C₄²	—	43	1	no	—
139	C₂×C₄²	—	43	1	no	—
140	C₂×C₄²	—	43	2	no	—
141	C₂×C₄²	—	43	2	no	—
142	C₂×C₄²	—	86	2	no	—
143	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
144	#97	—	2	2	no	—
145	#97	—	43	4	no	—
146	#97	—	86	2	no	—
147	#97	C₄×D₄₃	1	4	yes	—
148	#95	—	2	2	no	—
149	#95	—	86	2	no	—
150	#95	—	86	2	no	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	43	1	no	—
153	C₂²×C₄	—	43	1	no	—
154	C₂²×C₄	—	43	2	no	—
155	C₂²×C₄	—	86	2	no	—
156	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×D₄₃	1	2	yes	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	86	4	no	—
162	C₂×C₈	—	172	2	no	—
163	C₂×C₈	—	172	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	43	4	no	—
166	C₄⋊C₄	—	86	2	no	—
167	C₄⋊C₄	—	86	4	no	—
168	C₄⋊C₄	—	86	4	no	—
169	C₄⋊C₄	—	172	4	no	—
170	C₄²	—	86	2	no	—
171	C₄²	—	172	1	no	—
172	C₄²	—	172	2	no	—
173	C₄²	—	172	2	no	—
174	C₄²	—	172	4	no	—
175	C₄²	C₂×C₄×D₄₃	1	2	yes	—
176	C₄²	#10621900	1	1	yes	characteristic
177	C₄²	#10621902	1	1	yes	characteristic
178	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	43	2	no	—
187	C₂×C₄	—	86	2	no	—
188	C₂×C₄	—	86	1	no	—
189	C₂×C₄	—	86	2	no	—
190	C₂×C₄	—	86	2	no	—
191	C₂×C₄	—	86	2	no	—
192	C₂×C₄	—	86	2	no	—
193	C₂×C₄	—	86	4	no	—
194	C₂×C₄	—	172	2	no	—
195	C₂×C₄	—	172	2	no	—
196	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
197	C₂×C₄	#13063283	1	1	yes	characteristic
198	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
199	C₂×C₄	#13063287	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	172	4	no	—
202	C₂²	#14582746	1	1	yes	characteristic
203	C₂²	#101×D₄₃	1	1	yes	characteristic
204	C₂²	#14582880	1	1	yes	characteristic
205	C₂²	#14582914	1	2	yes	—
206	C₂²	#14582914	1	2	yes	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	—	8	2	no	—
212	C₄	—	86	2	no	—
213	C₄	—	172	2	no	—
214	C₄	—	172	1	no	—
215	C₄	—	172	4	no	—
216	C₂	#19133220	1	1	yes	characteristic
217	C₂	#19133223	1	2	yes	—
218	C₂	#19133231	1	1	yes	characteristic
219	C₂	#19134390	1	1	yes	characteristic
220	C₂	#19134404	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19137150

C₂

yes

maximal, characteristic

#19134343

C₂

yes

maximal

C₂×#14582640

C₂

yes

maximal

#19132476

C₂

yes

maximal, characteristic

C₄₃×#1295

C₂

yes

maximal, characteristic

C₂×#13063317

C₂²

yes

characteristic

C₂×#13063236

C₂²

yes

—

#14582795

C₄

yes

—

#14582781

—

C₄⋊C₄×Dic₄₃

C₄

yes

—

#14582750

—

C₄²×Dic₄₃

C₂²

yes

—

—

C₄

yes

complemented

C₂×#97×C₄₃

C₂²

yes

—

C₄₃×#382

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

#13063317

—

#13063317

—

#13063317

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×Dic₄₃⋊C₄

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₃

D₄

yes

—

C₂×C₄×Dic₄₃

D₄

yes

—

#13063236

—

#13063236

C₂×C₄

yes

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#97×C₄₃

—

#97×C₄₃

C₂×C₄

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₄₃

C₂²⋊C₄

yes

—

C₂²×Dic₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

yes

C₄²×C₄₃

yes

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₄²

yes

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

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×C₄²

yes

characteristic

C₂×C₄×C₄₃

#117

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

#101

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#8225

—

maximal, Sylow

C₂²×C₄₃

C₄×C₂²⋊C₄

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#421

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

100

C₄×C₄₃

—

101

C₄×C₄₃

—

102

Dic₄₃

—

103

Dic₄₃

—

104

Dic₄₃

—

105

Dic₄₃

—

106

C₄×#125

—

107

#1745

—

108

C₂×#330

—

109

#1295

—

110

#1295

D₄₃

yes

characteristic

111

C₂×C₄₃

#1534

yes

characteristic

112

C₂×C₄₃

#1535

yes

—

113

C₂×C₄₃

#1537

yes

characteristic

114

C₂×C₄₃

C₄×#101

yes

characteristic

115

C₂×C₄₃

#1759

yes

—

116

C₂×#125

—

117

C₂×#97

—

118

C₂×#97

C₂×D₄₃

yes

—

119

#389

—

120

#387

—

121

#382

C₂×D₄₃

yes

characteristic

122

C₄×C₄⋊C₄

—

123

C₄×C₄⋊C₄

—

124

C₄³

—

125

#336

—

126

#336

—

127

#330

—

128

C₄₃

#8225

yes

characteristic, complemented, Sylow

129

C₂²×C₈

—

130

C₂²×C₈

—

131

#125

—

132

#125

—

133

#125

—

134

C₂×C₄⋊C₄

—

135

C₂×C₄⋊C₄

—

136

C₂×C₄⋊C₄

—

137

C₂×C₄⋊C₄

C₄×D₄₃

yes

—

138

C₂×C₄²

—

139

C₂×C₄²

—

140

C₂×C₄²

—

141

C₂×C₄²

—

142

C₂×C₄²

—

143

C₂×C₄²

C₂²×D₄₃

yes

characteristic

144

#97

—

145

#97

—

146

#97

—

147

#97

C₄×D₄₃

yes

—

148

#95

—

149

#95

—

150

#95

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

154

C₂²×C₄

—

155

C₂²×C₄

—

156

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

157

C₂²×C₄

D₄×D₄₃

yes

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

172

—

163

C₂×C₈

—

172

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

172

—

170

C₄²

—

171

C₄²

—

172

—

172

C₄²

—

172

—

173

C₄²

—

172

—

174

C₄²

—

172

—

175

C₄²

C₂×C₄×D₄₃

yes

—

176

C₄²

#10621900

yes

characteristic

177

C₄²

#10621902

yes

characteristic

178

C₂³

C₂²⋊C₄×D₄₃

yes

characteristic

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

172

—

195

C₂×C₄

—

172

—

196

C₂×C₄

C₄²×D₄₃

yes

characteristic

197

C₂×C₄

#13063283

yes

characteristic

198

C₂×C₄

(C₂²⋊D₄₃)⋊C₄

yes

—

199

C₂×C₄

#13063287

yes

—

200

C₈

—

201

C₈

—

172

—

202

C₂²

#14582746

yes

characteristic

203

C₂²

#101×D₄₃

yes

characteristic

204

C₂²

#14582880

yes

characteristic

205

C₂²

#14582914

yes

—

206

C₂²

#14582914

yes

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

172

—

214

C₄

—

172

—

215

C₄

—

172

—

216

C₂

#19133220

yes

characteristic

217

C₂

#19133223

yes

—

218

C₂

#19133231

yes

characteristic

219

C₂

#19134390

yes

characteristic

220

C₂

#19134404

yes

—

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1295	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1295	D₄₃
hypocenter	C₄₃	#8225
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

subgroup

G/H

center

C₂³

C₂²⋊C₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1295

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1295

N	H ≅ G/N	type
15. #14582640	C₄	semidirect
128. C₄₃	#8225	semidirect

H ≅ G/N

type

15. #14582640

C₄

semidirect

128. C₄₃

#8225

semidirect

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

subgroup	index
#19137150	2
#19134343	2
C₂×#14582640	2
#19132476	2
C₄₃×#1295	2
#8225	43

subgroup

index

C₄₃×#1295

Subgroups — #24987522

Conjugacy classes of subgroups

Complements

Maximal subgroups