Group #25015687 — subgroups

Subgroups — #25015687

Order 11008.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19145119	C₂	1	1	yes	maximal, characteristic
2	#19136316	C₂	1	2	yes	maximal
3	#19135466	C₂	1	1	yes	maximal, characteristic
4	#19132701	C₂	1	2	yes	maximal
5	C₄₃×#1377	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
7	#14583456	—	2	1	no	—
8	#14583308	—	2	4	no	—
9	#14582937	—	2	1	no	—
10	#14582937	C₂²	1	2	yes	—
11	C₂×#13063257	C₂²	1	1	yes	characteristic
12	C₂×#13063250	—	2	1	no	—
13	#14582805	—	2	1	no	—
14	#14582704	—	2	4	no	—
15	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
16	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×Dic₈₆	—	2	1	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	2	2	no	—
20	C₂×Dic₁₇₂	—	2	1	no	—
21	C₂×Dic₁₇₂	Q₈	1	2	yes	complemented
22	#13063325	—	2	2	no	—
23	#13063317	—	2	2	no	—
24	#13063304	—	4	2	no	—
25	C₄×Dic₈₆	—	4	2	no	—
26	#13063284	—	2	2	no	—
27	C₂×C₄⋊Dic₄₃	—	2	1	no	—
28	#13063259	—	2	2	no	—
29	#13063259	D₄	1	4	yes	—
30	#13063257	—	2	2	no	—
31	#13063257	—	2	2	no	—
32	#13063257	—	2	2	no	—
33	#13063250	—	4	2	no	—
34	C₄₃⋊#95	—	4	2	no	—
35	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
38	M₃₂×C₄₃	—	2	2	no	—
39	M₃₂×C₄₃	—	2	2	no	—
40	C₂×C₁₆×C₄₃	—	2	2	no	—
41	#97×C₄₃	—	2	2	no	—
42	C₂²×C₄×C₄₃	D₈	1	2	yes	—
43	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₄₃	—	4	1	no	—
45	C₂²×Dic₄₃	—	4	1	no	—
46	C₂×Dic₈₆	—	2	2	no	—
47	C₂×Dic₈₆	—	4	1	no	—
48	C₂×Dic₈₆	—	4	1	no	—
49	C₂×Dic₈₆	—	4	1	no	—
50	C₂×Dic₈₆	—	4	1	no	—
51	C₂×Dic₈₆	—	4	2	no	—
52	C₂×Dic₈₆	—	4	1	no	—
53	C₂×C₈×C₄₃	—	2	1	no	—
54	C₂×C₈×C₄₃	—	2	1	no	—
55	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₄₃	#41	1	2	yes	—
59	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
60	C₄²×C₄₃	—	2	1	no	—
61	C₄²×C₄₃	—	2	1	no	—
62	C₂²⋊Dic₄₃	—	4	2	no	—
63	C₄⋊Dic₄₃	—	2	4	no	—
64	C₄⋊Dic₄₃	—	4	2	no	—
65	C₄⋊Dic₄₃	—	4	1	no	—
66	C₄⋊Dic₄₃	—	4	1	no	—
67	Dic₄₃⋊C₄	—	4	4	no	—
68	C₄×Dic₄₃	—	4	2	no	—
69	Dic₁₇₂	—	2	4	no	—
70	Dic₁₇₂	—	4	2	no	—
71	C₁₆×C₄₃	—	4	2	no	—
72	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₄₃	—	2	2	no	—
74	C₂×C₄×C₄₃	—	4	2	no	—
75	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
76	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
78	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
81	C₂×Dic₄₃	—	4	4	no	—
82	C₂×Dic₄₃	—	4	2	no	—
83	C₂×Dic₄₃	—	8	1	no	—
84	C₂×Dic₄₃	—	8	1	no	—
85	C₂×Dic₄₃	—	8	2	no	—
86	C₂×Dic₄₃	—	8	1	no	—
87	Dic₈₆	—	4	2	no	—
88	Dic₈₆	—	4	2	no	—
89	Dic₈₆	—	8	4	no	—
90	Dic₈₆	—	8	1	no	—
91	Dic₈₆	—	8	1	no	—
92	C₈×C₄₃	—	2	2	no	—
93	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
94	C₈×C₄₃	#124	1	2	yes	—
95	#13368	—	43	1	no	maximal, Sylow
96	C₂²×C₄₃	—	2	2	no	—
97	C₂²×C₄₃	#447	1	1	yes	characteristic
98	C₂²×C₄₃	#474	1	1	yes	characteristic
99	C₂²×C₄₃	#486	1	1	yes	characteristic
100	C₄×C₄₃	—	4	2	no	—
101	C₄×C₄₃	#393	1	1	yes	characteristic
102	C₄×C₄₃	#448	1	1	yes	characteristic
103	C₄×C₄₃	#476	1	1	yes	characteristic
104	C₄×C₄₃	#489	1	1	yes	characteristic
105	Dic₄₃	—	8	4	no	—
106	Dic₄₃	—	8	2	no	—
107	Dic₄₃	—	16	1	no	—
108	#3162	—	43	1	no	—
109	#2141	—	43	2	no	—
110	#1971	—	43	1	no	—
111	#1386	—	43	2	no	—
112	#1377	D₄₃	1	1	yes	characteristic
113	C₂×C₄₃	—	2	1	no	—
114	C₂×C₄₃	—	2	1	no	—
115	C₂×C₄₃	#2023	1	1	yes	characteristic
116	C₂×C₄₃	#2191	1	2	yes	—
117	C₂²×Q₁₆	—	43	1	no	—
118	#533	—	86	1	no	—
119	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
120	#501	—	86	4	no	—
121	#427	—	43	2	no	—
122	#427	—	86	1	no	—
123	C₂×#106	—	43	1	no	—
124	C₂×#103	—	86	1	no	—
125	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
126	#391	—	86	1	no	—
127	#358	—	86	4	no	—
128	C₄₃	#13368	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₈	—	86	1	no	—
130	C₂×Q₁₆	—	43	2	no	—
131	C₂×Q₁₆	—	86	2	no	—
132	C₂×Q₁₆	—	86	2	no	—
133	C₂×Q₁₆	—	86	1	no	—
134	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	86	2	no	—
137	#125	—	86	2	no	—
138	C₂²⋊Q₈	—	172	2	no	—
139	C₄×Q₈	—	172	2	no	—
140	#117	—	86	2	no	—
141	C₂×C₄⋊C₄	—	86	1	no	—
142	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
143	M₃₂	—	2	2	no	—
144	M₃₂	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	#107	—	43	4	no	—
147	#107	—	86	2	no	—
148	#106	—	86	2	no	—
149	#106	—	86	2	no	—
150	#106	—	86	2	no	—
151	#103	—	172	2	no	—
152	#97	—	2	2	no	—
153	#95	—	172	2	no	—
154	C₂×Q₈	—	86	2	no	—
155	C₂×Q₈	—	172	2	no	—
156	C₂×Q₈	—	172	1	no	—
157	C₂×Q₈	—	172	1	no	—
158	C₂×Q₈	—	172	1	no	—
159	C₂×Q₈	—	172	1	no	—
160	C₂×Q₈	—	172	1	no	—
161	C₂²×C₄	—	172	1	no	—
162	C₂²×C₄	—	172	1	no	—
163	C₂²×C₄	D₃₄₄	1	2	yes	—
164	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
165	Q₁₆	—	86	4	no	—
166	Q₁₆	—	172	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	#10621873	1	2	yes	—
170	C₂×C₈	#10621898	1	2	yes	—
171	C₂×C₈	D₄×D₄₃	1	2	yes	—
172	C₂×C₈	#10621900	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
174	C₄⋊C₄	—	86	4	no	—
175	C₄⋊C₄	—	172	4	no	—
176	C₄⋊C₄	—	172	2	no	—
177	C₄⋊C₄	—	172	1	no	—
178	C₄⋊C₄	—	172	1	no	—
179	C₂²⋊C₄	—	172	2	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	172	2	no	—
183	C₁₆	—	4	2	no	—
184	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
185	Q₈	—	344	4	no	—
186	Q₈	—	344	1	no	—
187	Q₈	—	344	1	no	—
188	Q₈	—	172	2	no	—
189	Q₈	—	172	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	344	1	no	—
193	C₂×C₄	—	344	1	no	—
194	C₂×C₄	—	344	2	no	—
195	C₂×C₄	—	344	1	no	—
196	C₂×C₄	—	172	2	no	—
197	C₂×C₄	—	172	4	no	—
198	C₂×C₄	#13063295	1	1	yes	characteristic
199	C₂×C₄	#13063305	1	1	yes	characteristic
200	C₂×C₄	#13063313	1	1	yes	characteristic
201	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
202	C₂×C₄	#13063353	1	1	yes	characteristic
203	C₂×C₄	#13063358	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	#13063303	1	2	yes	—
206	C₈	#13063311	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	#14583035	1	1	yes	characteristic
209	C₂²	#14583180	1	1	yes	characteristic
210	C₂²	#14583246	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	344	4	no	—
213	C₄	—	344	2	no	—
214	C₄	—	688	1	no	—
215	C₄	#14582812	1	1	yes	characteristic
216	C₄	#14583040	1	1	yes	characteristic
217	C₄	#14583190	1	1	yes	characteristic
218	C₄	#14583259	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	#19135732	1	1	yes	characteristic
222	C₂	#19136548	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145119

C₂

yes

maximal, characteristic

#19136316

C₂

yes

maximal

#19135466

C₂

yes

maximal, characteristic

#19132701

C₂

yes

maximal

C₄₃×#1377

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₂

C₂²

yes

—

—

—

C₂²

yes

—

C₂×#13063257

C₂²

yes

—

—

—

C₂×M₃₂×C₄₃

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₂²×Dic₈₆

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

Q₈

yes

—

—

—

C₄×Dic₈₆

—

#13063284

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063259

D₄

yes

—

—

—

—

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

M₃₂×C₄₃

—

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

characteristic

C₄²×C₄₃

—

C₄²×C₄₃

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

C₈×C₄₃

#124

yes

—

#13368

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

100

C₄×C₄₃

—

101

C₄×C₄₃

#393

yes

characteristic

102

C₄×C₄₃

#448

yes

characteristic

103

C₄×C₄₃

#476

yes

characteristic

104

C₄×C₄₃

#489

yes

characteristic

105

Dic₄₃

—

106

Dic₄₃

—

107

Dic₄₃

—

108

#3162

—

109

#2141

—

110

#1971

—

111

#1386

—

112

#1377

D₄₃

yes

characteristic

113

C₂×C₄₃

—

114

C₂×C₄₃

—

115

C₂×C₄₃

#2023

yes

characteristic

116

C₂×C₄₃

#2191

yes

—

117

C₂²×Q₁₆

—

118

#533

—

119

C₂×M₃₂

C₂×D₄₃

yes

—

120

#501

—

121

#427

—

122

#427

—

123

C₂×#106

—

124

C₂×#103

—

125

C₂×#97

C₂×D₄₃

yes

characteristic

126

#391

—

127

#358

—

128

C₄₃

#13368

yes

characteristic, complemented, Sylow

129

C₂²×Q₈

—

130

C₂×Q₁₆

—

131

C₂×Q₁₆

—

132

C₂×Q₁₆

—

133

C₂×Q₁₆

—

134

C₂²×C₈

D₁₇₂

yes

characteristic

135

C₂²×C₈

C₂²×D₄₃

yes

characteristic

136

C₄⋊Q₈

—

137

#125

—

138

C₂²⋊Q₈

—

172

—

139

C₄×Q₈

—

172

—

140

#117

—

141

C₂×C₄⋊C₄

—

142

C₂×C₄²

D₁₇₂

yes

characteristic

143

M₃₂

—

144

M₃₂

—

145

C₂×C₁₆

—

146

#107

—

147

#107

—

148

#106

—

149

#106

—

150

#106

—

151

#103

—

172

—

152

#97

—

153

#95

—

172

—

154

C₂×Q₈

—

155

C₂×Q₈

—

172

—

156

C₂×Q₈

—

172

—

157

C₂×Q₈

—

172

—

158

C₂×Q₈

—

172

—

159

C₂×Q₈

—

172

—

160

C₂×Q₈

—

172

—

161

C₂²×C₄

—

172

—

162

C₂²×C₄

—

172

—

163

C₂²×C₄

D₃₄₄

yes

—

164

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

165

Q₁₆

—

166

Q₁₆

—

172

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

#10621873

yes

—

170

C₂×C₈

#10621898

yes

—

171

C₂×C₈

D₄×D₄₃

yes

—

172

C₂×C₈

#10621900

yes

characteristic

173

C₂×C₈

Q₈×D₄₃

yes

characteristic

174

C₄⋊C₄

—

175

C₄⋊C₄

—

172

—

176

C₄⋊C₄

—

172

—

177

C₄⋊C₄

—

172

—

178

C₄⋊C₄

—

172

—

179

C₂²⋊C₄

—

172

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

172

—

183

C₁₆

—

184

C₂³

C₂×D₃₄₄

yes

characteristic

185

Q₈

—

344

—

186

Q₈

—

344

—

187

Q₈

—

344

—

188

Q₈

—

172

—

189

Q₈

—

172

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

344

—

193

C₂×C₄

—

344

—

194

C₂×C₄

—

344

—

195

C₂×C₄

—

344

—

196

C₂×C₄

—

172

—

197

C₂×C₄

—

172

—

198

C₂×C₄

#13063295

yes

characteristic

199

C₂×C₄

#13063305

yes

characteristic

200

C₂×C₄

#13063313

yes

characteristic

201

C₂×C₄

C₂×#10621873

yes

characteristic

202

C₂×C₄

#13063353

yes

characteristic

203

C₂×C₄

#13063358

yes

characteristic

204

C₈

—

205

C₈

#13063303

yes

—

206

C₈

#13063311

yes

—

207

C₂²

—

208

C₂²

#14583035

yes

characteristic

209

C₂²

#14583180

yes

characteristic

210

C₂²

#14583246

yes

characteristic

211

C₄

—

212

C₄

—

344

—

213

C₄

—

344

—

214

C₄

—

688

—

215

C₄

#14582812

yes

characteristic

216

C₄

#14583040

yes

characteristic

217

C₄

#14583190

yes

characteristic

218

C₄

#14583259

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

#19135732

yes

characteristic

222

C₂

#19136548

yes

—

	subgroup	G/H
center	C₂²	#14583035
commutator, cosocle	C₂²×C₈×C₄₃	C₂³
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1377	C₂
socle	C₂²×C₄₃	#447
prime radical	C₂³×C₄₃	C₂×D₈
hypercenter	#1377	D₄₃
hypocenter	C₄₃	#13368
LCS	G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

subgroup

G/H

center

C₂²

#14583035

commutator, cosocle

C₂²×C₈×C₄₃

C₂³

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1377

C₂

socle

C₂²×C₄₃

#447

prime radical

C₂³×C₄₃

C₂×D₈

D₄₃

C₄₃

G ▷ C₂²×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1377

N	H ≅ G/N	type
21. C₂×Dic₁₇₂	Q₈	semidirect
128. C₄₃	#13368	semidirect

H ≅ G/N

type

21. C₂×Dic₁₇₂

Q₈

semidirect

128. C₄₃

#13368

semidirect

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

subgroup	index
#19145119	2
#19136316	2
#19135466	2
#19132701	2
C₄₃×#1377	2
#13368	43

subgroup

index

C₄₃×#1377

Subgroups — #25015687

Conjugacy classes of subgroups

Complements

Maximal subgroups