Group #17955895 — subgroups

Subgroups — #17955895

Order 4352.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#14544725	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
3	#14544352	C₂	1	2	yes	maximal
4	#14543681	C₂	1	2	yes	maximal
5	#14527531	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587536	—	2	1	no	—
7	C₂×#10587536	—	2	1	no	—
8	#11763813	C₂²	1	2	yes	—
9	#11763810	C₂²	1	1	yes	characteristic
10	C₂×C₈×#595	—	2	1	no	—
11	C₄×C₄⋊#595	C₂²	1	1	yes	characteristic
12	#11763674	—	2	1	no	—
13	#11763671	—	2	2	no	—
14	#11763670	—	2	2	no	—
15	#11762861	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762562	C₄	1	1	yes	characteristic, complemented
18	#11762559	C₄	1	2	yes	complemented
19	C₁₇×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#595	—	2	1	no	—
21	C₂×C₄×#595	—	2	1	no	—
22	C₂×#63243	—	2	2	no	—
23	C₂×#63242	—	2	1	no	—
24	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
26	C₄²×D₁₇	C₂³	1	1	yes	characteristic
27	(C₂×#595)⋊C₄	Q₈	1	2	yes	—
28	#10587544	—	2	1	no	—
29	C₄⋊#3624	—	2	2	no	—
30	C₄⋊#3624	—	2	2	no	—
31	C₄⋊#3624	D₄	1	1	yes	characteristic
32	C₄⋊#3624	D₄	1	1	yes	characteristic
33	#10587536	—	4	2	no	—
34	#10587536	—	4	2	no	—
35	C₈×#595	—	4	2	no	—
36	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
38	#10587491	—	2	2	no	—
39	C₈×Dic₁₇	—	2	2	no	—
40	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
41	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
42	C₄⋊#3622	C₂×C₄	1	2	yes	—
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	C₂²×#595	—	2	2	no	—
45	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
48	C₂×#3624	—	2	1	no	—
49	C₂×#3624	—	2	1	no	—
50	C₂×#3624	—	4	2	no	—
51	C₄⋊#595	—	2	2	no	—
52	C₄⋊#595	—	2	2	no	—
53	C₄×#595	—	4	1	no	—
54	C₄×#595	—	4	1	no	—
55	#63243	—	4	2	no	—
56	#63243	—	4	2	no	—
57	#63242	—	4	1	no	—
58	#63242	—	4	1	no	—
59	C₂×C₈×C₁₇	—	2	2	no	—
60	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
63	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
64	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
65	C₄×Dic₁₇	#41	1	1	yes	characteristic
66	C₂×#3622	—	2	2	no	—
67	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
69	C₈×D₁₇	—	2	1	no	—
70	C₈×D₁₇	—	2	1	no	—
71	C₈×D₁₇	SD₁₆	1	2	yes	—
72	C₈×D₁₇	SD₁₆	1	2	yes	—
73	#9628	—	17	1	no	maximal, Sylow
74	C₂²×D₁₇	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×#595	—	2	4	no	—
76	C₂×#595	—	4	2	no	—
77	C₂×#595	—	4	2	no	—
78	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
81	C₂×Dic₁₇	#121	1	1	yes	characteristic
82	C₂×Dic₁₇	#125	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
84	C₄×D₁₇	—	2	1	no	—
85	C₄×D₁₇	—	2	1	no	—
86	C₄×D₁₇	—	4	1	no	—
87	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₄×D₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₄×D₁₇	#137	1	1	yes	characteristic
91	#3624	—	4	1	no	—
92	#3624	—	4	1	no	—
93	#3624	—	8	2	no	—
94	C₈×C₁₇	—	2	1	no	—
95	C₈×C₁₇	—	4	2	no	—
96	C₈×C₁₇	#106	1	2	yes	—
97	#3622	—	2	1	no	—
98	#3622	—	4	2	no	—
99	#3622	#106	1	2	yes	—
100	#1848	—	17	1	no	—
101	C₈×C₄⋊C₄	—	17	1	no	—
102	#1570	—	17	2	no	—
103	#1564	—	17	1	no	—
104	#1303	—	17	2	no	—
105	C₂²×C₁₇	#389	1	1	yes	characteristic
106	C₂×D₁₇	—	2	1	no	—
107	C₂×D₁₇	—	2	1	no	—
108	C₂×D₁₇	#475	1	1	yes	characteristic
109	C₂×D₁₇	#476	1	1	yes	characteristic
110	#595	—	4	4	no	—
111	C₄×C₁₇	—	2	1	no	—
112	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
113	C₄×C₁₇	#428	1	1	yes	characteristic
114	C₄×C₁₇	#443	1	1	yes	characteristic
115	C₄×C₁₇	#443	1	1	yes	characteristic
116	Dic₁₇	—	2	1	no	—
117	Dic₁₇	—	2	1	no	—
118	Dic₁₇	#460	1	1	yes	characteristic
119	Dic₁₇	#461	1	1	yes	characteristic
120	#431	—	17	2	no	—
121	C₂×#108	—	34	1	no	—
122	C₂×#108	—	34	1	no	—
123	#423	—	17	2	no	—
124	C₂×C₄⋊C₈	—	17	1	no	—
125	C₂×C₄×C₈	—	17	1	no	—
126	C₂×C₄×C₈	—	34	1	no	—
127	C₄×C₄⋊C₄	—	17	1	no	—
128	#336	—	34	1	no	—
129	#335	—	17	1	no	—
130	#335	#595	1	1	yes	characteristic, complemented
131	#334	—	17	2	no	—
132	#334	—	34	2	no	—
133	#334	—	34	2	no	—
134	C₂×C₁₇	#1945	1	1	yes	characteristic
135	C₂×C₁₇	#1948	1	1	yes	characteristic
136	C₂×C₁₇	#1953	1	1	yes	characteristic
137	D₁₇	—	2	1	no	—
138	D₁₇	—	2	1	no	—
139	C₂×M₁₆	—	34	2	no	—
140	C₂²×C₈	—	17	1	no	—
141	C₂²×C₈	—	17	1	no	—
142	C₂²×C₈	—	34	1	no	—
143	C₂×C₄⋊C₄	—	34	1	no	—
144	C₂×C₄²	—	17	1	no	—
145	C₂×C₄²	—	34	1	no	—
146	#108	—	68	2	no	—
147	#108	—	68	2	no	—
148	C₄⋊C₈	—	17	1	no	—
149	C₄⋊C₈	—	17	1	no	—
150	C₄⋊C₈	—	17	2	no	—
151	C₄⋊C₈	—	34	1	no	—
152	C₄⋊C₈	—	34	2	no	—
153	C₄⋊C₈	—	34	2	no	—
154	C₄⋊C₈	C₂×#595	1	2	yes	—
155	C₂²⋊C₈	—	34	2	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	—	17	1	no	—
159	C₄×C₈	—	34	2	no	—
160	C₄×C₈	—	68	2	no	—
161	C₄×C₈	C₂×#595	1	1	yes	characteristic
162	#95	—	17	2	no	—
163	C₁₇	#9628	1	1	yes	characteristic, complemented, Sylow
164	C₂²×C₄	—	17	1	no	—
165	C₂²×C₄	—	17	1	no	—
166	C₂²×C₄	—	17	1	no	—
167	C₂²×C₄	—	34	2	no	—
168	M₁₆	—	68	2	no	—
169	M₁₆	—	68	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	2	no	—
173	C₂×C₈	—	17	2	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	34	1	no	—
176	C₂×C₈	—	34	1	no	—
177	C₂×C₈	—	34	1	no	—
178	C₂×C₈	—	34	1	no	—
179	C₂×C₈	—	34	2	no	—
180	C₂×C₈	—	68	1	no	—
181	C₂×C₈	—	68	1	no	—
182	C₂×C₈	—	68	2	no	—
183	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
184	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
185	C₄⋊C₄	—	34	2	no	—
186	C₄⋊C₄	—	34	2	no	—
187	C₄²	—	17	1	no	—
188	C₄²	—	17	1	no	—
189	C₄²	—	17	1	no	—
190	C₄²	—	68	1	no	—
191	C₄²	—	68	1	no	—
192	C₄²	C₂²×#595	1	1	yes	characteristic
193	C₂³	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	17	1	no	—
198	C₂×C₄	—	17	1	no	—
199	C₂×C₄	—	17	1	no	—
200	C₂×C₄	—	17	1	no	—
201	C₂×C₄	—	34	1	no	—
202	C₂×C₄	—	34	1	no	—
203	C₂×C₄	—	34	4	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	68	2	no	—
206	C₂×C₄	—	68	2	no	—
207	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
208	C₂×C₄	D₄×#595	1	1	yes	characteristic
209	C₂×C₄	Q₈×#595	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	4	2	no	—
212	C₈	—	17	2	no	—
213	C₈	—	34	1	no	—
214	C₈	—	68	1	no	—
215	C₈	—	68	1	no	—
216	C₈	—	68	2	no	—
217	C₈	—	136	2	no	—
218	C₈	#10587533	1	2	yes	—
219	C₂²	—	17	1	no	—
220	C₂²	—	17	1	no	—
221	C₂²	—	34	1	no	—
222	C₂²	—	34	1	no	—
223	C₂²	#11763760	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	17	1	no	—
226	C₄	—	17	1	no	—
227	C₄	—	34	1	no	—
228	C₄	—	34	1	no	—
229	C₄	—	68	4	no	—
230	C₄	C₂×#10587533	1	1	yes	characteristic
231	C₄	#11763820	1	1	yes	characteristic
232	C₄	#11763851	1	1	yes	characteristic
233	C₄	#11763853	1	1	yes	characteristic
234	C₂	—	34	1	no	—
235	C₂	—	34	1	no	—
236	C₂	#14544892	1	1	yes	characteristic
237	C₂	#14544896	1	1	yes	characteristic
238	C₂	#14544904	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544725

C₂

yes

maximal, characteristic

C₈×C₄⋊#595

C₂

yes

maximal, characteristic

#14544352

C₂

yes

maximal

#14543681

C₂

yes

maximal

#14527531

C₂

yes

maximal, characteristic

C₂×#10587536

—

C₂×#10587536

—

#11763813

C₂²

yes

—

#11763810

C₂²

yes

characteristic

C₂×C₈×#595

—

C₄×C₄⋊#595

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic, complemented

#11762559

C₄

yes

complemented

C₁₇×#335

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄×#595

—

C₂×#63243

—

C₂×#63242

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂³

yes

characteristic

(C₂×#595)⋊C₄

Q₈

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#3624

D₄

yes

—

—

—

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

C₈×Dic₁₇

—

C₈×Dic₁₇

Q₈

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄⋊#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

C₂×C₄×D₁₇

yes

—

—

—

—

—

—

—

—

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

C₄×Dic₁₇

yes

—

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

SD₁₆

yes

—

C₈×D₁₇

SD₁₆

yes

—

#9628

—

maximal, Sylow

C₂²×D₁₇

C₂²⋊Q₈

yes

—

—

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×Q₈

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

C₄⋊Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

—

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#106

yes

—

—

—

yes

—

100

#1848

—

101

C₈×C₄⋊C₄

—

102

#1570

—

103

#1564

—

104

#1303

—

105

C₂²×C₁₇

#389

yes

characteristic

106

C₂×D₁₇

—

107

C₂×D₁₇

—

108

C₂×D₁₇

#475

yes

characteristic

109

C₂×D₁₇

#476

yes

characteristic

110

#595

—

111

C₄×C₁₇

—

112

C₄×C₁₇

C₂×#106

yes

characteristic

113

C₄×C₁₇

#428

yes

characteristic

114

C₄×C₁₇

#443

yes

characteristic

115

C₄×C₁₇

#443

yes

characteristic

116

Dic₁₇

—

117

Dic₁₇

—

118

Dic₁₇

#460

yes

characteristic

119

Dic₁₇

#461

yes

characteristic

120

#431

—

121

C₂×#108

—

122

C₂×#108

—

123

#423

—

124

C₂×C₄⋊C₈

—

125

C₂×C₄×C₈

—

126

C₂×C₄×C₈

—

127

C₄×C₄⋊C₄

—

128

#336

—

129

#335

—

130

#335

#595

yes

characteristic, complemented

131

#334

—

132

#334

—

133

#334

—

134

C₂×C₁₇

#1945

yes

characteristic

135

C₂×C₁₇

#1948

yes

characteristic

136

C₂×C₁₇

#1953

yes

characteristic

137

D₁₇

—

138

D₁₇

—

139

C₂×M₁₆

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₂²×C₈

—

143

C₂×C₄⋊C₄

—

144

C₂×C₄²

—

145

C₂×C₄²

—

146

#108

—

147

#108

—

148

C₄⋊C₈

—

149

C₄⋊C₈

—

150

C₄⋊C₈

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

C₂×#595

yes

—

155

C₂²⋊C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

—

160

C₄×C₈

—

161

C₄×C₈

C₂×#595

yes

characteristic

162

#95

—

163

C₁₇

#9628

yes

characteristic, complemented, Sylow

164

C₂²×C₄

—

165

C₂²×C₄

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

M₁₆

—

169

M₁₆

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

C₄⋊#595

yes

characteristic

184

C₂×C₈

C₄⋊#595

yes

characteristic

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄²

—

188

C₄²

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

C₂²×#595

yes

characteristic

193

C₂³

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

208

C₂×C₄

D₄×#595

yes

characteristic

209

C₂×C₄

Q₈×#595

yes

characteristic

210

C₈

—

211

C₈

—

212

C₈

—

213

C₈

—

214

C₈

—

215

C₈

—

216

C₈

—

217

C₈

—

136

—

218

C₈

#10587533

yes

—

219

C₂²

—

220

C₂²

—

221

C₂²

—

222

C₂²

—

223

C₂²

#11763760

yes

characteristic

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

C₂×#10587533

yes

characteristic

231

C₄

#11763820

yes

characteristic

232

C₄

#11763851

yes

characteristic

233

C₄

#11763853

yes

characteristic

234

C₂

—

235

C₂

—

236

C₂

#14544892

yes

characteristic

237

C₂

#14544896

yes

characteristic

238

C₂

#14544904

yes

characteristic

	subgroup	G/H
center	C₂²	#11763760
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#389
prime radical	C₂²×D₁₇	C₂²⋊Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	#9628
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#11763760

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#335

C₄

socle

C₂²×C₁₇

#389

prime radical

C₂²×D₁₇

C₂²⋊Q₈

cosocle

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
17. #11762562	C₄	semidirect
18. #11762559	C₄	semidirect
19. C₁₇×#335	C₄	semidirect
130. #335	#595	semidirect
163. C₁₇	#9628	semidirect

H ≅ G/N

type

17. #11762562

C₄

semidirect

18. #11762559

C₄

semidirect

19. C₁₇×#335

C₄

semidirect

130. #335

#595

semidirect

163. C₁₇

#9628

semidirect

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

subgroup	index
#14544725	2
C₈×C₄⋊#595	2
#14544352	2
#14543681	2
#14527531	2
#9628	17

subgroup

index

C₈×C₄⋊#595

Subgroups — #17955895

Conjugacy classes of subgroups

Complements

Maximal subgroups