Group #17962257 — subgroups

Subgroups — #17962257

Order 4352.

294 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 92 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	D₁₇⋊#365	C₂	1	2	yes	maximal
2	#14544721	C₂	1	1	yes	maximal, characteristic
3	#14544586	C₂	1	1	yes	maximal, characteristic
4	#14543856	C₂	1	2	yes	maximal
5	D₁₇×#364	C₂	1	1	yes	maximal, characteristic
6	D₁₇⋊#108	C₂²	1	1	yes	characteristic
7	#11763819	C₂²	1	1	yes	characteristic
8	C₂×#10587534	—	2	1	no	—
9	C₂×#10587533	C₂²	1	2	yes	—
10	#11763772	—	2	1	no	—
11	#11763753	—	2	1	no	—
12	#11763734	—	2	2	no	—
13	#11763733	—	2	2	no	—
14	#11763680	—	2	1	no	—
15	M₃₂×D₁₇	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762630	C₄	1	1	yes	characteristic, complemented
18	#11762629	C₄	1	2	yes	complemented
19	C₁₇×#364	C₄	1	1	yes	characteristic, complemented
20	#10587677	—	2	1	no	—
21	C₂×C₄⋊#595	—	2	2	no	—
22	D₁₇⋊M₁₆	—	2	1	no	—
23	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	C₄²×D₁₇	D₄	1	1	yes	characteristic
26	(C₂×#595)⋊C₄	—	4	2	no	—
27	#10587536	—	2	1	no	—
28	#10587535	D₄	1	1	yes	characteristic
29	#10587535	D₄	1	1	yes	characteristic
30	#10587534	—	4	1	no	—
31	#10587534	—	4	1	no	—
32	#10587534	Q₈	1	2	yes	—
33	#10587533	—	2	1	no	—
34	#10587533	—	2	2	no	—
35	#10587533	—	2	2	no	—
36	#10587532	—	4	1	no	—
37	C₈×#595	—	4	1	no	—
38	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
42	#10587483	C₂×C₄	1	2	yes	—
43	C₄×#3622	Q₈	1	1	yes	characteristic
44	#10587469	—	2	2	no	—
45	C₁₆×D₁₇	—	2	2	no	—
46	C₂²×#595	—	4	2	no	—
47	C₂×C₄×D₁₇	—	2	1	no	—
48	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²⋊#595	—	4	1	no	—
50	#63247	—	4	1	no	—
51	C₄⋊#595	—	2	2	no	—
52	C₄⋊#595	—	4	2	no	—
53	C₄⋊#595	—	4	2	no	—
54	C₄×#595	—	4	1	no	—
55	#63243	—	2	1	no	—
56	#63243	—	2	1	no	—
57	#63242	—	4	1	no	—
58	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₁₇	C₄⋊C₄	1	1	yes	characteristic
61	C₄×Dic₁₇	—	2	1	no	—
62	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
64	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
65	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
66	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
67	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
68	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
69	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
70	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
71	C₈×D₁₇	#41	1	1	yes	characteristic
72	C₈×D₁₇	#41	1	1	yes	characteristic
73	C₁₆×C₁₇	—	2	2	no	—
74	#63214	—	2	2	no	—
75	#13187	—	17	1	no	maximal, Sylow
76	C₂²×D₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×#595	—	4	2	no	—
78	C₂×#595	—	8	4	no	—
79	C₂×#595	—	8	2	no	—
80	C₂×C₄×C₁₇	—	2	1	no	—
81	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₁₇	—	2	1	no	—
83	C₂×Dic₁₇	#137	1	1	yes	characteristic
84	C₄×D₁₇	—	4	2	no	—
85	C₄×D₁₇	#121	1	1	yes	characteristic
86	C₄×D₁₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
89	#3624	—	4	1	no	—
90	#3624	—	4	1	no	—
91	C₈×C₁₇	#106	1	1	yes	characteristic
92	C₈×C₁₇	#106	1	1	yes	characteristic
93	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
94	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
95	#3622	#106	1	1	yes	characteristic
96	#3622	#106	1	1	yes	characteristic
97	#3622	#125	1	1	yes	characteristic
98	#3622	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×#365	—	17	2	no	—
100	C₂×#364	—	17	1	no	—
101	#1846	—	17	1	no	—
102	#1772	—	17	1	no	—
103	#1386	—	17	2	no	—
104	C₂²×C₁₇	#428	1	1	yes	characteristic
105	C₂×D₁₇	—	2	2	no	—
106	C₂×D₁₇	#460	1	1	yes	characteristic
107	C₂×D₁₇	#476	1	1	yes	characteristic
108	#595	—	8	2	no	—
109	#595	—	16	2	no	—
110	C₄×C₁₇	—	4	1	no	—
111	C₄×C₁₇	#389	1	1	yes	characteristic
112	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
113	Dic₁₇	—	4	1	no	—
114	Dic₁₇	#461	1	1	yes	characteristic
115	Dic₁₇	#475	1	1	yes	characteristic
116	C₂×M₃₂	—	17	2	no	—
117	C₂×#108	—	17	1	no	—
118	#427	—	17	1	no	—
119	C₂×#107	—	34	1	no	—
120	C₂×#106	—	17	2	no	—
121	#405	—	34	1	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	#382	—	34	1	no	—
124	#365	—	34	2	no	—
125	#365	—	34	2	no	—
126	#364	—	17	2	no	—
127	#364	—	17	1	no	—
128	#364	#595	1	1	yes	characteristic, complemented
129	#339	—	34	1	no	—
130	C₂×C₁₇	—	2	1	no	—
131	C₂×C₁₇	#1948	1	1	yes	characteristic
132	D₁₇	—	2	1	no	—
133	D₁₇	#2231	1	2	yes	—
134	C₂×M₁₆	—	34	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	#117	—	34	1	no	—
138	C₂×C₄⋊C₄	—	34	2	no	—
139	C₂×C₄²	—	17	1	no	—
140	M₃₂	—	17	2	no	—
141	M₃₂	—	34	2	no	—
142	M₃₂	C₂×#595	1	2	yes	—
143	C₂×C₁₆	—	34	2	no	—
144	#108	—	17	1	no	—
145	#108	—	17	1	no	—
146	#108	—	34	1	no	—
147	#107	—	17	2	no	—
148	#107	—	68	1	no	—
149	#107	—	68	1	no	—
150	#106	—	34	1	no	—
151	#106	—	34	2	no	—
152	#106	—	34	2	no	—
153	#97	—	68	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	68	1	no	—
158	C₄×C₈	C₂×#595	1	1	yes	characteristic
159	#95	—	68	2	no	—
160	C₁₇	#13187	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	34	1	no	—
163	C₂²×C₄	—	68	2	no	—
164	M₁₆	—	34	1	no	—
165	M₁₆	—	34	1	no	—
166	M₁₆	—	68	1	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	17	1	no	—
169	C₂×C₈	—	17	1	no	—
170	C₂×C₈	—	17	1	no	—
171	C₂×C₈	—	17	1	no	—
172	C₂×C₈	—	17	1	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	17	1	no	—
175	C₂×C₈	—	17	1	no	—
176	C₂×C₈	—	17	1	no	—
177	C₂×C₈	—	68	1	no	—
178	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
179	C₂×C₈	C₂²×#595	1	1	yes	characteristic
180	C₄⋊C₄	—	34	2	no	—
181	C₄⋊C₄	—	68	2	no	—
182	C₄⋊C₄	—	68	2	no	—
183	C₂²⋊C₄	—	68	1	no	—
184	C₄²	—	17	1	no	—
185	C₄²	—	34	1	no	—
186	C₄²	—	68	1	no	—
187	C₄²	C₄⋊#595	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	—	34	2	no	—
190	C₂³	—	17	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	34	1	no	—
198	C₂×C₄	—	68	2	no	—
199	C₂×C₄	—	68	2	no	—
200	C₂×C₄	—	136	2	no	—
201	C₂×C₄	—	136	4	no	—
202	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
203	C₈	—	17	1	no	—
204	C₈	—	17	1	no	—
205	C₈	—	17	1	no	—
206	C₈	—	17	1	no	—
207	C₈	—	68	1	no	—
208	C₈	—	68	1	no	—
209	C₈	#10587533	1	1	yes	characteristic
210	C₈	#10587533	1	1	yes	characteristic
211	C₈	D₄×#595	1	1	yes	characteristic
212	C₈	Q₈×#595	1	1	yes	characteristic
213	C₂²	—	17	1	no	—
214	C₂²	—	17	1	no	—
215	C₂²	—	34	2	no	—
216	C₂²	#11763820	1	1	yes	characteristic
217	C₄	—	4	1	no	—
218	C₄	—	272	2	no	—
219	C₄	—	17	1	no	—
220	C₄	—	17	1	no	—
221	C₄	—	68	1	no	—
222	C₄	—	136	2	no	—
223	C₄	#11763760	1	1	yes	characteristic
224	C₄	C₂×#10587533	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	17	2	no	—
227	C₂	—	34	1	no	—
228	C₂	#14544896	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₇⋊#365

C₂

yes

maximal

#14544721

C₂

yes

maximal, characteristic

#14544586

C₂

yes

maximal, characteristic

#14543856

C₂

yes

maximal

D₁₇×#364

C₂

yes

maximal, characteristic

D₁₇⋊#108

C₂²

yes

characteristic

#11763819

C₂²

yes

characteristic

C₂×#10587534

—

C₂×#10587533

C₂²

yes

—

—

—

—

—

—

M₃₂×D₁₇

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762630

C₄

yes

characteristic, complemented

#11762629

C₄

yes

complemented

C₁₇×#364

C₄

yes

characteristic, complemented

#10587677

—

C₂×C₄⋊#595

—

D₁₇⋊M₁₆

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

(C₂×#595)⋊C₄

—

#10587536

—

#10587535

D₄

yes

characteristic

#10587535

D₄

yes

—

—

Q₈

yes

—

—

—

—

—

—

C₁₇×M₃₂

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

#10587483

C₂×C₄

yes

—

C₄×#3622

Q₈

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂²×#595

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×D₄

yes

characteristic

C₂²⋊#595

—

—

—

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

C₂×D₄

yes

characteristic

C₈×D₁₇

C₂×Q₈

yes

C₈×D₁₇

yes

C₈×D₁₇

yes

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

—

—

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#137

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

#121

yes

characteristic

C₄×D₁₇

C₂²⋊Q₈

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

characteristic

#3624

—

#3624

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

C₄×D₄

yes

characteristic

C₈×C₁₇

C₄×Q₈

yes

yes

yes

yes

C₄⋊Q₈

yes

characteristic

C₂×#365

—

100

C₂×#364

—

101

#1846

—

102

#1772

—

103

#1386

—

104

C₂²×C₁₇

#428

yes

characteristic

105

C₂×D₁₇

—

106

C₂×D₁₇

#460

yes

characteristic

107

C₂×D₁₇

#476

yes

characteristic

108

#595

—

109

#595

—

110

C₄×C₁₇

—

111

C₄×C₁₇

#389

yes

characteristic

112

C₄×C₁₇

C₂×#106

yes

characteristic

113

Dic₁₇

—

114

Dic₁₇

#461

yes

characteristic

115

Dic₁₇

#475

yes

characteristic

116

C₂×M₃₂

—

117

C₂×#108

—

118

#427

—

119

C₂×#107

—

120

C₂×#106

—

121

#405

—

122

C₂×C₄×C₈

—

123

#382

—

124

#365

—

125

#365

—

126

#364

—

127

#364

—

128

#364

#595

yes

characteristic, complemented

129

#339

—

130

C₂×C₁₇

—

131

C₂×C₁₇

#1948

yes

characteristic

132

D₁₇

—

133

D₁₇

#2231

yes

—

134

C₂×M₁₆

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

#117

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

—

140

M₃₂

—

141

M₃₂

—

142

M₃₂

C₂×#595

yes

—

143

C₂×C₁₆

—

144

#108

—

145

#108

—

146

#108

—

147

#107

—

148

#107

—

149

#107

—

150

#106

—

151

#106

—

152

#106

—

153

#97

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

C₂×#595

yes

characteristic

159

#95

—

160

C₁₇

#13187

yes

characteristic, complemented, Sylow

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

M₁₆

—

165

M₁₆

—

166

M₁₆

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

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

C₄⋊#595

yes

characteristic

179

C₂×C₈

C₂²×#595

yes

characteristic

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₂²⋊C₄

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

C₄⋊#595

yes

characteristic

188

C₁₆

—

189

C₁₆

—

190

C₂³

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

136

—

201

C₂×C₄

—

136

—

202

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

#10587533

yes

characteristic

210

C₈

#10587533

yes

characteristic

211

C₈

D₄×#595

yes

characteristic

212

C₈

Q₈×#595

yes

characteristic

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

#11763820

yes

characteristic

217

C₄

—

218

C₄

—

272

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

136

—

223

C₄

#11763760

yes

characteristic

224

C₄

C₂×#10587533

yes

characteristic

225

C₂

—

226

C₂

—

227

C₂

—

228

C₂

#14544896

yes

characteristic

	subgroup	G/H
center	C₂	#14544896
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#364	C₄
socle	C₂×C₁₇	#1948
prime radical	C₂²×D₁₇	AGL(1,ℤ/8)
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#364	#595
hypocenter	C₁₇	#13187
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#14544896

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#364

C₄

C₂×C₁₇

C₂²×D₁₇

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
17. #11762630	C₄	semidirect
18. #11762629	C₄	semidirect
19. C₁₇×#364	C₄	semidirect
128. #364	#595	semidirect
160. C₁₇	#13187	semidirect

H ≅ G/N

type

17. #11762630

C₄

semidirect

18. #11762629

C₄

semidirect

19. C₁₇×#364

C₄

semidirect

128. #364

#595

semidirect

160. C₁₇

#13187

semidirect

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

subgroup	index
D₁₇⋊#365	2
#14544721	2
#14544586	2
#14543856	2
D₁₇×#364	2
#13187	17

subgroup

index

D₁₇⋊#365

D₁₇×#364

Subgroups — #17962257

Conjugacy classes of subgroups

Complements

Maximal subgroups