Group #17961024 — subgroups

Subgroups — #17961024

Order 4352.

334 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 128 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	#14544996	C₂	1	2	yes	maximal
2	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
3	#14544586	C₂	1	1	yes	maximal, characteristic
4	#14543840	C₂	1	2	yes	maximal
5	D₁₇×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587529	C₂²	1	1	yes	characteristic
7	C₂×#10587529	C₂²	1	1	yes	characteristic
8	D₁₇⋊#108	C₄	1	1	yes	characteristic
9	#11763819	C₄	1	1	yes	characteristic
10	#11763772	C₂²	1	2	yes	—
11	#11763696	—	2	2	no	—
12	#11763695	C₄	1	2	yes	complemented
13	C₄×#63216	C₄	1	2	yes	complemented
14	C₁₆×#595	—	2	2	no	—
15	#11763680	C₄	1	2	yes	—
16	M₃₂×D₁₇	C₂²	1	2	yes	—
17	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	#11762630	C₄	1	1	yes	characteristic, complemented
19	#11762629	C₄	1	2	yes	complemented
20	C₁₇×#364	C₄	1	1	yes	characteristic, complemented
21	#10587677	C₂×C₄	1	2	yes	—
22	D₁₇⋊M₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×#63216	C₂×C₄	1	2	yes	—
27	C₂×#63216	C₂×C₄	1	2	yes	—
28	#10587536	C₈	1	2	yes	—
29	#10587535	C₈	1	2	yes	—
30	#10587534	C₈	1	2	yes	—
31	#10587533	C₈	1	2	yes	—
32	#10587532	—	2	2	no	—
33	C₈×#595	—	2	2	no	—
34	#10587529	—	2	1	no	—
35	#10587529	—	2	1	no	—
36	#10587529	D₄	1	1	yes	characteristic
37	#10587529	D₄	1	1	yes	characteristic
38	#10587529	Q₈	1	2	yes	—
39	C₁₇×M₃₂	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
43	#10587483	C₂×C₄	1	2	yes	—
44	C₄×#3622	C₂×C₄	1	1	yes	characteristic
45	#10587469	—	2	2	no	—
46	C₁₆×D₁₇	—	2	2	no	—
47	C₂×C₄×D₁₇	—	2	1	no	—
48	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²⋊#595	—	2	2	no	—
50	#63247	—	2	2	no	—
51	C₄⋊#595	C₂×C₈	1	4	yes	—
52	C₄×#595	—	2	2	no	—
53	#63243	C₂×C₈	1	4	yes	—
54	#63242	—	2	2	no	—
55	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₄²	1	1	yes	characteristic
58	C₄×Dic₁₇	—	2	1	no	—
59	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
60	C₂×#3622	C₄²	1	1	yes	characteristic
61	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
62	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
63	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
64	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
65	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
66	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
67	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
68	C₈×D₁₇	#41	1	1	yes	characteristic
69	C₈×D₁₇	#41	1	1	yes	characteristic
70	#63216	—	2	2	no	—
71	#63216	C₄⋊C₄	1	2	yes	—
72	#63216	C₄⋊C₄	1	2	yes	—
73	C₁₆×C₁₇	—	2	2	no	—
74	#63214	—	2	2	no	—
75	#12664	—	17	1	no	maximal, Sylow
76	C₂²×D₁₇	#131	1	1	yes	characteristic
77	C₂×#595	—	2	4	no	—
78	C₂×C₄×C₁₇	—	2	1	no	—
79	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
80	C₂×Dic₁₇	—	2	1	no	—
81	C₂×Dic₁₇	#131	1	1	yes	characteristic
82	C₄×D₁₇	—	2	4	no	—
83	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
84	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
85	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
86	C₄×D₁₇	C₂²×C₈	1	1	yes	characteristic
87	#3624	—	2	4	no	—
88	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
89	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
90	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
91	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
92	#3622	C₄×C₈	1	1	yes	characteristic
93	#3622	C₄×C₈	1	1	yes	characteristic
94	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
95	#3622	#117	1	1	yes	characteristic
96	C₂×#364	—	17	1	no	—
97	#2108	—	17	2	no	—
98	C₂×C₄×C₁₆	—	17	1	no	—
99	#1772	—	17	1	no	—
100	#1379	—	17	2	no	—
101	C₂²×C₁₇	#405	1	1	yes	characteristic
102	C₂×D₁₇	—	2	2	no	—
103	C₂×D₁₇	C₈×D₄	1	1	yes	characteristic
104	C₂×D₁₇	C₈×Q₈	1	1	yes	characteristic
105	#595	—	4	4	no	—
106	C₄×C₁₇	—	2	2	no	—
107	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
108	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
109	Dic₁₇	—	2	2	no	—
110	Dic₁₇	C₈×D₄	1	1	yes	characteristic
111	Dic₁₇	C₈×Q₈	1	1	yes	characteristic
112	C₂×M₃₂	—	17	2	no	—
113	C₂²×C₁₆	—	17	1	no	—
114	C₂²×C₁₆	—	17	1	no	—
115	C₂×#108	—	17	1	no	—
116	#427	—	17	1	no	—
117	#405	—	17	2	no	—
118	C₂×C₄×C₈	—	17	1	no	—
119	#364	—	17	2	no	—
120	#364	—	17	1	no	—
121	#364	#595	1	1	yes	characteristic, complemented
122	#346	—	34	2	no	—
123	C₄×C₁₆	—	17	2	no	—
124	C₄×C₁₆	—	17	2	no	—
125	C₄×C₁₆	—	34	2	no	—
126	#339	—	17	2	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
129	D₁₇	—	2	1	no	—
130	D₁₇	#2170	1	2	yes	—
131	C₂×M₁₆	—	17	2	no	—
132	C₂²×C₈	—	17	1	no	—
133	C₂²×C₈	—	17	1	no	—
134	#117	—	17	2	no	—
135	C₂×C₄²	—	17	1	no	—
136	M₃₂	—	17	2	no	—
137	M₃₂	—	34	2	no	—
138	M₃₂	C₂×#595	1	2	yes	—
139	C₂×C₁₆	—	17	2	no	—
140	C₂×C₁₆	—	17	1	no	—
141	C₂×C₁₆	—	17	1	no	—
142	C₂×C₁₆	—	17	2	no	—
143	C₂×C₁₆	—	17	2	no	—
144	C₂×C₁₆	—	34	2	no	—
145	C₂×C₁₆	—	34	1	no	—
146	C₂×C₁₆	—	34	1	no	—
147	#108	—	17	2	no	—
148	#108	—	17	2	no	—
149	#107	—	17	2	no	—
150	#106	—	17	2	no	—
151	#97	—	34	2	no	—
152	C₄×C₈	—	17	1	no	—
153	C₄×C₈	—	17	1	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	34	2	no	—
156	C₄×C₈	C₂×#595	1	1	yes	characteristic
157	C₁₇	#12664	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	34	1	no	—
160	M₁₆	—	17	4	no	—
161	M₁₆	—	34	2	no	—
162	C₂×C₈	—	17	1	no	—
163	C₂×C₈	—	17	1	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	17	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	—	34	2	no	—
173	C₂×C₈	C₄×#595	1	1	yes	characteristic
174	C₂×C₈	C₂²×#595	1	1	yes	characteristic
175	C₄⋊C₄	—	17	4	no	—
176	C₂²⋊C₄	—	34	2	no	—
177	C₄²	—	17	1	no	—
178	C₄²	—	34	1	no	—
179	C₄²	—	34	2	no	—
180	C₄²	C₄×#595	1	1	yes	characteristic
181	C₁₆	—	2	2	no	—
182	C₁₆	—	17	2	no	—
183	C₁₆	—	17	2	no	—
184	C₁₆	—	34	2	no	—
185	C₁₆	—	34	2	no	—
186	C₂³	—	17	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	17	1	no	—
189	C₂×C₄	—	17	1	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	34	4	no	—
194	C₂×C₄	—	34	1	no	—
195	C₂×C₄	—	34	4	no	—
196	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
197	C₈	—	17	1	no	—
198	C₈	—	17	1	no	—
199	C₈	—	17	1	no	—
200	C₈	—	17	1	no	—
201	C₈	—	34	4	no	—
202	C₈	C₈×#595	1	1	yes	characteristic
203	C₈	C₈×#595	1	1	yes	characteristic
204	C₈	D₄×#595	1	1	yes	characteristic
205	C₈	Q₈×#595	1	1	yes	characteristic
206	C₂²	—	17	1	no	—
207	C₂²	—	17	1	no	—
208	C₂²	—	34	2	no	—
209	C₂²	#11763775	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	17	1	no	—
212	C₄	—	17	1	no	—
213	C₄	—	34	2	no	—
214	C₄	—	68	4	no	—
215	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
216	C₄	C₂×C₈×#595	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	17	2	no	—
219	C₂	—	34	1	no	—
220	C₂	#14544577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544996

C₂

yes

maximal

C₄×#10587529

C₂

yes

maximal, characteristic

#14544586

C₂

yes

maximal, characteristic

#14543840

C₂

yes

maximal

D₁₇×#364

C₂

yes

maximal, characteristic

C₂×#10587529

C₂²

yes

characteristic

C₂×#10587529

C₂²

yes

characteristic

D₁₇⋊#108

C₄

yes

characteristic

#11763819

C₄

yes

characteristic

#11763772

C₂²

yes

—

#11763696

—

#11763695

C₄

yes

complemented

C₄×#63216

C₄

yes

complemented

C₁₆×#595

—

#11763680

C₄

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

yes

—

D₁₇⋊M₁₆

C₂×C₄

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

—

#10587536

C₈

yes

—

#10587535

C₈

yes

—

#10587534

C₈

yes

—

#10587533

C₈

yes

—

—

—

—

—

D₄

yes

characteristic

#10587529

D₄

yes

characteristic

#10587529

Q₈

yes

—

C₁₇×M₃₂

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

#10587483

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²×C₄

yes

characteristic

C₂²⋊#595

—

#63247

—

C₄⋊#595

C₂×C₈

yes

—

C₄×#595

—

#63243

C₂×C₈

yes

—

#63242

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄²

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×D₄

yes

characteristic

C₈×D₁₇

C₂×Q₈

yes

C₈×D₁₇

yes

C₈×D₁₇

yes

—

C₄⋊C₄

yes

—

#63216

C₄⋊C₄

yes

—

C₁₆×C₁₇

—

—

—

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#131

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

C₄×D₁₇

C₄×Q₈

yes

characteristic

C₄×D₁₇

C₂²×C₈

yes

characteristic

C₄×D₁₇

C₂²×C₈

yes

characteristic

#3624

—

C₈×C₁₇

C₄×C₈

yes

characteristic

C₈×C₁₇

C₄×C₈

yes

characteristic

C₈×C₁₇

C₄×D₄

yes

characteristic

C₈×C₁₇

C₄×Q₈

yes

characteristic

#3622

C₄×C₈

yes

characteristic

#3622

C₄×C₈

yes

characteristic

#3622

C₂×C₄⋊C₄

yes

yes

—

—

C₂×C₄×C₁₆

—

#1772

—

100

#1379

—

101

C₂²×C₁₇

#405

yes

characteristic

102

C₂×D₁₇

—

103

C₂×D₁₇

C₈×D₄

yes

characteristic

104

C₂×D₁₇

C₈×Q₈

yes

characteristic

105

#595

—

106

C₄×C₁₇

—

107

C₄×C₁₇

C₄×C₄⋊C₄

yes

characteristic

108

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

109

Dic₁₇

—

110

Dic₁₇

C₈×D₄

yes

characteristic

111

Dic₁₇

C₈×Q₈

yes

characteristic

112

C₂×M₃₂

—

113

C₂²×C₁₆

—

114

C₂²×C₁₆

—

115

C₂×#108

—

116

#427

—

117

#405

—

118

C₂×C₄×C₈

—

119

#364

—

120

#364

—

121

#364

#595

yes

characteristic, complemented

122

#346

—

123

C₄×C₁₆

—

124

C₄×C₁₆

—

125

C₄×C₁₆

—

126

#339

—

127

C₂×C₁₇

—

128

C₂×C₁₇

C₈×C₄⋊C₄

yes

characteristic

129

D₁₇

—

130

D₁₇

#2170

yes

—

131

C₂×M₁₆

—

132

C₂²×C₈

—

133

C₂²×C₈

—

134

#117

—

135

C₂×C₄²

—

136

M₃₂

—

137

M₃₂

—

138

M₃₂

C₂×#595

yes

—

139

C₂×C₁₆

—

140

C₂×C₁₆

—

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

C₂×C₁₆

—

145

C₂×C₁₆

—

146

C₂×C₁₆

—

147

#108

—

148

#108

—

149

#107

—

150

#106

—

151

#97

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

C₂×#595

yes

characteristic

157

C₁₇

#12664

yes

characteristic, complemented, Sylow

158

C₂²×C₄

—

159

C₂²×C₄

—

160

M₁₆

—

161

M₁₆

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₂×C₈

—

167

C₂×C₈

—

168

C₂×C₈

—

169

C₂×C₈

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

C₄×#595

yes

characteristic

174

C₂×C₈

C₂²×#595

yes

characteristic

175

C₄⋊C₄

—

176

C₂²⋊C₄

—

177

C₄²

—

178

C₄²

—

179

C₄²

—

180

C₄²

C₄×#595

yes

characteristic

181

C₁₆

—

182

C₁₆

—

183

C₁₆

—

184

C₁₆

—

185

C₁₆

—

186

C₂³

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

C₂×C₄×#595

yes

characteristic

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

C₈×#595

yes

characteristic

203

C₈

C₈×#595

yes

characteristic

204

C₈

D₄×#595

yes

characteristic

205

C₈

Q₈×#595

yes

characteristic

206

C₂²

—

207

C₂²

—

208

C₂²

—

209

C₂²

#11763775

yes

characteristic

210

C₄

—

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

C₄⋊C₄×#595

yes

characteristic

216

C₄

C₂×C₈×#595

yes

characteristic

217

C₂

—

218

C₂

—

219

C₂

—

220

C₂

#14544577

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#364	C₄
socle	C₂×C₁₇	C₈×C₄⋊C₄
prime radical	C₂²×D₁₇	#131
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#364	#595
hypocenter	C₁₇	#12664
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#364

C₄

socle

C₂×C₁₇

C₈×C₄⋊C₄

prime radical

C₂²×D₁₇

#131

cosocle

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
12. #11763695	C₄	semidirect
13. C₄×#63216	C₄	semidirect
18. #11762630	C₄	semidirect
19. #11762629	C₄	semidirect
20. C₁₇×#364	C₄	semidirect
121. #364	#595	semidirect
157. C₁₇	#12664	semidirect

H ≅ G/N

type

12. #11763695

C₄

semidirect

13. C₄×#63216

C₄

semidirect

18. #11762630

C₄

semidirect

19. #11762629

C₄

semidirect

20. C₁₇×#364

C₄

semidirect

121. #364

#595

semidirect

157. C₁₇

#12664

semidirect

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

subgroup	index
#14544996	2
C₄×#10587529	2
#14544586	2
#14543840	2
D₁₇×#364	2
#12664	17

subgroup

index

D₁₇×#364

Subgroups — #17961024

Conjugacy classes of subgroups

Complements

Maximal subgroups