Group #17948581 — subgroups

Subgroups — #17948581

Order 4352.

286 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
3	D₁₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587529	—	2	1	no	—
5	#11763810	C₄	1	1	yes	characteristic
6	C₂×C₈×#595	—	2	1	no	—
7	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
8	#11763695	—	2	1	no	—
9	C₄×#63216	C₄	1	2	yes	complemented
10	#11763674	—	2	1	no	—
11	C₂×C₁₆×D₁₇	—	2	1	no	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
14	#11762627	—	2	1	no	—
15	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄⋊#595	C₈	1	2	yes	—
17	C₂×C₄×#595	—	2	1	no	—
18	C₂×#63242	—	2	1	no	—
19	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
22	(C₂×#595)⋊C₄	—	2	1	no	—
23	#10587544	—	2	1	no	—
24	C₄⋊#3624	C₈	1	2	yes	—
25	C₂×#63216	—	2	2	no	—
26	C₈×#595	—	4	1	no	—
27	C₈×#595	—	4	1	no	—
28	#10587529	—	2	2	no	—
29	#10587529	—	2	2	no	—
30	C₂×C₁₆×C₁₇	—	2	1	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	D₄	1	1	yes	characteristic
33	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
34	C₂×#63214	—	2	1	no	—
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	C₁₆×D₁₇	—	4	1	no	—
37	C₁₆×D₁₇	—	4	1	no	—
38	C₂²×#595	—	2	2	no	—
39	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
42	C₂×#3624	—	2	2	no	—
43	C₄⋊#595	—	2	2	no	—
44	C₄⋊#595	C₁₆	1	4	yes	complemented
45	C₄×#595	—	4	1	no	—
46	C₄×#595	—	4	1	no	—
47	#63242	—	4	1	no	—
48	#63242	—	4	1	no	—
49	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₁₇	C₄²	1	1	yes	characteristic
52	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
53	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
54	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
55	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
57	C₈×D₁₇	—	2	1	no	—
58	C₈×D₁₇	—	2	1	no	—
59	C₈×D₁₇	D₈	1	1	yes	characteristic
60	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
61	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
62	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
63	#63216	—	2	4	no	—
64	C₁₆×C₁₇	—	4	1	no	—
65	#63214	—	4	1	no	—
66	#7167	—	17	1	no	maximal, Sylow
67	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×#595	—	2	4	no	—
69	C₂×#595	—	4	2	no	—
70	C₂×#595	—	4	2	no	—
71	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
74	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
77	C₄×D₁₇	—	2	2	no	—
78	C₄×D₁₇	#102	1	1	yes	characteristic
79	C₄×D₁₇	#103	1	1	yes	characteristic
80	C₄×D₁₇	#104	1	1	yes	characteristic
81	C₄×D₁₇	#104	1	1	yes	characteristic
82	C₄×D₁₇	C₂×C₁₆	1	2	yes	—
83	C₄×D₁₇	M₃₂	1	2	yes	—
84	#3624	—	4	2	no	—
85	C₈×C₁₇	—	2	1	no	—
86	C₈×C₁₇	#102	1	1	yes	characteristic
87	C₈×C₁₇	#103	1	1	yes	characteristic
88	#3622	—	2	1	no	—
89	#3622	#106	1	1	yes	characteristic
90	#3622	#107	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	—	17	1	no	—
92	C₂×C₄×C₁₆	—	17	1	no	—
93	C₈×C₄⋊C₄	—	17	1	no	—
94	C₂²×C₁₇	#336	1	1	yes	characteristic
95	C₂×D₁₇	D₄⋊C₈	1	1	yes	characteristic
96	C₂×D₁₇	Q₈⋊C₈	1	1	yes	characteristic
97	C₂×D₁₇	C₂²⋊C₁₆	1	2	yes	—
98	C₂×D₁₇	#350	1	2	yes	—
99	#595	—	4	4	no	—
100	C₄×C₁₇	—	2	1	no	—
101	C₄×C₁₇	#339	1	1	yes	characteristic
102	C₄×C₁₇	#340	1	1	yes	characteristic
103	C₄×C₁₇	C₄×C₁₆	1	1	yes	characteristic
104	C₄×C₁₇	#346	1	1	yes	characteristic
105	Dic₁₇	—	2	1	no	—
106	Dic₁₇	D₄⋊C₈	1	1	yes	characteristic
107	Dic₁₇	Q₈⋊C₈	1	1	yes	characteristic
108	Dic₁₇	C₄⋊C₁₆	1	2	yes	—
109	C₂²×C₁₆	—	34	1	no	—
110	C₂²×C₁₆	—	34	1	no	—
111	C₂×C₄⋊C₈	—	17	1	no	—
112	C₂×C₄×C₈	—	17	1	no	—
113	C₂×C₄×C₈	—	34	1	no	—
114	C₄×C₄⋊C₄	—	17	1	no	—
115	C₄⋊C₁₆	—	17	1	no	—
116	C₄⋊C₁₆	—	34	1	no	—
117	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
118	C₄×C₁₆	—	17	2	no	—
119	C₄×C₁₆	—	34	1	no	—
120	#336	—	34	1	no	—
121	C₂×C₁₇	#1279	1	1	yes	characteristic
122	C₂×C₁₇	#1374	1	1	yes	characteristic
123	C₂×C₁₇	#1379	1	1	yes	characteristic
124	D₁₇	D₄⋊C₁₆	1	2	yes	—
125	D₁₇	Q₈⋊C₁₆	1	2	yes	—
126	C₂²×C₈	—	17	1	no	—
127	C₂²×C₈	—	17	1	no	—
128	C₂²×C₈	—	34	1	no	—
129	C₂×C₄⋊C₄	—	17	2	no	—
130	C₂×C₄²	—	17	1	no	—
131	C₂×C₄²	—	34	1	no	—
132	C₂×C₁₆	—	2	1	no	—
133	C₂×C₁₆	—	34	2	no	—
134	C₂×C₁₆	—	34	2	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₂×C₁₆	—	34	1	no	—
137	C₂×C₁₆	—	68	1	no	—
138	C₂×C₁₆	—	68	1	no	—
139	C₄⋊C₈	—	17	2	no	—
140	C₄⋊C₈	—	34	1	no	—
141	C₄×C₈	—	17	1	no	—
142	C₄×C₈	—	17	1	no	—
143	C₄×C₈	—	17	1	no	—
144	C₄×C₈	—	68	1	no	—
145	C₄×C₈	—	68	1	no	—
146	C₄×C₈	C₂×#595	1	1	yes	characteristic
147	#95	—	34	1	no	—
148	C₁₇	#7167	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	17	1	no	—
150	C₂²×C₄	—	17	1	no	—
151	C₂²×C₄	—	17	1	no	—
152	C₂²×C₄	—	34	2	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	17	1	no	—
157	C₂×C₈	—	17	1	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	34	1	no	—
160	C₂×C₈	—	34	1	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	—	68	1	no	—
163	C₂×C₈	—	68	1	no	—
164	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
166	C₄⋊C₄	—	17	4	no	—
167	C₄⋊C₄	—	34	2	no	—
168	C₄²	—	17	1	no	—
169	C₄²	—	17	1	no	—
170	C₄²	—	17	1	no	—
171	C₄²	—	68	1	no	—
172	C₄²	—	68	1	no	—
173	C₄²	C₄×#595	1	1	yes	characteristic
174	C₁₆	—	4	1	no	—
175	C₁₆	—	34	4	no	—
176	C₁₆	—	68	1	no	—
177	C₂³	—	17	1	no	—
178	C₂×C₄	—	17	1	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	17	1	no	—
181	C₂×C₄	—	17	1	no	—
182	C₂×C₄	—	17	1	no	—
183	C₂×C₄	—	17	1	no	—
184	C₂×C₄	—	17	2	no	—
185	C₂×C₄	—	17	2	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	34	2	no	—
188	C₂×C₄	—	34	4	no	—
189	C₂×C₄	—	68	2	no	—
190	C₂×C₄	—	68	2	no	—
191	C₂×C₄	C₈×#595	1	1	yes	characteristic
192	C₂×C₄	#10587532	1	1	yes	characteristic
193	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	17	1	no	—
196	C₈	—	17	1	no	—
197	C₈	—	34	1	no	—
198	C₈	—	68	2	no	—
199	C₈	#10587547	1	1	yes	characteristic
200	C₈	#10587549	1	1	yes	characteristic
201	C₂²	—	17	1	no	—
202	C₂²	—	17	1	no	—
203	C₂²	—	17	2	no	—
204	C₂²	—	17	2	no	—
205	C₂²	#11763674	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	17	1	no	—
208	C₄	—	17	1	no	—
209	C₄	—	17	2	no	—
210	C₄	—	34	1	no	—
211	C₄	—	68	4	no	—
212	C₄	#11763681	1	1	yes	characteristic
213	C₄	#11763682	1	1	yes	characteristic
214	C₄	C₁₆×#595	1	1	yes	characteristic
215	C₄	#11763696	1	1	yes	characteristic
216	C₂	—	17	2	no	—
217	C₂	—	17	2	no	—
218	C₂	#14543618	1	1	yes	characteristic
219	C₂	#14543821	1	1	yes	characteristic
220	C₂	#14543840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587529

C₂

yes

maximal, characteristic

C₈×C₄⋊#595

C₂

yes

maximal, characteristic

D₁₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×#10587529

—

#11763810

C₄

yes

characteristic

C₂×C₈×#595

—

C₄×C₄⋊#595

C₄

yes

characteristic

#11763695

—

C₄×#63216

C₄

yes

complemented

#11763674

—

C₂×C₁₆×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊#63214

C₄

yes

characteristic, complemented

#11762627

—

C₁₇×C₄⋊C₁₆

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

C₈

yes

—

C₂×C₄×#595

—

C₂×#63242

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

(C₂×#595)⋊C₄

—

#10587544

—

C₄⋊#3624

C₈

yes

—

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

—

C₄×#3622

C₂×C₄

yes

characteristic

C₁₆×D₁₇

—

C₁₆×D₁₇

—

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

—

—

C₁₆

yes

—

—

—

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

characteristic

#63216

—

C₁₆×C₁₇

—

#63214

—

#7167

—

maximal, Sylow

C₂²×D₁₇

C₂²⋊C₈

yes

—

—

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

C₂×C₁₆

yes

—

C₄×D₁₇

M₃₂

yes

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

yes

C₂×C₄⋊C₁₆

—

C₂×C₄×C₁₆

—

C₈×C₄⋊C₄

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

D₄⋊C₈

yes

characteristic

C₂×D₁₇

Q₈⋊C₈

yes

characteristic

C₂×D₁₇

C₂²⋊C₁₆

yes

—

C₂×D₁₇

#350

yes

—

#595

—

100

C₄×C₁₇

—

101

C₄×C₁₇

#339

yes

characteristic

102

C₄×C₁₇

#340

yes

characteristic

103

C₄×C₁₇

C₄×C₁₆

yes

characteristic

104

C₄×C₁₇

#346

yes

characteristic

105

Dic₁₇

—

106

Dic₁₇

D₄⋊C₈

yes

characteristic

107

Dic₁₇

Q₈⋊C₈

yes

characteristic

108

Dic₁₇

C₄⋊C₁₆

yes

—

109

C₂²×C₁₆

—

110

C₂²×C₁₆

—

111

C₂×C₄⋊C₈

—

112

C₂×C₄×C₈

—

113

C₂×C₄×C₈

—

114

C₄×C₄⋊C₄

—

115

C₄⋊C₁₆

—

116

C₄⋊C₁₆

—

117

C₄⋊C₁₆

#595

yes

characteristic, complemented

118

C₄×C₁₆

—

119

C₄×C₁₆

—

120

#336

—

121

C₂×C₁₇

#1279

yes

characteristic

122

C₂×C₁₇

#1374

yes

characteristic

123

C₂×C₁₇

#1379

yes

characteristic

124

D₁₇

D₄⋊C₁₆

yes

—

125

D₁₇

Q₈⋊C₁₆

yes

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂×#595

yes

characteristic

147

#95

—

148

C₁₇

#7167

yes

characteristic, complemented, Sylow

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

—

160

C₂×C₈

—

161

C₂×C₈

—

162

C₂×C₈

—

163

C₂×C₈

—

164

C₂×C₈

C₄⋊#595

yes

characteristic

165

C₂×C₈

C₂²⋊#595

yes

characteristic

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₄²

C₄×#595

yes

characteristic

174

C₁₆

—

175

C₁₆

—

176

C₁₆

—

177

C₂³

—

178

C₂×C₄

—

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

C₈×#595

yes

characteristic

192

C₂×C₄

#10587532

yes

characteristic

193

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

#10587547

yes

characteristic

200

C₈

#10587549

yes

characteristic

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

—

205

C₂²

#11763674

yes

characteristic

206

C₄

—

207

C₄

—

208

C₄

—

209

C₄

—

210

C₄

—

211

C₄

—

212

C₄

#11763681

yes

characteristic

213

C₄

#11763682

yes

characteristic

214

C₄

C₁₆×#595

yes

characteristic

215

C₄

#11763696

yes

characteristic

216

C₂

—

217

C₂

—

218

C₂

#14543618

yes

characteristic

219

C₂

#14543821

yes

characteristic

220

C₂

#14543840

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#595
commutator	C₄×C₁₇	C₄×C₁₆
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×C₄⋊C₁₆	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#7167
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊#595

commutator

C₄×C₁₇

C₄×C₁₆

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×C₄⋊C₁₆

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₂²⋊C₈

cosocle

C₄×C₈×D₁₇

C₂²

C₄⋊C₁₆

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
9. C₄×#63216	C₄	semidirect
13. C₄⋊#63214	C₄	semidirect
15. C₁₇×C₄⋊C₁₆	C₄	semidirect
44. C₄⋊#595	C₁₆	semidirect
117. C₄⋊C₁₆	#595	semidirect
148. C₁₇	#7167	semidirect

H ≅ G/N

type

9. C₄×#63216

C₄

semidirect

13. C₄⋊#63214

C₄

semidirect

15. C₁₇×C₄⋊C₁₆

C₄

semidirect

44. C₄⋊#595

C₁₆

semidirect

117. C₄⋊C₁₆

#595

semidirect

148. C₁₇

#7167

semidirect

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

subgroup	index
C₄×#10587529	2
C₈×C₄⋊#595	2
D₁₇×C₄⋊C₁₆	2
#7167	17

subgroup

index

C₄×#10587529

C₈×C₄⋊#595

D₁₇×C₄⋊C₁₆

#7167

Subgroups — #17948581

Conjugacy classes of subgroups

Complements

Maximal subgroups