Group #17955894 — subgroups

Subgroups — #17955894

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	#14544723	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	#14527529	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587535	—	2	1	no	—
7	C₂×#10587535	—	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	#11762561	C₄	1	2	yes	complemented
18	#11762560	C₄	1	1	yes	characteristic, complemented
19	C₁₇×#334	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	2	yes	—
32	#10587535	—	4	2	no	—
33	#10587535	—	4	2	no	—
34	C₈×#595	—	4	2	no	—
35	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
37	#10587491	—	2	2	no	—
38	C₈×Dic₁₇	—	2	2	no	—
39	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
40	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
41	C₄⋊#3622	C₂×C₄	1	2	yes	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂²×#595	—	2	2	no	—
44	C₂×C₄×D₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
47	C₂×#3624	—	2	2	no	—
48	C₂×#3624	—	4	2	no	—
49	C₄⋊#595	—	2	4	no	—
50	C₄×#595	—	4	1	no	—
51	C₄×#595	—	4	1	no	—
52	#63243	—	4	2	no	—
53	#63243	—	4	2	no	—
54	#63242	—	4	1	no	—
55	#63242	—	4	1	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₁₇	#41	1	1	yes	characteristic
63	C₂×#3622	—	2	2	no	—
64	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
65	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
66	C₈×D₁₇	—	2	1	no	—
67	C₈×D₁₇	—	2	1	no	—
68	C₈×D₁₇	SD₁₆	1	2	yes	—
69	C₈×D₁₇	SD₁₆	1	2	yes	—
70	#9628	—	17	1	no	maximal, Sylow
71	C₂²×D₁₇	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×#595	—	2	4	no	—
73	C₂×#595	—	4	2	no	—
74	C₂×#595	—	4	2	no	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
78	C₂×Dic₁₇	#121	1	1	yes	characteristic
79	C₂×Dic₁₇	#125	1	1	yes	characteristic
80	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₄×D₁₇	—	2	2	no	—
82	C₄×D₁₇	—	4	1	no	—
83	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
84	C₄×D₁₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₄×D₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₄×D₁₇	#137	1	1	yes	characteristic
87	#3624	—	4	2	no	—
88	#3624	—	8	2	no	—
89	C₈×C₁₇	—	2	1	no	—
90	C₈×C₁₇	—	4	2	no	—
91	C₈×C₁₇	#106	1	2	yes	—
92	#3622	—	2	1	no	—
93	#3622	—	4	2	no	—
94	#3622	#106	1	2	yes	—
95	#1848	—	17	1	no	—
96	C₈×C₄⋊C₄	—	17	1	no	—
97	#1570	—	17	2	no	—
98	#1564	—	17	1	no	—
99	#1303	—	17	2	no	—
100	C₂²×C₁₇	#389	1	1	yes	characteristic
101	C₂×D₁₇	—	2	1	no	—
102	C₂×D₁₇	—	2	1	no	—
103	C₂×D₁₇	#475	1	1	yes	characteristic
104	C₂×D₁₇	#476	1	1	yes	characteristic
105	#595	—	4	4	no	—
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	C₂×#106	1	1	yes	characteristic
108	C₄×C₁₇	#428	1	1	yes	characteristic
109	C₄×C₁₇	#443	1	2	yes	—
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	—	2	1	no	—
112	Dic₁₇	#460	1	1	yes	characteristic
113	Dic₁₇	#461	1	1	yes	characteristic
114	#431	—	17	2	no	—
115	C₂×#108	—	34	1	no	—
116	C₂×#108	—	34	1	no	—
117	#423	—	17	2	no	—
118	C₂×C₄⋊C₈	—	17	1	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	C₂×C₄×C₈	—	34	1	no	—
121	C₄×C₄⋊C₄	—	17	1	no	—
122	#336	—	34	1	no	—
123	#335	—	17	2	no	—
124	#334	—	17	1	no	—
125	#334	—	34	2	no	—
126	#334	—	34	2	no	—
127	#334	#595	1	1	yes	characteristic, complemented
128	C₂×C₁₇	#1945	1	1	yes	characteristic
129	C₂×C₁₇	#1948	1	1	yes	characteristic
130	C₂×C₁₇	#1953	1	1	yes	characteristic
131	D₁₇	—	2	1	no	—
132	D₁₇	—	2	1	no	—
133	C₂×M₁₆	—	34	2	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	34	1	no	—
137	C₂×C₄⋊C₄	—	34	1	no	—
138	C₂×C₄²	—	17	1	no	—
139	C₂×C₄²	—	34	1	no	—
140	#108	—	68	2	no	—
141	#108	—	68	2	no	—
142	C₄⋊C₈	—	17	2	no	—
143	C₄⋊C₈	—	17	2	no	—
144	C₄⋊C₈	—	34	1	no	—
145	C₄⋊C₈	—	34	2	no	—
146	C₄⋊C₈	—	34	2	no	—
147	C₄⋊C₈	C₂×#595	1	2	yes	—
148	C₂²⋊C₈	—	34	2	no	—
149	C₄×C₈	—	17	1	no	—
150	C₄×C₈	—	17	1	no	—
151	C₄×C₈	—	17	1	no	—
152	C₄×C₈	—	34	2	no	—
153	C₄×C₈	—	68	2	no	—
154	C₄×C₈	C₂×#595	1	1	yes	characteristic
155	#95	—	17	2	no	—
156	C₁₇	#9628	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	—	17	1	no	—
159	C₂²×C₄	—	17	1	no	—
160	C₂²×C₄	—	34	2	no	—
161	M₁₆	—	68	2	no	—
162	M₁₆	—	68	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	2	no	—
166	C₂×C₈	—	17	2	no	—
167	C₂×C₈	—	17	1	no	—
168	C₂×C₈	—	34	1	no	—
169	C₂×C₈	—	34	1	no	—
170	C₂×C₈	—	34	2	no	—
171	C₂×C₈	—	34	2	no	—
172	C₂×C₈	—	68	1	no	—
173	C₂×C₈	—	68	1	no	—
174	C₂×C₈	—	68	2	no	—
175	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
177	C₄⋊C₄	—	34	4	no	—
178	C₄²	—	17	1	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	68	1	no	—
182	C₄²	—	68	1	no	—
183	C₄²	C₂²×#595	1	1	yes	characteristic
184	C₂³	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	17	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	—	34	2	no	—
193	C₂×C₄	—	34	4	no	—
194	C₂×C₄	—	68	1	no	—
195	C₂×C₄	—	68	2	no	—
196	C₂×C₄	—	68	2	no	—
197	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
198	C₂×C₄	D₄×#595	1	1	yes	characteristic
199	C₂×C₄	Q₈×#595	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	4	2	no	—
202	C₈	—	17	2	no	—
203	C₈	—	34	1	no	—
204	C₈	—	68	2	no	—
205	C₈	—	68	2	no	—
206	C₈	—	136	2	no	—
207	C₈	#10587533	1	2	yes	—
208	C₂²	—	17	1	no	—
209	C₂²	—	17	1	no	—
210	C₂²	—	34	1	no	—
211	C₂²	—	34	1	no	—
212	C₂²	#11763760	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	17	1	no	—
215	C₄	—	17	1	no	—
216	C₄	—	34	1	no	—
217	C₄	—	34	1	no	—
218	C₄	—	68	4	no	—
219	C₄	C₂×#10587533	1	1	yes	characteristic
220	C₄	#11763820	1	1	yes	characteristic
221	C₄	#11763852	1	2	yes	—
222	C₂	—	34	1	no	—
223	C₂	—	34	1	no	—
224	C₂	#14544891	1	1	yes	characteristic
225	C₂	#14544896	1	1	yes	characteristic
226	C₂	#14544903	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544723

C₂

yes

maximal, characteristic

C₈×C₄⋊#595

C₂

yes

maximal, characteristic

#14544352

C₂

yes

maximal

#14543681

C₂

yes

maximal

#14527529

C₂

yes

maximal, characteristic

C₂×#10587535

—

C₂×#10587535

—

#11763813

C₂²

yes

—

#11763810

C₂²

yes

characteristic

C₂×C₈×#595

—

C₄×C₄⋊#595

C₂²

yes

—

—

—

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762561

C₄

yes

complemented

#11762560

C₄

yes

characteristic, complemented

C₁₇×#334

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

—

#10587535

—

#10587535

—

C₈×#595

—

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

C₂×SD₁₆

yes

characteristic

C₄×D₁₇

C₂×SD₁₆

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#106

yes

—

—

—

yes

—

#1848

—

C₈×C₄⋊C₄

—

#1570

—

#1564

—

#1303

—

100

C₂²×C₁₇

#389

yes

characteristic

101

C₂×D₁₇

—

102

C₂×D₁₇

—

103

C₂×D₁₇

#475

yes

characteristic

104

C₂×D₁₇

#476

yes

characteristic

105

#595

—

106

C₄×C₁₇

—

107

C₄×C₁₇

C₂×#106

yes

characteristic

108

C₄×C₁₇

#428

yes

characteristic

109

C₄×C₁₇

#443

yes

—

110

Dic₁₇

—

111

Dic₁₇

—

112

Dic₁₇

#460

yes

characteristic

113

Dic₁₇

#461

yes

characteristic

114

#431

—

115

C₂×#108

—

116

C₂×#108

—

117

#423

—

118

C₂×C₄⋊C₈

—

119

C₂×C₄×C₈

—

120

C₂×C₄×C₈

—

121

C₄×C₄⋊C₄

—

122

#336

—

123

#335

—

124

#334

—

125

#334

—

126

#334

—

127

#334

#595

yes

characteristic, complemented

128

C₂×C₁₇

#1945

yes

characteristic

129

C₂×C₁₇

#1948

yes

characteristic

130

C₂×C₁₇

#1953

yes

characteristic

131

D₁₇

—

132

D₁₇

—

133

C₂×M₁₆

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂×C₄⋊C₄

—

138

C₂×C₄²

—

139

C₂×C₄²

—

140

#108

—

141

#108

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

C₂×#595

yes

—

148

C₂²⋊C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

C₂×#595

yes

characteristic

155

#95

—

156

C₁₇

#9628

yes

characteristic, complemented, Sylow

157

C₂²×C₄

—

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

M₁₆

—

162

M₁₆

—

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

—

174

C₂×C₈

—

175

C₂×C₈

C₄⋊#595

yes

characteristic

176

C₂×C₈

C₄⋊#595

yes

characteristic

177

C₄⋊C₄

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

183

C₄²

C₂²×#595

yes

characteristic

184

C₂³

—

185

C₂×C₄

—

186

C₂×C₄

—

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

—

197

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

198

C₂×C₄

D₄×#595

yes

characteristic

199

C₂×C₄

Q₈×#595

yes

characteristic

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

136

—

207

C₈

#10587533

yes

—

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

—

212

C₂²

#11763760

yes

characteristic

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

—

219

C₄

C₂×#10587533

yes

characteristic

220

C₄

#11763820

yes

characteristic

221

C₄

#11763852

yes

—

222

C₂

—

223

C₂

—

224

C₂

#14544891

yes

characteristic

225

C₂

#14544896

yes

characteristic

226

C₂

#14544903

yes

characteristic

	subgroup	G/H
center	C₂²	#11763760
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#334	C₄
socle	C₂²×C₁₇	#389
prime radical	C₂²×D₁₇	C₂²⋊Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	#334	#595
hypocenter	C₁₇	#9628
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#11763760

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#334

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 ◁ #334

N	H ≅ G/N	type
17. #11762561	C₄	semidirect
18. #11762560	C₄	semidirect
19. C₁₇×#334	C₄	semidirect
127. #334	#595	semidirect
156. C₁₇	#9628	semidirect

H ≅ G/N

type

17. #11762561

C₄

semidirect

18. #11762560

C₄

semidirect

19. C₁₇×#334

C₄

semidirect

127. #334

#595

semidirect

156. C₁₇

#9628

semidirect

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

subgroup	index
#14544723	2
C₈×C₄⋊#595	2
#14544352	2
#14543681	2
#14527529	2
#9628	17

subgroup

index

C₈×C₄⋊#595

Subgroups — #17955894

Conjugacy classes of subgroups

Complements

Maximal subgroups