Group #17955859 — subgroups

Subgroups — #17955859

Order 4352.

306 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 126 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	#14544335	C₂	1	2	yes	maximal
2	C₂×#11763672	C₂	1	1	yes	maximal, characteristic
3	#14543987	C₂	1	1	yes	maximal, characteristic
4	#14543630	C₂	1	2	yes	maximal
5	#108×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	#11763825	C₂²	1	1	yes	characteristic
7	#11763825	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#3624	C₂²	1	2	yes	—
9	#11763672	—	2	2	no	—
10	#11763672	—	2	1	no	—
11	#11763672	—	2	1	no	—
12	#11763670	—	2	2	no	—
13	#11763650	—	2	1	no	—
14	C₈×#3624	—	2	1	no	—
15	C₂×#10587490	C₄	1	1	yes	characteristic
16	M₁₆×Dic₁₇	C₂²	1	2	yes	—
17	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
18	#11762573	C₄	1	2	yes	—
19	C₂×C₁₇×#108	C₄	1	1	yes	characteristic
20	C₂²×#3624	—	2	2	no	—
21	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₇	C₂³	1	1	yes	characteristic
24	C₂×#63223	C₂×C₄	1	2	yes	—
25	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊#3624	—	2	1	no	—
27	C₂²⋊#3624	Q₈	1	2	yes	—
28	C₄⋊#3624	—	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	C₄×#3624	—	2	1	no	—
34	C₄×#3624	—	2	1	no	—
35	C₁₇×#108	C₈	1	4	yes	complemented
36	#10587490	—	2	2	no	—
37	#10587486	—	2	2	no	—
38	C₈×Dic₁₇	—	2	2	no	—
39	C₈×Dic₁₇	D₄	1	1	yes	characteristic
40	C₈×Dic₁₇	D₄	1	1	yes	characteristic
41	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
42	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×Dic₁₇	#41	1	1	yes	characteristic
46	C₂×#3624	—	2	1	no	—
47	C₂×#3624	—	2	1	no	—
48	C₂×#3624	—	2	2	no	—
49	C₂×#3624	—	4	2	no	—
50	C₂×#3624	—	8	2	no	—
51	M₁₆×C₁₇	—	2	2	no	—
52	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
58	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄×Dic₁₇	C₂×Q₈	1	1	yes	characteristic
61	C₄×Dic₁₇	#41	1	1	yes	characteristic
62	#63223	—	2	2	no	—
63	#63223	—	2	2	no	—
64	C₂×#3622	—	2	1	no	—
65	C₂×#3622	—	2	1	no	—
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	#9608	—	17	1	no	maximal, Sylow
70	C₂³×C₁₇	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
77	C₂×Dic₁₇	—	2	1	no	—
78	C₂×Dic₁₇	—	2	1	no	—
79	C₂×Dic₁₇	—	4	1	no	—
80	C₂×Dic₁₇	#121	1	1	yes	characteristic
81	C₂×Dic₁₇	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×Dic₁₇	#125	1	1	yes	characteristic
83	C₂×Dic₁₇	C₄⋊Q₈	1	1	yes	characteristic
84	#3624	—	4	2	no	—
85	#3624	—	4	1	no	—
86	#3624	—	4	1	no	—
87	#3624	—	8	2	no	—
88	C₈×C₁₇	—	2	4	no	—
89	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
90	C₈×C₁₇	C₄⋊C₈	1	2	yes	—
91	#3622	—	2	1	no	—
92	#3622	—	2	1	no	—
93	#3622	—	4	2	no	—
94	C₄×#108	—	17	1	no	—
95	#1565	—	17	2	no	—
96	C₂×#335	—	17	1	no	—
97	#1450	—	17	1	no	—
98	#1283	—	17	2	no	—
99	C₂²×C₁₇	—	2	1	no	—
100	C₂²×C₁₇	—	2	1	no	—
101	C₂²×C₁₇	#389	1	1	yes	characteristic
102	C₂²×C₁₇	C₈×Q₈	1	1	yes	characteristic
103	C₂²×C₁₇	#446	1	1	yes	characteristic
104	C₄×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₄×C₁₇	#424	1	1	yes	characteristic
106	C₄×C₁₇	C₈×D₄	1	1	yes	characteristic
107	C₄×C₁₇	#436	1	1	yes	characteristic
108	Dic₁₇	—	2	1	no	—
109	Dic₁₇	—	2	1	no	—
110	Dic₁₇	—	2	1	no	—
111	Dic₁₇	#472	1	1	yes	characteristic
112	Dic₁₇	#473	1	1	yes	characteristic
113	#431	—	17	1	no	—
114	#431	—	17	1	no	—
115	C₂×#108	—	17	1	no	—
116	C₂×#108	#595	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	17	2	no	—
118	C₄×M₁₆	—	17	2	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	#341	—	17	2	no	—
121	#335	—	34	1	no	—
122	#335	—	34	1	no	—
123	#335	—	34	2	no	—
124	#334	—	34	2	no	—
125	#322	—	34	1	no	—
126	C₈²	—	34	1	no	—
127	C₂×C₁₇	—	2	1	no	—
128	C₂×C₁₇	—	2	1	no	—
129	C₂×C₁₇	#1939	1	1	yes	characteristic
130	C₂×C₁₇	#1952	1	1	yes	characteristic
131	C₂×C₁₇	#1953	1	1	yes	characteristic
132	C₂×M₁₆	—	17	2	no	—
133	C₂×M₁₆	C₂×#595	1	2	yes	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	34	2	no	—
136	C₂²×C₈	C₂×#595	1	1	yes	characteristic
137	C₂×C₄²	—	17	1	no	—
138	#108	—	34	2	no	—
139	#108	#3624	1	4	yes	complemented
140	C₄⋊C₈	—	17	1	no	—
141	C₄⋊C₈	—	17	1	no	—
142	C₄⋊C₈	—	34	1	no	—
143	C₄⋊C₈	—	34	2	no	—
144	C₄⋊C₈	—	34	2	no	—
145	C₂²⋊C₈	—	17	2	no	—
146	C₂²⋊C₈	—	34	1	no	—
147	#97	—	34	2	no	—
148	C₄×C₈	—	17	1	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	—	34	1	no	—
154	C₄×C₈	—	34	1	no	—
155	C₁₇	#9608	1	1	yes	characteristic, complemented, Sylow
156	C₂²×C₄	—	17	1	no	—
157	C₂²×C₄	—	17	1	no	—
158	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
159	M₁₆	—	2	2	no	—
160	M₁₆	—	34	2	no	—
161	M₁₆	—	34	2	no	—
162	M₁₆	C₂×#3624	1	4	yes	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	17	1	no	—
165	C₂×C₈	—	17	1	no	—
166	C₂×C₈	—	34	1	no	—
167	C₂×C₈	—	34	1	no	—
168	C₂×C₈	—	34	2	no	—
169	C₂×C₈	—	34	1	no	—
170	C₂×C₈	—	34	1	no	—
171	C₂×C₈	—	34	2	no	—
172	C₂×C₈	—	68	2	no	—
173	C₂×C₈	—	136	2	no	—
174	C₂×C₈	#63243	1	2	yes	—
175	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
176	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
177	C₂×C₈	C₂×#3624	1	2	yes	—
178	C₄²	—	17	1	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₂³	Q₈×#595	1	1	yes	characteristic
183	C₂×C₄	—	17	1	no	—
184	C₂×C₄	—	17	1	no	—
185	C₂×C₄	—	17	1	no	—
186	C₂×C₄	—	17	1	no	—
187	C₂×C₄	—	34	1	no	—
188	C₂×C₄	—	34	1	no	—
189	C₂×C₄	—	68	1	no	—
190	C₂×C₄	C₂×#63243	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
192	C₂×C₄	#10587678	1	1	yes	characteristic
193	C₂×C₄	D₄×#595	1	1	yes	characteristic
194	C₂×C₄	#10587680	1	1	yes	characteristic
195	C₂×C₄	C₂²×#3624	1	1	yes	characteristic
196	C₈	—	2	4	no	—
197	C₈	—	34	1	no	—
198	C₈	—	34	1	no	—
199	C₈	—	68	1	no	—
200	C₈	—	68	1	no	—
201	C₈	—	68	2	no	—
202	C₈	—	68	2	no	—
203	C₈	—	136	2	no	—
204	C₈	C₄⋊#3624	1	2	yes	—
205	C₈	C₄⋊#3624	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11763760	1	1	yes	characteristic
209	C₂²	Q₈×#3624	1	1	yes	characteristic
210	C₂²	#11763859	1	1	yes	characteristic
211	C₄	—	17	1	no	—
212	C₄	—	17	1	no	—
213	C₄	—	34	1	no	—
214	C₄	—	34	1	no	—
215	C₄	—	34	1	no	—
216	C₄	C₂×C₄⋊#3624	1	1	yes	characteristic
217	C₄	#11763816	1	1	yes	characteristic
218	C₄	D₄×#3624	1	1	yes	characteristic
219	C₄	#11763843	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#14544881	1	1	yes	characteristic
223	C₂	#14544902	1	1	yes	characteristic
224	C₂	#14544904	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14544335

C₂

yes

maximal

C₂×#11763672

C₂

yes

maximal, characteristic

#14543987

C₂

yes

maximal, characteristic

#14543630

C₂

yes

maximal

#108×Dic₁₇

C₂

yes

maximal, characteristic

#11763825

C₂²

yes

characteristic

#11763825

C₂²

yes

characteristic

C₂×C₄⋊#3624

C₂²

yes

—

—

—

—

—

—

—

C₄

yes

characteristic

M₁₆×Dic₁₇

C₂²

yes

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762573

C₄

yes

—

C₂×C₁₇×#108

C₄

yes

characteristic

C₂²×#3624

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#63223

C₂×C₄

yes

—

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²⋊#3624

—

C₂²⋊#3624

Q₈

yes

—

—

—

—

D₄

yes

characteristic

C₄⋊#3624

D₄

yes

characteristic

C₄×#3624

—

C₄×#3624

—

C₁₇×#108

C₈

yes

complemented

#10587490

—

#10587486

—

C₈×Dic₁₇

—

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×Dic₁₇

C₂×Q₈

yes

C₂²×Dic₁₇

yes

—

—

—

—

—

M₁₆×C₁₇

—

M₁₆×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₄×Dic₁₇

C₂×D₄

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

#9608

—

maximal, Sylow

C₂³×C₁₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

#121

yes

characteristic

C₂×Dic₁₇

C₂²⋊Q₈

yes

characteristic

C₂×Dic₁₇

#125

yes

characteristic

C₂×Dic₁₇

C₄⋊Q₈

yes

—

—

—

—

C₈×C₁₇

—

C₈×C₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

C₄⋊C₈

yes

—

—

—

—

—

—

—

—

—

C₂²×C₁₇

—

100

C₂²×C₁₇

—

101

C₂²×C₁₇

#389

yes

characteristic

102

C₂²×C₁₇

C₈×Q₈

yes

characteristic

103

C₂²×C₁₇

#446

yes

characteristic

104

C₄×C₁₇

C₂×C₄⋊C₈

yes

characteristic

105

C₄×C₁₇

#424

yes

characteristic

106

C₄×C₁₇

C₈×D₄

yes

characteristic

107

C₄×C₁₇

#436

yes

characteristic

108

Dic₁₇

—

109

Dic₁₇

—

110

Dic₁₇

—

111

Dic₁₇

#472

yes

characteristic

112

Dic₁₇

#473

yes

characteristic

113

#431

—

114

#431

—

115

C₂×#108

—

116

C₂×#108

#595

yes

characteristic

117

C₂×C₄⋊C₈

—

118

C₄×M₁₆

—

119

C₂×C₄×C₈

—

120

#341

—

121

#335

—

122

#335

—

123

#335

—

124

#334

—

125

#322

—

126

C₈²

—

127

C₂×C₁₇

—

128

C₂×C₁₇

—

129

C₂×C₁₇

#1939

yes

characteristic

130

C₂×C₁₇

#1952

yes

characteristic

131

C₂×C₁₇

#1953

yes

characteristic

132

C₂×M₁₆

—

133

C₂×M₁₆

C₂×#595

yes

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

C₂²×C₈

C₂×#595

yes

characteristic

137

C₂×C₄²

—

138

#108

—

139

#108

#3624

yes

complemented

140

C₄⋊C₈

—

141

C₄⋊C₈

—

142

C₄⋊C₈

—

143

C₄⋊C₈

—

144

C₄⋊C₈

—

145

C₂²⋊C₈

—

146

C₂²⋊C₈

—

147

#97

—

148

C₄×C₈

—

149

C₄×C₈

—

150

C₄×C₈

—

151

C₄×C₈

—

152

C₄×C₈

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₁₇

#9608

yes

characteristic, complemented, Sylow

156

C₂²×C₄

—

157

C₂²×C₄

—

158

C₂²×C₄

C₂²×#595

yes

characteristic

159

M₁₆

—

160

M₁₆

—

161

M₁₆

—

162

M₁₆

C₂×#3624

yes

—

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

—

136

—

174

C₂×C₈

#63243

yes

—

175

C₂×C₈

C₄⋊#595

yes

characteristic

176

C₂×C₈

C₄⋊#595

yes

characteristic

177

C₂×C₈

C₂×#3624

yes

—

178

C₄²

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₂³

Q₈×#595

yes

characteristic

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

—

187

C₂×C₄

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

C₂×#63243

yes

characteristic

191

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

192

C₂×C₄

#10587678

yes

characteristic

193

C₂×C₄

D₄×#595

yes

characteristic

194

C₂×C₄

#10587680

yes

characteristic

195

C₂×C₄

C₂²×#3624

yes

characteristic

196

C₈

—

197

C₈

—

198

C₈

—

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

136

—

204

C₈

C₄⋊#3624

yes

—

205

C₈

C₄⋊#3624

yes

—

206

C₂²

—

207

C₂²

—

208

C₂²

#11763760

yes

characteristic

209

C₂²

Q₈×#3624

yes

characteristic

210

C₂²

#11763859

yes

characteristic

211

C₄

—

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

C₂×C₄⋊#3624

yes

characteristic

217

C₄

#11763816

yes

characteristic

218

C₄

D₄×#3624

yes

characteristic

219

C₄

#11763843

yes

characteristic

220

C₂

—

221

C₂

—

222

C₂

#14544881

yes

characteristic

223

C₂

#14544902

yes

characteristic

224

C₂

#14544904

yes

characteristic

	subgroup	G/H
center	C₂²	#11763760
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₂×C₁₇×#108	C₄
socle	C₂²×C₁₇	#389
prime radical	C₂³×C₁₇	C₄×Q₈
cosocle	C₂×C₄×Dic₁₇	C₂³
hypercenter	C₂×#108	#595
hypocenter	C₁₇	#9608
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#11763760

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₂×C₁₇×#108

C₄

socle

C₂²×C₁₇

#389

prime radical

C₂³×C₁₇

C₄×Q₈

cosocle

C₂×C₄×Dic₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
35. C₁₇×#108	C₈	semidirect
139. #108	#3624	semidirect
155. C₁₇	#9608	semidirect

H ≅ G/N

type

35. C₁₇×#108

C₈

semidirect

139. #108

#3624

semidirect

155. C₁₇

#9608

semidirect

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

subgroup	index
#14544335	2
C₂×#11763672	2
#14543987	2
#14543630	2
#108×Dic₁₇	2
#9608	17

subgroup

index

#108×Dic₁₇

Subgroups — #17955859

Conjugacy classes of subgroups

Complements

Maximal subgroups