Group #17961042 — subgroups

Subgroups — #17961042

Order 4352.

296 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 112 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	#14545123	C₂	1	1	yes	maximal, characteristic
2	#14545003	C₂	1	2	yes	maximal
3	#14544586	C₂	1	1	yes	maximal, characteristic
4	#14543833	C₂	1	2	yes	maximal
5	#14530377	C₂	1	1	yes	maximal, characteristic
6	#11763865	C₂²	1	2	yes	—
7	D₁₇⋊#108	C₄	1	1	yes	characteristic
8	#11763819	C₄	1	1	yes	characteristic
9	#11763772	C₂²	1	2	yes	—
10	#11763732	C₄	1	2	yes	complemented
11	#11763731	C₄	1	2	yes	complemented
12	#11763700	—	2	2	no	—
13	#11763699	—	2	2	no	—
14	#11763680	C₄	1	2	yes	—
15	C₂×#10587469	C₂²	1	1	yes	characteristic
16	C₂×#10587469	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
18	#11762585	C₄	1	1	yes	characteristic, complemented
19	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic, complemented
20	C₁₆×Dic₁₇	C₄	1	1	yes	characteristic, complemented
21	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
22	#10587677	C₂×C₄	1	2	yes	—
23	D₁₇⋊M₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
26	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587538	C₂×C₄	1	4	yes	—
28	#10587536	—	2	1	no	—
29	#10587535	—	2	1	no	—
30	#10587534	—	2	1	no	—
31	#10587533	—	2	1	no	—
32	#10587532	—	2	2	no	—
33	C₈×#595	—	2	2	no	—
34	#10587530	—	2	2	no	—
35	#10587529	—	2	2	no	—
36	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
41	C₂×#63214	C₂×C₄	1	1	yes	characteristic
42	C₂×#63214	C₂×C₄	1	1	yes	characteristic
43	C₄×#3622	C₂×C₄	1	1	yes	characteristic
44	#10587469	—	2	1	no	—
45	#10587469	—	2	1	no	—
46	#10587469	D₄	1	2	yes	—
47	#10587469	Q₈	1	2	yes	—
48	C₂×C₄×D₁₇	—	2	1	no	—
49	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊#595	—	2	2	no	—
51	#63247	—	2	2	no	—
52	C₄⋊#595	—	2	2	no	—
53	C₄×#595	—	2	2	no	—
54	#63243	—	2	2	no	—
55	#63242	—	2	2	no	—
56	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₄²	1	1	yes	characteristic
59	C₄×Dic₁₇	—	2	1	no	—
60	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
61	C₂×#3622	C₄²	1	1	yes	characteristic
62	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₁₇	M₁₆	1	1	yes	characteristic
64	C₈×D₁₇	M₁₆	1	1	yes	characteristic
65	C₈×D₁₇	M₁₆	1	1	yes	characteristic
66	C₈×D₁₇	M₁₆	1	1	yes	characteristic
67	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
68	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
69	C₈×D₁₇	#41	1	1	yes	characteristic
70	C₈×D₁₇	#41	1	1	yes	characteristic
71	#63216	—	2	4	no	—
72	C₁₆×C₁₇	—	2	1	no	—
73	C₁₆×C₁₇	C₄⋊C₄	1	2	yes	—
74	#63214	—	2	1	no	—
75	#63214	C₄⋊C₄	1	2	yes	—
76	#12667	—	17	1	no	maximal, Sylow
77	C₂²×D₁₇	#131	1	1	yes	characteristic
78	C₂×#595	—	4	2	no	—
79	C₂×C₄×C₁₇	—	2	1	no	—
80	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×Dic₁₇	—	2	1	no	—
82	C₂×Dic₁₇	#131	1	1	yes	characteristic
83	C₄×D₁₇	—	4	2	no	—
84	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
85	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
86	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
87	C₄×D₁₇	C₂×M₁₆	1	1	yes	characteristic
88	#3624	—	4	2	no	—
89	C₈×C₁₇	#97	1	1	yes	characteristic
90	C₈×C₁₇	#97	1	1	yes	characteristic
91	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₈×C₁₇	#117	1	1	yes	characteristic
93	#3622	#97	1	1	yes	characteristic
94	#3622	#97	1	1	yes	characteristic
95	#3622	C₄×D₄	1	1	yes	characteristic
96	#3622	C₄×Q₈	1	1	yes	characteristic
97	C₂×#364	—	17	1	no	—
98	#2110	—	17	2	no	—
99	C₄×M₃₂	—	17	1	no	—
100	#1772	—	17	1	no	—
101	#1378	—	17	2	no	—
102	C₂²×C₁₇	#405	1	1	yes	characteristic
103	C₂×D₁₇	—	4	1	no	—
104	C₂×D₁₇	#436	1	1	yes	characteristic
105	C₂×D₁₇	#446	1	1	yes	characteristic
106	#595	—	8	2	no	—
107	C₄×C₁₇	—	4	1	no	—
108	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
110	Dic₁₇	—	2	2	no	—
111	Dic₁₇	#436	1	1	yes	characteristic
112	Dic₁₇	#446	1	1	yes	characteristic
113	C₂×M₃₂	—	17	1	no	—
114	C₂×M₃₂	—	17	1	no	—
115	C₂×M₃₂	—	17	2	no	—
116	C₂×#108	—	17	1	no	—
117	#427	—	17	1	no	—
118	#405	—	17	2	no	—
119	C₂×C₄×C₈	—	17	1	no	—
120	#364	—	17	2	no	—
121	#364	—	17	2	no	—
122	#347	—	34	2	no	—
123	#347	—	34	2	no	—
124	#346	—	17	1	no	—
125	#346	#595	1	1	yes	characteristic, complemented
126	C₄×C₁₆	—	17	1	no	—
127	C₄×C₁₆	—	17	1	no	—
128	#339	—	17	2	no	—
129	C₂×C₁₇	—	2	1	no	—
130	C₂×C₁₇	#1770	1	1	yes	characteristic
131	D₁₇	—	2	1	no	—
132	D₁₇	—	2	1	no	—
133	C₂×M₁₆	—	17	2	no	—
134	C₂²×C₈	—	17	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	#117	—	17	2	no	—
137	C₂×C₄²	—	17	1	no	—
138	M₃₂	—	17	2	no	—
139	M₃₂	—	17	2	no	—
140	M₃₂	—	17	4	no	—
141	M₃₂	—	34	1	no	—
142	M₃₂	—	34	1	no	—
143	M₃₂	—	34	2	no	—
144	C₂×C₁₆	—	17	1	no	—
145	C₂×C₁₆	—	17	1	no	—
146	C₂×C₁₆	—	34	2	no	—
147	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
148	C₂×C₁₆	C₂×#595	1	1	yes	characteristic
149	#108	—	34	1	no	—
150	#108	—	34	1	no	—
151	#107	—	34	1	no	—
152	#106	—	34	1	no	—
153	#97	—	34	2	no	—
154	C₄×C₈	—	17	1	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	34	2	no	—
158	C₄×C₈	C₂×#595	1	1	yes	characteristic
159	C₁₇	#12667	1	1	yes	characteristic, complemented, Sylow
160	C₂²×C₄	—	17	1	no	—
161	C₂²×C₄	—	34	1	no	—
162	M₁₆	—	34	2	no	—
163	M₁₆	—	34	2	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	—	17	1	no	—
173	C₂×C₈	—	17	1	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	C₄×#595	1	1	yes	characteristic
176	C₂×C₈	C₂²×#595	1	1	yes	characteristic
177	C₄⋊C₄	—	34	2	no	—
178	C₂²⋊C₄	—	34	2	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	34	1	no	—
181	C₄²	—	34	2	no	—
182	C₄²	C₄×#595	1	1	yes	characteristic
183	C₁₆	—	2	1	no	—
184	C₁₆	—	17	2	no	—
185	C₁₆	—	34	4	no	—
186	C₁₆	—	34	1	no	—
187	C₁₆	C₄⋊#595	1	2	yes	—
188	C₂³	—	17	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	17	1	no	—
191	C₂×C₄	—	17	1	no	—
192	C₂×C₄	—	17	1	no	—
193	C₂×C₄	—	17	1	no	—
194	C₂×C₄	—	17	1	no	—
195	C₂×C₄	—	34	1	no	—
196	C₂×C₄	—	68	2	no	—
197	C₂×C₄	—	68	2	no	—
198	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
199	C₈	—	17	1	no	—
200	C₈	—	17	1	no	—
201	C₈	—	17	1	no	—
202	C₈	—	17	1	no	—
203	C₈	—	68	2	no	—
204	C₈	#10587532	1	1	yes	characteristic
205	C₈	#10587532	1	1	yes	characteristic
206	C₈	C₂×C₄⋊#595	1	1	yes	characteristic
207	C₈	#10587677	1	1	yes	characteristic
208	C₂²	—	17	1	no	—
209	C₂²	—	17	1	no	—
210	C₂²	—	68	1	no	—
211	C₂²	#11763772	1	1	yes	characteristic
212	C₄	—	4	1	no	—
213	C₄	—	17	1	no	—
214	C₄	—	17	1	no	—
215	C₄	—	34	2	no	—
216	C₄	—	136	2	no	—
217	C₄	C₄×C₄⋊#595	1	1	yes	characteristic
218	C₄	C₂×#10587532	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	34	1	no	—
221	C₂	—	34	1	no	—
222	C₂	#14544578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545123

C₂

yes

maximal, characteristic

#14545003

C₂

yes

maximal

#14544586

C₂

yes

maximal, characteristic

#14543833

C₂

yes

maximal

#14530377

C₂

yes

maximal, characteristic

#11763865

C₂²

yes

—

D₁₇⋊#108

C₄

yes

characteristic

#11763819

C₄

yes

characteristic

#11763772

C₂²

yes

—

#11763732

C₄

yes

complemented

#11763731

C₄

yes

—

—

C₄

yes

—

C₂×#10587469

C₂²

yes

characteristic

C₂×#10587469

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic, complemented

C₁₆×Dic₁₇

C₄

yes

characteristic, complemented

C₁₆×Dic₁₇

C₄

yes

characteristic, complemented

C₁₇×#346

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

#10587538

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

—

—

D₄

yes

—

#10587469

Q₈

yes

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²×C₄

yes

characteristic

C₂²⋊#595

—

—

—

—

—

—

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

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

C₂×D₄

yes

characteristic

C₈×D₁₇

C₂×Q₈

yes

C₈×D₁₇

yes

C₈×D₁₇

yes

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₄⋊C₄

yes

—

#63214

—

#63214

C₄⋊C₄

yes

—

—

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

C₂×M₁₆

yes

characteristic

C₄×D₁₇

C₂×M₁₆

yes

characteristic

#3624

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

C₂×C₄⋊C₄

yes

C₈×C₁₇

yes

yes

yes

C₄×D₄

yes

characteristic

#3622

C₄×Q₈

yes

characteristic

C₂×#364

—

#2110

—

C₄×M₃₂

—

100

#1772

—

101

#1378

—

102

C₂²×C₁₇

#405

yes

characteristic

103

C₂×D₁₇

—

104

C₂×D₁₇

#436

yes

characteristic

105

C₂×D₁₇

#446

yes

characteristic

106

#595

—

107

C₄×C₁₇

—

108

C₄×C₁₇

C₄×C₄⋊C₄

yes

characteristic

109

C₄×C₁₇

C₂×#97

yes

characteristic

110

Dic₁₇

—

111

Dic₁₇

#436

yes

characteristic

112

Dic₁₇

#446

yes

characteristic

113

C₂×M₃₂

—

114

C₂×M₃₂

—

115

C₂×M₃₂

—

116

C₂×#108

—

117

#427

—

118

#405

—

119

C₂×C₄×C₈

—

120

#364

—

121

#364

—

122

#347

—

123

#347

—

124

#346

—

125

#346

#595

yes

characteristic, complemented

126

C₄×C₁₆

—

127

C₄×C₁₆

—

128

#339

—

129

C₂×C₁₇

—

130

C₂×C₁₇

#1770

yes

characteristic

131

D₁₇

—

132

D₁₇

—

133

C₂×M₁₆

—

134

C₂²×C₈

—

135

C₂²×C₈

—

136

#117

—

137

C₂×C₄²

—

138

M₃₂

—

139

M₃₂

—

140

M₃₂

—

141

M₃₂

—

142

M₃₂

—

143

M₃₂

—

144

C₂×C₁₆

—

145

C₂×C₁₆

—

146

C₂×C₁₆

—

147

C₂×C₁₆

C₂×#595

yes

characteristic

148

C₂×C₁₆

C₂×#595

yes

characteristic

149

#108

—

150

#108

—

151

#107

—

152

#106

—

153

#97

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

C₂×#595

yes

characteristic

159

C₁₇

#12667

yes

characteristic, complemented, Sylow

160

C₂²×C₄

—

161

C₂²×C₄

—

162

M₁₆

—

163

M₁₆

—

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

C₂²×#595

yes

characteristic

177

C₄⋊C₄

—

178

C₂²⋊C₄

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

C₄×#595

yes

characteristic

183

C₁₆

—

184

C₁₆

—

185

C₁₆

—

186

C₁₆

—

187

C₁₆

C₄⋊#595

yes

—

188

C₂³

—

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

—

198

C₂×C₄

C₂×C₄×#595

yes

characteristic

199

C₈

—

200

C₈

—

201

C₈

—

202

C₈

—

203

C₈

—

204

C₈

#10587532

yes

characteristic

205

C₈

#10587532

yes

characteristic

206

C₈

C₂×C₄⋊#595

yes

characteristic

207

C₈

#10587677

yes

characteristic

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

#11763772

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

—

136

—

217

C₄

C₄×C₄⋊#595

yes

characteristic

218

C₄

C₂×#10587532

yes

characteristic

219

C₂

—

220

C₂

—

221

C₂

—

222

C₂

#14544578

yes

characteristic

	subgroup	G/H
center	C₈	#10587677
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×#346	C₄
socle	C₂×C₁₇	#1770
prime radical	C₂²×D₁₇	#131
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	#346	#595
hypocenter	C₁₇	#12667
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

subgroup

G/H

center

C₈

#10587677

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×#346

C₄

C₂×C₁₇

C₂²×D₁₇

C₂×C₈×D₁₇

C₂³

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

N	H ≅ G/N	type
10. #11763732	C₄	semidirect
11. #11763731	C₄	semidirect
18. #11762585	C₄	semidirect
19. C₁₆×Dic₁₇	C₄	semidirect
20. C₁₆×Dic₁₇	C₄	semidirect
21. C₁₇×#346	C₄	semidirect
125. #346	#595	semidirect
159. C₁₇	#12667	semidirect

H ≅ G/N

type

10. #11763732

C₄

semidirect

11. #11763731

C₄

semidirect

18. #11762585

C₄

semidirect

19. C₁₆×Dic₁₇

C₄

semidirect

20. C₁₆×Dic₁₇

C₄

semidirect

21. C₁₇×#346

C₄

semidirect

125. #346

#595

semidirect

159. C₁₇

#12667

semidirect

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

subgroup	index
#14545123	2
#14545003	2
#14544586	2
#14543833	2
#14530377	2
#12667	17

subgroup

index

Subgroups — #17961042

Conjugacy classes of subgroups

Complements

Maximal subgroups