Group #17952405 — subgroups

Subgroups — #17952405

Order 4352.

358 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 138 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×#10587535	C₂	1	1	yes	maximal, characteristic
2	#14544334	C₂	1	1	yes	maximal, characteristic
3	C₈×#63243	C₂	1	2	yes	maximal
4	#14543616	C₂	1	2	yes	maximal
5	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
6	#11763828	C₂²	1	2	yes	—
7	C₂×#10587535	C₄	1	2	yes	—
8	C₄×#63243	C₂²	1	2	yes	—
9	#11763685	C₄	1	2	yes	—
10	#11763672	C₄	1	2	yes	—
11	#11763671	C₄	1	2	yes	—
12	#11763651	—	2	2	no	—
13	C₈×#3624	—	2	2	no	—
14	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762531	C₄	1	1	yes	characteristic
18	#11762531	C₄	1	1	yes	characteristic
19	#11762530	C₄	1	1	yes	characteristic
20	C₁₇×#322	C₄	1	1	yes	characteristic
21	C₂×#63243	C₂×C₄	1	4	yes	—
22	C₂×C₈×D₁₇	—	2	1	no	—
23	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	Q₈	1	2	yes	—
26	C₄²×D₁₇	C₂³	1	1	yes	characteristic
27	#10587543	—	2	2	no	—
28	#10587542	—	2	2	no	—
29	C₄⋊#3624	C₂×C₄	1	4	yes	—
30	C₄×#3624	—	2	2	no	—
31	C₄×#3624	—	2	2	no	—
32	#10587535	C₈	1	8	yes	complemented
33	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	C₈×Dic₁₇	—	2	1	no	—
37	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
38	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	D₄	1	2	yes	—
40	C₄×#3622	C₂×C₄	1	1	yes	characteristic
41	C₄×#3622	C₂×C₄	1	1	yes	characteristic
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
46	C₂×#3624	—	2	4	no	—
47	C₂×#3624	—	2	4	no	—
48	#63243	—	2	4	no	—
49	#63243	C₂×C₈	1	8	yes	—
50	C₂×C₈×C₁₇	—	2	1	no	—
51	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₄×Dic₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₁₇	#41	1	1	yes	characteristic
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	C₄²	1	1	yes	characteristic
60	C₂×#3622	C₄²	1	1	yes	characteristic
61	C₂×#3622	C₄⋊C₄	1	2	yes	—
62	C₈×D₁₇	—	2	2	no	—
63	C₈×D₁₇	—	4	2	no	—
64	C₈×D₁₇	—	4	2	no	—
65	C₈×D₁₇	C₂×C₈	1	4	yes	—
66	#8294	—	17	1	no	maximal, Sylow
67	C₂²×D₁₇	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
71	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
72	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
74	C₄×D₁₇	—	2	2	no	—
75	C₄×D₁₇	—	2	2	no	—
76	C₄×D₁₇	C₂²×C₈	1	2	yes	—
77	C₄×D₁₇	#131	1	2	yes	—
78	#3624	—	2	8	no	—
79	#3624	—	4	4	no	—
80	C₈×C₁₇	—	2	1	no	—
81	C₈×C₁₇	—	2	2	no	—
82	C₈×C₁₇	—	4	1	no	—
83	C₈×C₁₇	C₄×C₈	1	2	yes	—
84	#3622	—	2	1	no	—
85	#3622	—	2	2	no	—
86	#3622	—	4	1	no	—
87	#3622	C₄×C₈	1	2	yes	—
88	C₄×#108	—	17	1	no	—
89	#1564	—	17	1	no	—
90	C₈×M₁₆	—	17	2	no	—
91	C₂×#322	—	17	1	no	—
92	#1278	—	17	2	no	—
93	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂×D₁₇	—	2	1	no	—
95	C₂×D₁₇	—	2	1	no	—
96	C₂×D₁₇	C₈×Q₈	1	2	yes	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₁₇	#405	1	1	yes	characteristic
100	C₄×C₁₇	#430	1	2	yes	—
101	Dic₁₇	—	2	1	no	—
102	Dic₁₇	C₈×D₄	1	2	yes	—
103	Dic₁₇	#444	1	2	yes	—
104	#431	—	17	2	no	—
105	C₂×#108	—	17	2	no	—
106	C₄×M₁₆	—	17	2	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	1	no	—
110	#341	—	17	2	no	—
111	#335	—	17	2	no	—
112	#334	—	17	2	no	—
113	#322	—	17	1	no	—
114	#322	—	17	1	no	—
115	#322	—	17	1	no	—
116	#322	—	34	2	no	—
117	#322	#595	1	1	yes	characteristic
118	C₈²	—	34	2	no	—
119	C₂×C₁₇	C₈×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₁₇	#1773	1	1	yes	characteristic
121	C₂×C₁₇	#1778	1	1	yes	characteristic
122	D₁₇	—	2	2	no	—
123	C₂×M₁₆	—	17	4	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	—	17	1	no	—
126	C₂²×C₈	—	17	2	no	—
127	C₂²×C₈	—	34	1	no	—
128	C₂×C₄²	—	17	1	no	—
129	#108	—	17	8	no	—
130	C₄⋊C₈	—	17	4	no	—
131	C₂²⋊C₈	—	34	2	no	—
132	#97	—	34	2	no	—
133	C₄×C₈	—	17	1	no	—
134	C₄×C₈	—	17	1	no	—
135	C₄×C₈	—	17	2	no	—
136	C₄×C₈	—	17	1	no	—
137	C₄×C₈	—	17	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	34	1	no	—
140	C₄×C₈	—	34	2	no	—
141	C₄×C₈	—	34	2	no	—
142	C₄×C₈	C₂×#595	1	1	yes	characteristic
143	C₄×C₈	C₂×#595	1	1	yes	characteristic
144	C₄×C₈	C₂×#595	1	1	yes	characteristic
145	C₁₇	#8294	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	17	1	no	—
147	C₂²×C₄	—	17	1	no	—
148	C₂²×C₄	—	17	1	no	—
149	M₁₆	—	17	8	no	—
150	M₁₆	—	34	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	4	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	2	no	—
156	C₂×C₈	—	34	2	no	—
157	C₂×C₈	—	34	4	no	—
158	C₂×C₈	—	34	4	no	—
159	C₂×C₈	—	34	1	no	—
160	C₂×C₈	—	68	2	no	—
161	C₂×C₈	—	68	2	no	—
162	C₂×C₈	C₄×#595	1	1	yes	characteristic
163	C₂×C₈	C₄×#595	1	1	yes	characteristic
164	C₂×C₈	C₄⋊#595	1	2	yes	—
165	C₄²	—	17	1	no	—
166	C₄²	—	17	1	no	—
167	C₄²	—	17	1	no	—
168	C₄²	C₂²×#595	1	1	yes	characteristic
169	C₂³	—	17	1	no	—
170	C₂×C₄	—	17	1	no	—
171	C₂×C₄	—	17	1	no	—
172	C₂×C₄	—	17	2	no	—
173	C₂×C₄	—	17	2	no	—
174	C₂×C₄	—	17	1	no	—
175	C₂×C₄	—	34	2	no	—
176	C₂×C₄	—	34	2	no	—
177	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
179	C₂×C₄	#10587677	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	17	2	no	—
184	C₈	—	34	8	no	—
185	C₈	—	34	1	no	—
186	C₈	—	34	2	no	—
187	C₈	—	68	1	no	—
188	C₈	—	68	4	no	—
189	C₈	C₈×#595	1	2	yes	—
190	C₂²	—	17	2	no	—
191	C₂²	—	34	1	no	—
192	C₂²	—	34	1	no	—
193	C₂²	C₄×C₄⋊#595	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	17	2	no	—
196	C₄	—	17	2	no	—
197	C₄	—	34	1	no	—
198	C₄	C₂×C₈×#595	1	1	yes	characteristic
199	C₄	#11763772	1	1	yes	characteristic
200	C₄	#11763824	1	2	yes	—
201	C₂	—	34	2	no	—
202	C₂	C₈×C₄⋊#595	1	1	yes	characteristic
203	C₂	#14544590	1	1	yes	characteristic
204	C₂	#14544604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587535

C₂

yes

maximal, characteristic

#14544334

C₂

yes

maximal, characteristic

C₈×#63243

C₂

yes

maximal

#14543616

C₂

yes

maximal

D₁₇×#322

C₂

yes

maximal, characteristic

#11763828

C₂²

yes

—

C₂×#10587535

C₄

yes

—

C₄×#63243

C₂²

yes

—

#11763685

C₄

yes

—

#11763672

C₄

yes

—

#11763671

C₄

yes

—

#11763651

—

C₈×#3624

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

C₄

yes

characteristic

#11762531

C₄

yes

characteristic

#11762530

C₄

yes

characteristic

C₁₇×#322

C₄

yes

characteristic

C₂×#63243

C₂×C₄

yes

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

—

C₄²×D₁₇

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#3624

—

C₄×#3624

—

#10587535

C₈

yes

complemented

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₄×#3622

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×Q₈

yes

C₂×C₄×D₁₇

yes

—

—

—

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

C₄×Dic₁₇

yes

—

C₄²

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

#8294

—

maximal, Sylow

C₂²×D₁₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

characteristic

C₂×Dic₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

C₄×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂²×C₈

yes

—

C₄×D₁₇

#131

yes

—

#3624

—

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

—

—

—

—

C₄×C₈

yes

—

C₄×#108

—

#1564

—

C₈×M₁₆

—

C₂×#322

—

#1278

—

C₂²×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₈×Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

C₄×C₁₇

#405

yes

characteristic

100

C₄×C₁₇

#430

yes

—

101

Dic₁₇

—

102

Dic₁₇

C₈×D₄

yes

—

103

Dic₁₇

#444

yes

—

104

#431

—

105

C₂×#108

—

106

C₄×M₁₆

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

—

110

#341

—

111

#335

—

112

#334

—

113

#322

—

114

#322

—

115

#322

—

116

#322

—

117

#322

#595

yes

characteristic

118

C₈²

—

119

C₂×C₁₇

C₈×C₄⋊C₄

yes

characteristic

120

C₂×C₁₇

#1773

yes

characteristic

121

C₂×C₁₇

#1778

yes

characteristic

122

D₁₇

—

123

C₂×M₁₆

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂×C₄²

—

129

#108

—

130

C₄⋊C₈

—

131

C₂²⋊C₈

—

132

#97

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

C₂×#595

yes

characteristic

143

C₄×C₈

C₂×#595

yes

characteristic

144

C₄×C₈

C₂×#595

yes

characteristic

145

C₁₇

#8294

yes

characteristic, complemented, Sylow

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

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

C₄×#595

yes

characteristic

163

C₂×C₈

C₄×#595

yes

characteristic

164

C₂×C₈

C₄⋊#595

yes

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²×#595

yes

characteristic

169

C₂³

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

C₂×C₄×#595

yes

characteristic

178

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

179

C₂×C₄

#10587677

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

C₈×#595

yes

—

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

C₄×C₄⋊#595

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

C₂×C₈×#595

yes

characteristic

199

C₄

#11763772

yes

characteristic

200

C₄

#11763824

yes

—

201

C₂

—

202

C₂

C₈×C₄⋊#595

yes

characteristic

203

C₂

#14544590

yes

characteristic

204

C₂

#14544604

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#322	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	C₄×Q₈
cosocle	C₄²×D₁₇	C₂³
hypercenter	#322	#595
hypocenter	C₁₇	#8294
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#322

C₄

socle

C₂²×C₁₇

C₄×C₄⋊C₄

prime radical

C₂²×D₁₇

C₄×Q₈

cosocle

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
32. #10587535	C₈	semidirect
145. C₁₇	#8294	semidirect

H ≅ G/N

type

32. #10587535

C₈

semidirect

145. C₁₇

#8294

semidirect

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

subgroup	index
C₄×#10587535	2
#14544334	2
C₈×#63243	2
#14543616	2
D₁₇×#322	2
#8294	17

subgroup

index

D₁₇×#322

Subgroups — #17952405

Conjugacy classes of subgroups

Complements

Maximal subgroups