Group #17952301 — subgroups

Subgroups — #17952301

Order 4352.

390 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 148 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×#10587532	C₂	1	1	yes	maximal, characteristic
2	#14544332	C₂	1	1	yes	maximal, characteristic
3	#14543990	C₂	1	2	yes	maximal
4	#14543611	C₂	1	2	yes	maximal
5	#14527531	C₂	1	1	yes	maximal, characteristic
6	#11763828	C₂²	1	2	yes	—
7	C₂×#10587532	C₄	1	2	yes	—
8	C₂×#10587532	C₄	1	2	yes	—
9	C₄×#63242	C₂²	1	1	yes	characteristic
10	C₄²×#595	C₂²	1	1	yes	characteristic
11	#11763673	C₄	1	2	yes	complemented
12	#11763670	C₄	1	2	yes	complemented
13	#11763651	—	2	2	no	—
14	C₈×#3624	—	2	2	no	—
15	#11762861	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	#11762562	C₄	1	1	yes	characteristic, complemented
18	#11762559	C₄	1	2	yes	complemented
19	C₁₇×#335	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×#595	—	2	1	no	—
21	C₂×C₄×#595	—	2	1	no	—
22	C₂×C₄×#595	C₂×C₄	1	2	yes	—
23	C₂×#63242	C₂×C₄	1	2	yes	—
24	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×D₁₇	C₂³	1	1	yes	characteristic
27	#10587543	—	2	2	no	—
28	C₄⋊#3624	C₂×C₄	1	4	yes	—
29	C₄×#3624	—	2	2	no	—
30	C₄×#3624	D₄	1	2	yes	—
31	C₄×#3624	Q₈	1	2	yes	—
32	#10587532	—	2	4	no	—
33	#10587532	C₈	1	8	yes	complemented
34	C₁₇×C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
36	#10587491	—	2	2	no	—
37	C₈×Dic₁₇	—	2	2	no	—
38	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
40	C₄⋊#3622	C₂×C₄	1	2	yes	—
41	C₄×#3622	C₂×C₄	1	1	yes	characteristic
42	C₂²×#595	—	2	2	no	—
43	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
45	C₂×C₄×D₁₇	#41	1	1	yes	characteristic
46	C₂×#3624	—	2	4	no	—
47	C₂×#3624	C₄⋊C₄	1	4	yes	—
48	C₄×#595	—	2	2	no	—
49	C₄×#595	—	4	1	no	—
50	C₄×#595	—	4	1	no	—
51	C₄×#595	—	4	2	no	—
52	C₄×#595	C₂×C₈	1	4	yes	—
53	#63242	C₂×C₈	1	4	yes	—
54	#63242	M₁₆	1	4	yes	—
55	C₂×C₈×C₁₇	—	2	2	no	—
56	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
58	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₄×Dic₁₇	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	C₂×#3622	—	2	2	no	—
63	C₂×#3622	C₄²	1	1	yes	characteristic
64	C₂×#3622	C₄²	1	1	yes	characteristic
65	C₈×D₁₇	—	2	2	no	—
66	C₈×D₁₇	C₂×C₈	1	4	yes	—
67	#8268	—	17	1	no	maximal, Sylow
68	C₂²×D₁₇	#117	1	1	yes	characteristic
69	C₂×#595	—	2	4	no	—
70	C₂×#595	—	4	2	no	—
71	C₂×#595	—	4	2	no	—
72	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
75	C₂×Dic₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₁₇	C₄×D₄	1	1	yes	characteristic
77	C₂×Dic₁₇	C₄×Q₈	1	1	yes	characteristic
78	C₄×D₁₇	—	2	1	no	—
79	C₄×D₁₇	—	2	1	no	—
80	C₄×D₁₇	—	2	2	no	—
81	C₄×D₁₇	C₂²×C₈	1	2	yes	—
82	C₄×D₁₇	C₂×M₁₆	1	2	yes	—
83	#3624	—	4	4	no	—
84	#3624	C₄⋊C₈	1	8	yes	—
85	C₈×C₁₇	—	2	1	no	—
86	C₈×C₁₇	—	4	2	no	—
87	C₈×C₁₇	C₄×C₈	1	2	yes	—
88	#3622	—	2	1	no	—
89	#3622	—	4	2	no	—
90	#3622	C₄×C₈	1	2	yes	—
91	C₄×#97	—	17	1	no	—
92	#1564	—	17	1	no	—
93	#1564	—	17	1	no	—
94	#1450	—	17	2	no	—
95	#1276	—	17	2	no	—
96	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂×D₁₇	—	2	1	no	—
98	C₂×D₁₇	—	2	1	no	—
99	C₂×D₁₇	#431	1	2	yes	—
100	#595	—	4	4	no	—
101	C₄×C₁₇	—	2	1	no	—
102	C₄×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
103	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
104	C₄×C₁₇	#444	1	1	yes	characteristic
105	C₄×C₁₇	#444	1	1	yes	characteristic
106	Dic₁₇	—	2	1	no	—
107	Dic₁₇	C₂×C₄⋊C₈	1	2	yes	—
108	Dic₁₇	#444	1	2	yes	—
109	#431	—	17	2	no	—
110	#431	—	17	2	no	—
111	C₂×#97	—	17	2	no	—
112	C₂×#97	—	17	2	no	—
113	C₂×C₄×C₈	—	17	1	no	—
114	C₂×C₄×C₈	—	17	1	no	—
115	C₄³	—	17	1	no	—
116	#335	—	17	2	no	—
117	#335	—	17	1	no	—
118	#335	#595	1	1	yes	characteristic, complemented
119	#334	—	17	2	no	—
120	#334	—	17	2	no	—
121	#322	—	34	2	no	—
122	C₈²	—	34	2	no	—
123	C₂×C₁₇	C₄×C₄⋊C₈	1	1	yes	characteristic
124	C₂×C₁₇	#1773	1	1	yes	characteristic
125	C₂×C₁₇	#1773	1	1	yes	characteristic
126	D₁₇	—	2	2	no	—
127	C₂²×C₈	—	17	2	no	—
128	C₂²×C₈	—	17	1	no	—
129	C₂²×C₈	—	17	1	no	—
130	C₂×C₄²	—	17	1	no	—
131	C₂×C₄²	—	17	2	no	—
132	C₂×C₄²	—	34	1	no	—
133	C₂×C₄²	—	34	1	no	—
134	C₄⋊C₈	—	17	4	no	—
135	C₄⋊C₈	—	17	2	no	—
136	C₄⋊C₈	C₂×#595	1	2	yes	—
137	C₂²⋊C₈	—	34	2	no	—
138	C₂²⋊C₈	—	34	2	no	—
139	#97	—	17	8	no	—
140	#97	—	34	4	no	—
141	C₄×C₈	—	17	2	no	—
142	C₄×C₈	—	17	2	no	—
143	C₄×C₈	—	17	1	no	—
144	C₄×C₈	—	17	1	no	—
145	C₄×C₈	—	17	1	no	—
146	C₄×C₈	—	34	2	no	—
147	C₄×C₈	—	34	2	no	—
148	C₄×C₈	C₂×#595	1	1	yes	characteristic
149	C₁₇	#8268	1	1	yes	characteristic, complemented, Sylow
150	C₂²×C₄	—	17	1	no	—
151	C₂²×C₄	—	17	1	no	—
152	C₂²×C₄	—	17	1	no	—
153	C₂²×C₄	—	34	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	17	4	no	—
156	C₂×C₈	—	17	4	no	—
157	C₂×C₈	—	17	4	no	—
158	C₂×C₈	—	17	1	no	—
159	C₂×C₈	—	17	4	no	—
160	C₂×C₈	—	17	1	no	—
161	C₂×C₈	—	34	2	no	—
162	C₂×C₈	—	34	4	no	—
163	C₂×C₈	—	34	2	no	—
164	C₂×C₈	C₄×#595	1	1	yes	characteristic
165	C₂×C₈	C₄×#595	1	1	yes	characteristic
166	C₄²	—	17	1	no	—
167	C₄²	—	17	1	no	—
168	C₄²	—	17	1	no	—
169	C₄²	—	17	4	no	—
170	C₄²	—	34	2	no	—
171	C₄²	—	68	2	no	—
172	C₄²	—	68	1	no	—
173	C₄²	—	68	1	no	—
174	C₄²	C₂²×#595	1	1	yes	characteristic
175	C₂³	—	17	1	no	—
176	C₂×C₄	—	17	1	no	—
177	C₂×C₄	—	17	1	no	—
178	C₂×C₄	—	17	2	no	—
179	C₂×C₄	—	17	2	no	—
180	C₂×C₄	—	17	1	no	—
181	C₂×C₄	—	34	2	no	—
182	C₂×C₄	—	34	1	no	—
183	C₂×C₄	—	34	1	no	—
184	C₂×C₄	—	34	4	no	—
185	C₂×C₄	—	68	2	no	—
186	C₂×C₄	—	68	2	no	—
187	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
188	C₂×C₄	D₄×#595	1	1	yes	characteristic
189	C₂×C₄	Q₈×#595	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	4	2	no	—
192	C₈	—	17	8	no	—
193	C₈	—	17	2	no	—
194	C₈	—	34	1	no	—
195	C₈	—	68	4	no	—
196	C₈	—	68	2	no	—
197	C₈	C₈×#595	1	2	yes	—
198	C₂²	—	17	2	no	—
199	C₂²	—	34	1	no	—
200	C₂²	—	34	1	no	—
201	C₂²	C₄⋊C₄×#595	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	17	2	no	—
204	C₄	—	17	2	no	—
205	C₄	—	34	1	no	—
206	C₄	—	68	4	no	—
207	C₄	C₂×C₈×#595	1	1	yes	characteristic
208	C₄	M₁₆×#595	1	1	yes	characteristic
209	C₄	#11763855	1	1	yes	characteristic
210	C₄	#11763856	1	1	yes	characteristic
211	C₂	—	34	2	no	—
212	C₂	C₄⋊C₈×#595	1	1	yes	characteristic
213	C₂	#14544591	1	1	yes	characteristic
214	C₂	#14544593	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587532

C₂

yes

maximal, characteristic

#14544332

C₂

yes

maximal, characteristic

#14543990

C₂

yes

maximal

#14543611

C₂

yes

maximal

#14527531

C₂

yes

maximal, characteristic

#11763828

C₂²

yes

—

C₂×#10587532

C₄

yes

—

C₂×#10587532

C₄

yes

—

C₄×#63242

C₂²

yes

characteristic

C₄²×#595

C₂²

yes

characteristic

#11763673

C₄

yes

complemented

#11763670

C₄

yes

—

—

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic, complemented

#11762559

C₄

yes

complemented

C₁₇×#335

C₄

yes

characteristic, complemented

C₂×C₄×#595

—

C₂×C₄×#595

—

C₂×C₄×#595

C₂×C₄

yes

—

C₂×#63242

C₂×C₄

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₄²×D₁₇

C₂³

yes

characteristic

#10587543

—

C₄⋊#3624

C₂×C₄

yes

—

C₄×#3624

—

C₄×#3624

D₄

yes

—

C₄×#3624

Q₈

yes

—

#10587532

—

#10587532

C₈

yes

complemented

C₁₇×C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587491

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₄⋊#3622

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

C₂²×#595

—

C₂×C₄×D₁₇

C₂²×C₄

yes

C₂×C₄×D₁₇

yes

C₂×C₄×D₁₇

yes

—

C₄⋊C₄

yes

—

—

—

—

—

C₂×C₈

yes

—

#63242

C₂×C₈

yes

—

#63242

M₁₆

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₂×D₄

yes

characteristic

C₄×Dic₁₇

C₂×Q₈

yes

characteristic

C₂×#3622

—

C₂×#3622

C₄²

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

—

C₂²×D₁₇

yes

—

—

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×Q₈

yes

characteristic

C₂×Dic₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₁₇

C₄×D₄

yes

characteristic

C₂×Dic₁₇

C₄×Q₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂²×C₈

yes

—

C₄×D₁₇

C₂×M₁₆

yes

—

#3624

—

#3624

C₄⋊C₈

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

—

#3622

—

#3622

—

#3622

C₄×C₈

yes

—

C₄×#97

—

—

—

—

—

C₂²×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

#431

yes

—

100

#595

—

101

C₄×C₁₇

—

102

C₄×C₁₇

C₂×C₄×C₈

yes

characteristic

103

C₄×C₁₇

C₄×M₁₆

yes

characteristic

104

C₄×C₁₇

#444

yes

characteristic

105

C₄×C₁₇

#444

yes

characteristic

106

Dic₁₇

—

107

Dic₁₇

C₂×C₄⋊C₈

yes

—

108

Dic₁₇

#444

yes

—

109

#431

—

110

#431

—

111

C₂×#97

—

112

C₂×#97

—

113

C₂×C₄×C₈

—

114

C₂×C₄×C₈

—

115

C₄³

—

116

#335

—

117

#335

—

118

#335

#595

yes

characteristic, complemented

119

#334

—

120

#334

—

121

#322

—

122

C₈²

—

123

C₂×C₁₇

C₄×C₄⋊C₈

yes

characteristic

124

C₂×C₁₇

#1773

yes

characteristic

125

C₂×C₁₇

#1773

yes

characteristic

126

D₁₇

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂×C₄²

—

131

C₂×C₄²

—

132

C₂×C₄²

—

133

C₂×C₄²

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

C₄⋊C₈

C₂×#595

yes

—

137

C₂²⋊C₈

—

138

C₂²⋊C₈

—

139

#97

—

140

#97

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₄×C₈

—

148

C₄×C₈

C₂×#595

yes

characteristic

149

C₁₇

#8268

yes

characteristic, complemented, Sylow

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

—

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

C₄×#595

yes

characteristic

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

—

170

C₄²

—

171

C₄²

—

172

C₄²

—

173

C₄²

—

174

C₄²

C₂²×#595

yes

characteristic

175

C₂³

—

176

C₂×C₄

—

177

C₂×C₄

—

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

C₂×C₄×#595

yes

characteristic

188

C₂×C₄

D₄×#595

yes

characteristic

189

C₂×C₄

Q₈×#595

yes

characteristic

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

—

195

C₈

—

196

C₈

—

197

C₈

C₈×#595

yes

—

198

C₂²

—

199

C₂²

—

200

C₂²

—

201

C₂²

C₄⋊C₄×#595

yes

characteristic

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

C₂×C₈×#595

yes

characteristic

208

C₄

M₁₆×#595

yes

characteristic

209

C₄

#11763855

yes

characteristic

210

C₄

#11763856

yes

characteristic

211

C₂

—

212

C₂

C₄⋊C₈×#595

yes

characteristic

213

C₂

#14544591

yes

characteristic

214

C₂

#14544593

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#595
commutator	C₄×C₁₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂²×D₁₇	#117
cosocle	C₄²×D₁₇	C₂³
hypercenter	#335	#595
hypocenter	C₁₇	#8268
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×#595

commutator

C₄×C₁₇

C₂×C₄×C₈

Frattini

C₄²

C₂²×#595

Fitting

C₁₇×#335

C₄

socle

C₂²×C₁₇

C₄×C₄⋊C₄

prime radical

C₂²×D₁₇

#117

cosocle

C₄²×D₁₇

C₂³

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
11. #11763673	C₄	semidirect
12. #11763670	C₄	semidirect
17. #11762562	C₄	semidirect
18. #11762559	C₄	semidirect
19. C₁₇×#335	C₄	semidirect
33. #10587532	C₈	semidirect
118. #335	#595	semidirect
149. C₁₇	#8268	semidirect

H ≅ G/N

type

11. #11763673

C₄

semidirect

12. #11763670

C₄

semidirect

17. #11762562

C₄

semidirect

18. #11762559

C₄

semidirect

19. C₁₇×#335

C₄

semidirect

33. #10587532

C₈

semidirect

118. #335

#595

semidirect

149. C₁₇

#8268

semidirect

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

subgroup	index
C₄×#10587532	2
#14544332	2
#14543990	2
#14543611	2
#14527531	2
#8268	17

subgroup

index

Subgroups — #17952301

Conjugacy classes of subgroups

Complements

Maximal subgroups