Group #17961037 — subgroups

Subgroups — #17961037

Order 4352.

318 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₂×#11763699	C₂	1	2	yes	maximal
2	C₄×#10587530	C₂	1	1	yes	maximal, characteristic
3	#14544578	C₂	1	1	yes	maximal, characteristic
4	#14543824	C₂	1	2	yes	maximal
5	#14530538	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587530	C₂²	1	1	yes	characteristic
7	C₂×#10587530	C₂²	1	1	yes	characteristic
8	#11763810	C₄	1	1	yes	characteristic
9	C₂×#10587532	C₂²	1	2	yes	—
10	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
11	#11763699	—	2	4	no	—
12	#11763698	C₄	1	2	yes	complemented
13	C₄×#63216	C₄	1	2	yes	complemented
14	#11763674	C₄	1	2	yes	—
15	C₂×#10587469	C₂²	1	2	yes	—
16	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
17	C₄⋊#63214	C₄	1	1	yes	characteristic, complemented
18	#11762627	C₄	1	2	yes	complemented
19	C₁₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄⋊#595	—	2	1	no	—
21	C₂×C₄×#595	C₂×C₄	1	2	yes	—
22	C₂×#63242	C₂×C₄	1	2	yes	—
23	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂³	1	1	yes	characteristic
25	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
26	(C₂×#595)⋊C₄	—	2	1	no	—
27	#10587544	—	2	1	no	—
28	C₄⋊#3624	—	2	1	no	—
29	C₂×#63216	C₂×C₄	1	2	yes	—
30	C₂×#63216	C₂×C₄	1	2	yes	—
31	#10587532	—	2	2	no	—
32	#10587532	—	2	2	no	—
33	#10587530	—	2	1	no	—
34	#10587530	—	2	1	no	—
35	#10587530	D₄	1	2	yes	—
36	#10587530	Q₈	1	2	yes	—
37	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
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	2	yes	—
42	C₄×#3622	C₂×C₄	1	1	yes	characteristic
43	#10587469	—	2	2	no	—
44	#10587469	—	2	2	no	—
45	C₂²×#595	—	2	2	no	—
46	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
47	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
48	C₂×C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×#3624	—	2	2	no	—
50	C₄⋊#595	—	4	2	no	—
51	C₄×#595	—	2	2	no	—
52	C₄×#595	—	2	2	no	—
53	#63242	—	2	2	no	—
54	#63242	—	2	2	no	—
55	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₇	C₄²	1	1	yes	characteristic
58	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
59	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
60	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
61	C₂×#3622	C₄²	1	1	yes	characteristic
62	C₂×#3622	C₂²×C₄	1	1	yes	characteristic
63	C₈×D₁₇	—	2	1	no	—
64	C₈×D₁₇	—	2	1	no	—
65	C₈×D₁₇	C₂×D₄	1	1	yes	characteristic
66	C₈×D₁₇	C₂×Q₈	1	1	yes	characteristic
67	C₈×D₁₇	#41	1	1	yes	characteristic
68	C₈×D₁₇	#41	1	1	yes	characteristic
69	#63216	—	2	2	no	—
70	#63216	C₄⋊C₄	1	4	yes	complemented
71	C₁₆×C₁₇	—	2	2	no	—
72	#63214	—	2	2	no	—
73	#12666	—	17	1	no	maximal, Sylow
74	C₂²×D₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×#595	—	4	2	no	—
76	C₂×#595	—	4	2	no	—
77	C₂×#595	—	4	2	no	—
78	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
81	C₂×Dic₁₇	#97	1	1	yes	characteristic
82	C₂×Dic₁₇	#97	1	1	yes	characteristic
83	C₂×Dic₁₇	C₂×M₁₆	1	1	yes	characteristic
84	C₄×D₁₇	—	2	2	no	—
85	C₄×D₁₇	—	4	1	no	—
86	C₄×D₁₇	C₄×D₄	1	1	yes	characteristic
87	C₄×D₁₇	C₄×Q₈	1	1	yes	characteristic
88	C₄×D₁₇	#131	1	1	yes	characteristic
89	C₄×D₁₇	#131	1	1	yes	characteristic
90	#3624	—	4	2	no	—
91	C₈×C₁₇	—	2	1	no	—
92	C₈×C₁₇	C₄×D₄	1	1	yes	characteristic
93	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
94	#3622	—	2	1	no	—
95	#3622	C₂×C₄⋊C₄	1	1	yes	characteristic
96	#3622	#117	1	1	yes	characteristic
97	#2150	—	17	1	no	—
98	C₂×#347	—	17	2	no	—
99	C₄×M₃₂	—	17	1	no	—
100	#1770	—	17	1	no	—
101	#1375	—	17	2	no	—
102	C₂²×C₁₇	C₂×#97	1	1	yes	characteristic
103	C₂×D₁₇	—	2	1	no	—
104	C₂×D₁₇	—	2	1	no	—
105	C₂×D₁₇	#436	1	1	yes	characteristic
106	C₂×D₁₇	#446	1	1	yes	characteristic
107	#595	—	8	2	no	—
108	C₄×C₁₇	—	2	1	no	—
109	C₄×C₁₇	#347	1	2	yes	complemented
110	C₄×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₄×C₁₇	#405	1	1	yes	characteristic
112	Dic₁₇	—	2	1	no	—
113	Dic₁₇	—	2	1	no	—
114	Dic₁₇	#436	1	1	yes	characteristic
115	Dic₁₇	#446	1	1	yes	characteristic
116	C₂×M₃₂	—	17	1	no	—
117	C₂×M₃₂	—	17	1	no	—
118	C₂×M₃₂	—	17	2	no	—
119	C₂×C₄⋊C₈	—	17	1	no	—
120	C₂×#97	—	17	2	no	—
121	C₂×C₄×C₈	—	17	1	no	—
122	C₄×C₄⋊C₄	—	17	1	no	—
123	C₄⋊C₁₆	—	17	2	no	—
124	C₄⋊C₁₆	—	17	1	no	—
125	C₄⋊C₁₆	#595	1	1	yes	characteristic, complemented
126	#347	—	34	4	no	—
127	#346	—	17	2	no	—
128	C₄×C₁₆	—	17	2	no	—
129	#336	—	17	2	no	—
130	C₂×C₁₇	#1770	1	1	yes	characteristic
131	C₂×C₁₇	C₂×#347	1	1	yes	characteristic
132	C₂×C₁₇	#2110	1	1	yes	characteristic
133	D₁₇	—	2	1	no	—
134	D₁₇	—	2	1	no	—
135	C₂²×C₈	—	17	1	no	—
136	C₂²×C₈	—	17	1	no	—
137	C₂²×C₈	—	17	2	no	—
138	C₂×C₄⋊C₄	—	34	1	no	—
139	C₂×C₄²	—	17	1	no	—
140	C₂×C₄²	—	17	2	no	—
141	M₃₂	—	17	2	no	—
142	M₃₂	—	17	2	no	—
143	M₃₂	—	34	2	no	—
144	M₃₂	—	34	2	no	—
145	M₃₂	—	34	1	no	—
146	M₃₂	—	34	1	no	—
147	C₂×C₁₆	—	17	2	no	—
148	C₂×C₁₆	—	17	2	no	—
149	C₂×C₁₆	—	17	2	no	—
150	C₂×C₁₆	C₂×#595	1	2	yes	—
151	C₄⋊C₈	—	34	1	no	—
152	C₄⋊C₈	—	34	1	no	—
153	#97	—	34	2	no	—
154	#97	—	34	2	no	—
155	C₄×C₈	—	17	1	no	—
156	C₄×C₈	—	17	1	no	—
157	C₄×C₈	—	17	1	no	—
158	C₄×C₈	C₂×#595	1	1	yes	characteristic
159	#95	—	34	1	no	—
160	C₁₇	#12666	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₄	—	17	1	no	—
162	C₂²×C₄	—	17	1	no	—
163	C₂²×C₄	—	17	1	no	—
164	C₂²×C₄	—	34	2	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	—	34	1	no	—
172	C₂×C₈	—	34	1	no	—
173	C₂×C₈	—	34	2	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	—	34	2	no	—
176	C₂×C₈	C₄×#595	1	1	yes	characteristic
177	C₂×C₈	C₂²×#595	1	1	yes	characteristic
178	C₄⋊C₄	—	68	2	no	—
179	C₄²	—	17	1	no	—
180	C₄²	—	17	1	no	—
181	C₄²	—	17	1	no	—
182	C₄²	—	34	2	no	—
183	C₄²	—	34	2	no	—
184	C₄²	C₄×#595	1	1	yes	characteristic
185	C₁₆	—	2	2	no	—
186	C₁₆	—	17	4	no	—
187	C₁₆	—	34	2	no	—
188	C₁₆	—	34	2	no	—
189	C₂³	—	17	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	—	17	1	no	—
196	C₂×C₄	—	17	1	no	—
197	C₂×C₄	—	34	2	no	—
198	C₂×C₄	—	68	1	no	—
199	C₂×C₄	—	68	2	no	—
200	C₂×C₄	—	68	2	no	—
201	C₂×C₄	—	68	2	no	—
202	C₂×C₄	#10587532	1	1	yes	characteristic
203	C₂×C₄	#10587532	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	17	1	no	—
207	C₈	—	17	1	no	—
208	C₈	—	34	1	no	—
209	C₈	—	68	2	no	—
210	C₈	D₄×#595	1	1	yes	characteristic
211	C₈	Q₈×#595	1	1	yes	characteristic
212	C₂²	—	17	1	no	—
213	C₂²	—	17	1	no	—
214	C₂²	—	34	1	no	—
215	C₂²	—	34	1	no	—
216	C₂²	C₂×#10587532	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	17	1	no	—
219	C₄	—	17	1	no	—
220	C₄	—	34	1	no	—
221	C₄	—	34	1	no	—
222	C₄	—	136	2	no	—
223	C₄	#11763699	1	2	yes	complemented
224	C₄	C₄⋊C₄×#595	1	1	yes	characteristic
225	C₄	#11763775	1	1	yes	characteristic
226	C₂	—	34	1	no	—
227	C₂	—	34	1	no	—
228	C₂	#14544581	1	1	yes	characteristic
229	C₂	C₂×#11763699	1	1	yes	characteristic
230	C₂	#14545004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763699

C₂

yes

maximal

C₄×#10587530

C₂

yes

maximal, characteristic

#14544578

C₂

yes

maximal, characteristic

#14543824

C₂

yes

maximal

#14530538

C₂

yes

maximal, characteristic

C₂×#10587530

C₂²

yes

characteristic

C₂×#10587530

C₂²

yes

characteristic

#11763810

C₄

yes

characteristic

C₂×#10587532

C₂²

yes

—

C₄×C₄⋊#595

C₄

yes

characteristic

#11763699

—

#11763698

C₄

yes

complemented

C₄×#63216

C₄

yes

complemented

#11763674

C₄

yes

—

C₂×#10587469

C₂²

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₄⋊#63214

C₄

yes

characteristic, complemented

#11762627

C₄

yes

complemented

C₁₇×C₄⋊C₁₆

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄×#595

C₂×C₄

yes

—

C₂×#63242

C₂×C₄

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

C₂³

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

(C₂×#595)⋊C₄

—

#10587544

—

C₄⋊#3624

—

C₂×#63216

C₂×C₄

yes

—

C₂×#63216

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

#10587530

Q₈

yes

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

characteristic

#10587469

—

#10587469

—

C₂²×#595

—

C₂×C₄×D₁₇

M₁₆

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

characteristic

C₂×C₄×D₁₇

C₂²×C₄

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

C₄²

yes

characteristic

C₂×#3622

C₂²×C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

C₂×D₄

yes

characteristic

C₈×D₁₇

C₂×Q₈

yes

C₈×D₁₇

yes

C₈×D₁₇

yes

—

C₄⋊C₄

yes

complemented

C₁₆×C₁₇

—

#63214

—

#12666

—

maximal, Sylow

C₂²×D₁₇

C₂×M₁₆

yes

—

—

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×C₄²

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

C₂×M₁₆

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₄×D₄

yes

characteristic

C₄×D₁₇

C₄×Q₈

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄×D₄

yes

characteristic

C₈×C₁₇

C₄×Q₈

yes

characteristic

#3622

—

#3622

C₂×C₄⋊C₄

yes

yes

—

—

C₄×M₃₂

—

100

#1770

—

101

#1375

—

102

C₂²×C₁₇

C₂×#97

yes

characteristic

103

C₂×D₁₇

—

104

C₂×D₁₇

—

105

C₂×D₁₇

#436

yes

characteristic

106

C₂×D₁₇

#446

yes

characteristic

107

#595

—

108

C₄×C₁₇

—

109

C₄×C₁₇

#347

yes

complemented

110

C₄×C₁₇

C₄×C₄⋊C₄

yes

characteristic

111

C₄×C₁₇

#405

yes

characteristic

112

Dic₁₇

—

113

Dic₁₇

—

114

Dic₁₇

#436

yes

characteristic

115

Dic₁₇

#446

yes

characteristic

116

C₂×M₃₂

—

117

C₂×M₃₂

—

118

C₂×M₃₂

—

119

C₂×C₄⋊C₈

—

120

C₂×#97

—

121

C₂×C₄×C₈

—

122

C₄×C₄⋊C₄

—

123

C₄⋊C₁₆

—

124

C₄⋊C₁₆

—

125

C₄⋊C₁₆

#595

yes

characteristic, complemented

126

#347

—

127

#346

—

128

C₄×C₁₆

—

129

#336

—

130

C₂×C₁₇

#1770

yes

characteristic

131

C₂×C₁₇

C₂×#347

yes

characteristic

132

C₂×C₁₇

#2110

yes

characteristic

133

D₁₇

—

134

D₁₇

—

135

C₂²×C₈

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₂×C₄⋊C₄

—

139

C₂×C₄²

—

140

C₂×C₄²

—

141

M₃₂

—

142

M₃₂

—

143

M₃₂

—

144

M₃₂

—

145

M₃₂

—

146

M₃₂

—

147

C₂×C₁₆

—

148

C₂×C₁₆

—

149

C₂×C₁₆

—

150

C₂×C₁₆

C₂×#595

yes

—

151

C₄⋊C₈

—

152

C₄⋊C₈

—

153

#97

—

154

#97

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₄×C₈

C₂×#595

yes

characteristic

159

#95

—

160

C₁₇

#12666

yes

characteristic, complemented, Sylow

161

C₂²×C₄

—

162

C₂²×C₄

—

163

C₂²×C₄

—

164

C₂²×C₄

—

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

—

176

C₂×C₈

C₄×#595

yes

characteristic

177

C₂×C₈

C₂²×#595

yes

characteristic

178

C₄⋊C₄

—

179

C₄²

—

180

C₄²

—

181

C₄²

—

182

C₄²

—

183

C₄²

—

184

C₄²

C₄×#595

yes

characteristic

185

C₁₆

—

186

C₁₆

—

187

C₁₆

—

188

C₁₆

—

189

C₂³

—

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

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

#10587532

yes

characteristic

203

C₂×C₄

#10587532

yes

characteristic

204

C₂×C₄

C₂×C₄×#595

yes

characteristic

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

209

C₈

—

210

C₈

D₄×#595

yes

characteristic

211

C₈

Q₈×#595

yes

characteristic

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

C₂×#10587532

yes

characteristic

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

136

—

223

C₄

#11763699

yes

complemented

224

C₄

C₄⋊C₄×#595

yes

characteristic

225

C₄

#11763775

yes

characteristic

226

C₂

—

227

C₂

—

228

C₂

#14544581

yes

characteristic

229

C₂

C₂×#11763699

yes

characteristic

230

C₂

#14545004

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#595
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₈	C₂²×#595
Fitting	C₁₇×C₄⋊C₁₆	C₄
socle	C₂²×C₁₇	C₂×#97
prime radical	C₂²×D₁₇	C₂×M₁₆
cosocle	C₂×C₈×D₁₇	C₂³
hypercenter	C₄⋊C₁₆	#595
hypocenter	C₁₇	#12666
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₄×#595

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₈

C₂²×#595

Fitting

C₁₇×C₄⋊C₁₆

C₄

socle

C₂²×C₁₇

C₂×#97

prime radical

C₂²×D₁₇

C₂×M₁₆

cosocle

C₂×C₈×D₁₇

C₂³

C₄⋊C₁₆

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
12. #11763698	C₄	semidirect
13. C₄×#63216	C₄	semidirect
17. C₄⋊#63214	C₄	semidirect
18. #11762627	C₄	semidirect
19. C₁₇×C₄⋊C₁₆	C₄	semidirect
70. #63216	C₄⋊C₄	semidirect
109. C₄×C₁₇	#347	semidirect
125. C₄⋊C₁₆	#595	semidirect
160. C₁₇	#12666	semidirect
223. C₄	#11763699	semidirect

H ≅ G/N

type

12. #11763698

C₄

semidirect

13. C₄×#63216

C₄

semidirect

17. C₄⋊#63214

C₄

semidirect

18. #11762627

C₄

semidirect

19. C₁₇×C₄⋊C₁₆

C₄

semidirect

70. #63216

C₄⋊C₄

semidirect

109. C₄×C₁₇

#347

semidirect

125. C₄⋊C₁₆

#595

semidirect

160. C₁₇

#12666

semidirect

223. C₄

#11763699

semidirect

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

subgroup	index
C₂×#11763699	2
C₄×#10587530	2
#14544578	2
#14543824	2
#14530538	2
#12666	17

subgroup

index

Subgroups — #17961037

Conjugacy classes of subgroups

Complements

Maximal subgroups