Group #16939355 — subgroups

Subgroups — #16939355

Order 4352.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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	#14529764	C₂	1	1	yes	maximal, characteristic
2	#14528937	C₂	1	1	yes	maximal, characteristic
3	#14527556	C₂	1	1	yes	maximal, characteristic
4	#14527188	C₂	1	1	yes	maximal, characteristic
5	#14527186	C₂	1	1	yes	maximal, characteristic
6	#14526612	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#1570	C₂	1	1	yes	maximal, characteristic
8	#11763366	C₂²	1	1	yes	characteristic
9	#11762862	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
11	C₂×#10587485	—	2	2	no	—
12	#11762655	C₂²	1	1	yes	characteristic
13	#11762570	—	2	2	no	—
14	#11762559	—	2	2	no	—
15	#11762541	—	2	2	no	—
16	#11762539	—	2	2	no	—
17	C₁₇×#431	C₂²	1	1	yes	characteristic
18	C₁₇×#423	C₂²	1	1	yes	characteristic
19	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₁₇×#334	—	2	2	no	—
21	C₂×M₁₆×C₁₇	—	2	1	no	—
22	C₂²×C₈×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	#10587612	—	2	1	no	—
25	#10587610	—	2	1	no	—
26	#10587610	—	2	1	no	—
27	C₂×C₄×Dic₁₇	D₄	1	2	yes	—
28	C₂²×#3622	—	2	1	no	—
29	#10587556	—	2	1	no	—
30	#10587555	Q₈	1	1	yes	characteristic
31	#10587555	Q₈	1	1	yes	characteristic
32	C₄×Dic₃₄	—	2	1	no	—
33	C₄×Dic₃₄	—	2	1	no	—
34	C₁₇×C₄⋊C₈	—	2	1	no	—
35	C₁₇×C₄⋊C₈	—	2	1	no	—
36	C₁₇×C₄⋊C₈	—	2	1	no	—
37	C₁₇×C₄⋊C₈	—	2	1	no	—
38	C₁₇×C₄⋊C₈	—	2	1	no	—
39	C₁₇×C₂²⋊C₈	—	2	1	no	—
40	C₄×C₈×C₁₇	D₄	1	2	yes	—
41	#10587502	—	2	1	no	—
42	#10587502	—	2	1	no	—
43	C₂²⋊#3622	—	2	1	no	—
44	#10587485	—	4	4	no	—
45	C₄⋊#3622	—	2	1	no	—
46	C₄⋊#3622	—	2	1	no	—
47	C₄⋊#3622	—	2	1	no	—
48	C₄×#3622	D₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
52	C₂²×Dic₁₇	—	2	2	no	—
53	C₂×Dic₃₄	—	2	1	no	—
54	C₂×Dic₃₄	—	2	1	no	—
55	M₁₆×C₁₇	—	4	2	no	—
56	C₂×C₈×C₁₇	—	2	2	no	—
57	C₂×C₈×C₁₇	—	4	1	no	—
58	C₂×C₈×C₁₇	—	4	1	no	—
59	C₂×C₈×C₁₇	—	4	2	no	—
60	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₇	#41	1	1	yes	characteristic
64	C₂²⋊Dic₁₇	—	4	1	no	—
65	C₄⋊Dic₁₇	—	2	1	no	—
66	C₄⋊Dic₁₇	—	2	1	no	—
67	C₄⋊Dic₁₇	—	2	1	no	—
68	C₄⋊Dic₁₇	—	2	1	no	—
69	#63225	—	4	1	no	—
70	#63225	—	4	1	no	—
71	C₄×Dic₁₇	—	2	2	no	—
72	C₄×Dic₁₇	—	4	1	no	—
73	C₄×Dic₁₇	SD₁₆	1	4	yes	—
74	C₂×#3622	—	2	2	no	—
75	C₂×#3622	—	4	1	no	—
76	C₂×#3622	—	4	2	no	—
77	#10670	—	17	1	no	maximal, Sylow
78	C₂³×C₁₇	#123	1	1	yes	characteristic
79	C₂×C₄×C₁₇	—	4	1	no	—
80	C₂×C₄×C₁₇	—	4	1	no	—
81	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
85	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
86	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
88	C₂×C₄×C₁₇	#137	1	2	yes	—
89	C₂×Dic₁₇	—	2	4	no	—
90	C₂×Dic₁₇	—	4	2	no	—
91	C₂×Dic₁₇	—	4	2	no	—
92	C₂×Dic₁₇	—	4	1	no	—
93	C₂×Dic₁₇	—	4	1	no	—
94	Dic₃₄	—	4	2	no	—
95	Dic₃₄	—	4	2	no	—
96	C₈×C₁₇	—	4	2	no	—
97	C₈×C₁₇	—	8	1	no	—
98	C₈×C₁₇	—	8	1	no	—
99	#3622	—	4	2	no	—
100	#3622	—	8	1	no	—
101	#1995	—	17	1	no	—
102	#1843	—	17	1	no	—
103	#1570	—	17	1	no	—
104	#1570	D₁₇	1	1	yes	characteristic
105	#1481	—	17	1	no	—
106	#1481	—	17	1	no	—
107	#1274	—	17	1	no	—
108	C₂²×C₁₇	—	2	1	no	—
109	C₂²×C₁₇	—	2	1	no	—
110	C₂²×C₁₇	#399	1	1	yes	characteristic
111	C₂²×C₁₇	#483	1	2	yes	—
112	C₄×C₁₇	—	2	1	no	—
113	C₄×C₁₇	—	2	1	no	—
114	C₄×C₁₇	—	2	1	no	—
115	C₄×C₁₇	—	2	1	no	—
116	C₄×C₁₇	#475	1	2	yes	—
117	C₄×C₁₇	#492	1	1	yes	characteristic
118	C₄×C₁₇	#497	1	1	yes	characteristic
119	Dic₁₇	—	4	4	no	—
120	Dic₁₇	—	8	1	no	—
121	Dic₁₇	—	8	1	no	—
122	#533	—	17	1	no	—
123	#431	—	17	1	no	—
124	#431	C₂×D₁₇	1	1	yes	characteristic
125	#423	C₂×D₁₇	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	17	1	no	—
127	C₂×C₄⋊C₈	—	34	2	no	—
128	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
129	#376	—	17	1	no	—
130	#339	—	34	2	no	—
131	#334	—	2	2	no	—
132	#334	—	34	2	no	—
133	Q₈⋊C₈	—	34	2	no	—
134	Q₈⋊C₈	—	34	2	no	—
135	C₂×C₁₇	—	2	2	no	—
136	C₂×C₁₇	#2096	1	1	yes	characteristic
137	C₂×C₁₇	#2098	1	1	yes	characteristic
138	C₂×C₁₇	#2098	1	1	yes	characteristic
139	C₂×M₁₆	—	2	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	34	1	no	—
142	C₄⋊Q₈	—	17	1	no	—
143	C₄⋊Q₈	—	17	1	no	—
144	#125	—	34	1	no	—
145	C₂²⋊Q₈	—	34	1	no	—
146	C₂²⋊Q₈	—	34	1	no	—
147	C₄×Q₈	—	34	1	no	—
148	C₄×Q₈	—	34	1	no	—
149	#117	—	34	1	no	—
150	C₂×C₄²	—	17	2	no	—
151	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	—	2	1	no	—
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	34	1	no	—
158	C₄⋊C₈	—	34	1	no	—
159	C₄⋊C₈	—	34	1	no	—
160	C₄⋊C₈	—	68	4	no	—
161	C₂²⋊C₈	—	2	1	no	—
162	C₂²⋊C₈	—	34	1	no	—
163	C₄×C₈	—	17	2	no	—
164	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
165	#95	—	34	1	no	—
166	#95	—	34	1	no	—
167	C₁₇	#10670	1	1	yes	characteristic, complemented, Sylow
168	C₂×Q₈	—	34	1	no	—
169	C₂×Q₈	—	34	1	no	—
170	C₂²×C₄	—	34	2	no	—
171	C₂²×C₄	#63252	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
173	C₂²×C₄	#63256	1	1	yes	characteristic
174	M₁₆	—	4	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	34	2	no	—
180	C₂×C₈	—	68	2	no	—
181	C₂×C₈	—	68	1	no	—
182	C₄⋊C₄	—	34	1	no	—
183	C₄⋊C₄	—	34	1	no	—
184	C₄⋊C₄	—	34	1	no	—
185	C₄⋊C₄	—	34	1	no	—
186	C₄⋊C₄	—	68	1	no	—
187	C₄⋊C₄	—	68	1	no	—
188	C₂²⋊C₄	—	68	1	no	—
189	C₄²	—	17	4	no	—
190	C₄²	—	34	2	no	—
191	C₄²	—	68	1	no	—
192	C₄²	#63252	1	1	yes	characteristic
193	C₄²	D₄×D₁₇	1	1	yes	characteristic
194	C₄²	Q₈×D₁₇	1	1	yes	characteristic
195	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
196	C₂³	#10587580	1	1	yes	characteristic
197	Q₈	—	68	2	no	—
198	Q₈	—	68	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	34	4	no	—
202	C₂×C₄	—	68	2	no	—
203	C₂×C₄	—	68	2	no	—
204	C₂×C₄	—	68	1	no	—
205	C₂×C₄	—	68	1	no	—
206	C₂×C₄	#10587580	1	1	yes	characteristic
207	C₂×C₄	#10587582	1	1	yes	characteristic
208	C₂×C₄	#10587582	1	1	yes	characteristic
209	C₂×C₄	SD₁₆×D₁₇	1	2	yes	—
210	C₂×C₄	#10587601	1	2	yes	—
211	C₂×C₄	#10587615	1	1	yes	characteristic
212	C₂×C₄	#10587626	1	1	yes	characteristic
213	C₂×C₄	#10587632	1	1	yes	characteristic
214	C₈	—	4	2	no	—
215	C₈	—	8	1	no	—
216	C₈	—	8	1	no	—
217	C₈	—	68	2	no	—
218	C₈	—	136	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	#11762744	1	1	yes	characteristic
222	C₂²	#11763141	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	68	4	no	—
228	C₄	—	136	1	no	—
229	C₄	—	136	1	no	—
230	C₄	#11763093	1	2	yes	—
231	C₄	#11763180	1	1	yes	characteristic
232	C₄	#11763200	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	#14530324	1	1	yes	characteristic
235	C₂	#14530336	1	1	yes	characteristic
236	C₂	#14530340	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14529764

C₂

yes

maximal, characteristic

#14528937

C₂

yes

maximal, characteristic

#14527556

C₂

yes

maximal, characteristic

#14527188

C₂

yes

maximal, characteristic

#14527186

C₂

yes

maximal, characteristic

#14526612

C₂

yes

maximal, characteristic

C₁₇×#1570

C₂

yes

maximal, characteristic

#11763366

C₂²

yes

characteristic

#11762862

C₂²

yes

characteristic

C₂×C₄⋊#3622

C₂²

yes

characteristic

C₂×#10587485

—

#11762655

C₂²

yes

—

—

—

—

C₁₇×#431

C₂²

yes

characteristic

C₁₇×#423

C₂²

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₁₇×#334

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

—

—

—

C₂×C₄×Dic₁₇

D₄

yes

—

C₂²×#3622

—

#10587556

—

#10587555

Q₈

yes

characteristic

#10587555

Q₈

yes

characteristic

C₄×Dic₃₄

—

C₄×Dic₃₄

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₂²⋊C₈

—

C₄×C₈×C₁₇

D₄

yes

—

#10587502

—

#10587502

—

C₂²⋊#3622

—

—

—

—

—

D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

C₂²×C₄×C₁₇

yes

C₂²×C₄×C₁₇

yes

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

C₄²×C₁₇

#41

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

SD₁₆

yes

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

100

#3622

—

101

#1995

—

102

#1843

—

103

#1570

—

104

#1570

D₁₇

yes

characteristic

105

#1481

—

106

#1481

—

107

#1274

—

108

C₂²×C₁₇

—

109

C₂²×C₁₇

—

110

C₂²×C₁₇

#399

yes

characteristic

111

C₂²×C₁₇

#483

yes

—

112

C₄×C₁₇

—

113

C₄×C₁₇

—

114

C₄×C₁₇

—

115

C₄×C₁₇

—

116

C₄×C₁₇

#475

yes

—

117

C₄×C₁₇

#492

yes

characteristic

118

C₄×C₁₇

#497

yes

characteristic

119

Dic₁₇

—

120

Dic₁₇

—

121

Dic₁₇

—

122

#533

—

123

#431

—

124

#431

C₂×D₁₇

yes

characteristic

125

#423

C₂×D₁₇

yes

characteristic

126

C₂×C₄⋊C₈

—

127

C₂×C₄⋊C₈

—

128

C₂×C₄⋊C₈

C₂×D₁₇

yes

characteristic

129

#376

—

130

#339

—

131

#334

—

132

#334

—

133

Q₈⋊C₈

—

134

Q₈⋊C₈

—

135

C₂×C₁₇

—

136

C₂×C₁₇

#2096

yes

characteristic

137

C₂×C₁₇

#2098

yes

characteristic

138

C₂×C₁₇

#2098

yes

characteristic

139

C₂×M₁₆

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

#125

—

145

C₂²⋊Q₈

—

146

C₂²⋊Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

#117

—

150

C₂×C₄²

—

151

C₂×C₄²

C₂²×D₁₇

yes

characteristic

152

C₄⋊C₈

—

153

C₄⋊C₈

—

154

C₄⋊C₈

—

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

C₄⋊C₈

—

158

C₄⋊C₈

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

C₂²⋊C₈

—

162

C₂²⋊C₈

—

163

C₄×C₈

—

164

C₄×C₈

C₂²⋊D₁₇

yes

—

165

#95

—

166

#95

—

167

C₁₇

#10670

yes

characteristic, complemented, Sylow

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂²×C₄

—

171

C₂²×C₄

#63252

yes

characteristic

172

C₂²×C₄

D₄×D₁₇

yes

characteristic

173

C₂²×C₄

#63256

yes

characteristic

174

M₁₆

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₂²⋊C₄

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

192

C₄²

#63252

yes

characteristic

193

C₄²

D₄×D₁₇

yes

characteristic

194

C₄²

Q₈×D₁₇

yes

characteristic

195

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

196

C₂³

#10587580

yes

characteristic

197

Q₈

—

198

Q₈

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

#10587580

yes

characteristic

207

C₂×C₄

#10587582

yes

characteristic

208

C₂×C₄

#10587582

yes

characteristic

209

C₂×C₄

SD₁₆×D₁₇

yes

—

210

C₂×C₄

#10587601

yes

—

211

C₂×C₄

#10587615

yes

characteristic

212

C₂×C₄

#10587626

yes

characteristic

213

C₂×C₄

#10587632

yes

characteristic

214

C₈

—

215

C₈

—

216

C₈

—

217

C₈

—

218

C₈

—

136

—

219

C₂²

—

220

C₂²

—

221

C₂²

#11762744

yes

characteristic

222

C₂²

#11763141

yes

—

223

C₄

—

224

C₄

—

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

136

—

229

C₄

—

136

—

230

C₄

#11763093

yes

—

231

C₄

#11763180

yes

characteristic

232

C₄

#11763200

yes

characteristic

233

C₂

—

234

C₂

#14530324

yes

characteristic

235

C₂

#14530336

yes

characteristic

236

C₂

#14530340

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587626
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1570	C₂
socle	C₂²×C₁₇	#399
prime radical	C₂³×C₁₇	#123
hypercenter	#1570	D₁₇
hypocenter	C₁₇	#10670
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

subgroup

G/H

center

C₂×C₄

#10587626

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1570

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1570

N	H ≅ G/N	type
167. C₁₇	#10670	semidirect

H ≅ G/N

type

167. C₁₇

#10670

semidirect

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

subgroup	index
#14529764	2
#14528937	2
#14527556	2
#14527188	2
#14527186	2
#14526612	2
C₁₇×#1570	2
#10670	17

subgroup

index

C₁₇×#1570

Subgroups — #16939355

Conjugacy classes of subgroups

Complements

Maximal subgroups