Group #16939073 — subgroups

Subgroups — #16939073

Order 4352.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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	#14528912	C₂	1	1	yes	maximal, characteristic
2	#14528789	C₂	1	2	yes	maximal
3	#14526668	C₂	1	2	yes	maximal
4	#14526619	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×#10587506	—	2	2	no	—
8	#11762679	C₂²	1	2	yes	—
9	#11762572	—	2	2	no	—
10	#11762571	—	2	2	no	—
11	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×#336	—	2	2	no	—
15	C₂²×Q₈×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	—	2	1	no	—
17	C₂²×C₈×C₁₇	D₄	1	2	yes	—
18	C₄×Q₈×C₁₇	—	2	2	no	—
19	C₄×Q₈×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	1	no	—
21	C₂×C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₇	—	2	1	no	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	C₂×C₄⋊Dic₁₇	—	2	2	no	—
27	C₂²×#3622	—	2	2	no	—
28	C₁₇×C₄⋊C₈	—	2	2	no	—
29	C₄×C₈×C₁₇	—	2	2	no	—
30	#10587506	—	4	4	no	—
31	C₄⋊#3622	—	2	2	no	—
32	C₄⋊#3622	—	2	2	no	—
33	C₂×Q₈×C₁₇	—	2	1	no	—
34	C₂×Q₈×C₁₇	—	2	2	no	—
35	C₂×Q₈×C₁₇	—	2	2	no	—
36	C₂×Q₈×C₁₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₁₇	—	2	1	no	—
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
42	C₂²×Dic₁₇	—	4	2	no	—
43	C₂×C₈×C₁₇	—	2	2	no	—
44	C₂×C₈×C₁₇	—	2	2	no	—
45	C₂×C₈×C₁₇	—	2	2	no	—
46	C₂×C₈×C₁₇	—	4	1	no	—
47	C₂×C₈×C₁₇	—	4	2	no	—
48	C₄⋊C₄×C₁₇	—	2	2	no	—
49	C₄⋊C₄×C₁₇	—	4	2	no	—
50	C₄²×C₁₇	—	4	1	no	—
51	C₄²×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	#41	1	2	yes	—
53	C₄²×C₁₇	#41	1	2	yes	—
54	C₄⋊Dic₁₇	—	4	2	no	—
55	C₄⋊Dic₁₇	—	4	2	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄⋊Dic₁₇	—	4	2	no	—
58	C₄⋊Dic₁₇	—	8	2	no	—
59	C₂×#3622	—	4	2	no	—
60	C₂×#3622	—	4	2	no	—
61	C₂×#3622	—	4	2	no	—
62	#10624	—	17	1	no	maximal, Sylow
63	C₂³×C₁₇	#121	1	1	yes	characteristic
64	Q₈×C₁₇	—	2	4	no	—
65	Q₈×C₁₇	—	4	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	4	1	no	—
69	C₂×C₄×C₁₇	—	4	1	no	—
70	C₂×C₄×C₁₇	—	4	1	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	#123	1	2	yes	—
73	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#126	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#135	1	2	yes	—
79	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄×C₁₇	#137	1	2	yes	—
81	C₂×Dic₁₇	—	8	2	no	—
82	C₂×Dic₁₇	—	8	2	no	—
83	C₂×Dic₁₇	—	8	2	no	—
84	C₈×C₁₇	—	4	2	no	—
85	C₈×C₁₇	—	8	1	no	—
86	#3622	—	8	2	no	—
87	#1984	D₁₇	1	1	yes	characteristic
88	#1834	—	17	1	no	—
89	#1803	—	17	2	no	—
90	#1296	—	17	2	no	—
91	#1277	—	17	1	no	—
92	C₂²×C₁₇	#397	1	1	yes	characteristic
93	C₂²×C₁₇	#464	1	2	yes	—
94	C₂²×C₁₇	#467	1	1	yes	characteristic
95	C₂²×C₁₇	#469	1	1	yes	characteristic
96	C₂²×C₁₇	#471	1	2	yes	—
97	C₄×C₁₇	—	2	2	no	—
98	C₄×C₁₇	—	4	2	no	—
99	C₄×C₁₇	—	8	1	no	—
100	C₄×C₁₇	#482	1	2	yes	—
101	C₄×C₁₇	#484	1	2	yes	—
102	C₄×C₁₇	#485	1	2	yes	—
103	C₄×C₁₇	#488	1	1	yes	characteristic
104	C₄×C₁₇	#491	1	1	yes	characteristic
105	Dic₁₇	—	16	2	no	—
106	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
107	C₂×C₄⋊C₈	—	17	2	no	—
108	C₂×C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
109	C₂×#103	—	34	2	no	—
110	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
111	#384	—	17	2	no	—
112	#340	—	34	2	no	—
113	#340	—	34	2	no	—
114	#336	—	2	2	no	—
115	C₂×C₁₇	#1684	1	2	yes	—
116	C₂×C₁₇	#1702	1	2	yes	—
117	C₂×C₁₇	#2074	1	1	yes	characteristic
118	C₂×C₁₇	#2076	1	1	yes	characteristic
119	C₂×C₁₇	#2079	1	1	yes	characteristic
120	C₂²×Q₈	D₆₈	1	1	yes	characteristic
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	34	2	no	—
123	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
124	C₄×Q₈	—	2	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	—	34	2	no	—
128	C₂×C₄⋊C₄	—	34	2	no	—
129	C₂×C₄⋊C₄	—	34	2	no	—
130	C₂×C₄⋊C₄	D₆₈	1	1	yes	characteristic
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	34	2	no	—
135	C₄⋊C₈	—	34	2	no	—
136	#103	—	68	4	no	—
137	C₄×C₈	—	2	2	no	—
138	C₁₇	#10624	1	1	yes	characteristic, complemented, Sylow
139	C₂×Q₈	—	2	1	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	Dic₆₈	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	1	no	—
145	C₂²×C₄	—	68	2	no	—
146	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
147	C₂²×C₄	#63252	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	68	2	no	—
155	C₂×C₈	—	68	2	no	—
156	C₂×C₈	—	68	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	68	2	no	—
160	C₄⋊C₄	—	68	2	no	—
161	C₄⋊C₄	—	68	2	no	—
162	C₄⋊C₄	—	68	2	no	—
163	C₄⋊C₄	—	136	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	#63254	1	2	yes	—
167	C₄²	#63254	1	2	yes	—
168	C₂³	#10587616	1	1	yes	characteristic
169	Q₈	—	2	4	no	—
170	Q₈	—	4	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	136	2	no	—
178	C₂×C₄	—	136	2	no	—
179	C₂×C₄	—	136	2	no	—
180	C₂×C₄	#10587565	1	2	yes	—
181	C₂×C₄	#10587572	1	2	yes	—
182	C₂×C₄	#10587590	1	1	yes	characteristic
183	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
184	C₂×C₄	#10587594	1	2	yes	—
185	C₂×C₄	#10587621	1	1	yes	characteristic
186	C₂×C₄	#10587622	1	1	yes	characteristic
187	C₂×C₄	#10587628	1	2	yes	—
188	C₂×C₄	#10587636	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	1	no	—
191	C₈	—	136	2	no	—
192	C₂²	#11762734	1	1	yes	characteristic
193	C₂²	#11763031	1	2	yes	—
194	C₂²	#11763050	1	1	yes	characteristic
195	C₂²	#11763062	1	1	yes	characteristic
196	C₂²	#11763072	1	2	yes	—
197	C₄	—	2	2	no	—
198	C₄	—	4	2	no	—
199	C₄	—	8	1	no	—
200	C₄	—	272	2	no	—
201	C₄	#11763134	1	2	yes	—
202	C₄	#11763145	1	2	yes	—
203	C₄	#11763147	1	2	yes	—
204	C₄	#11763163	1	1	yes	characteristic
205	C₄	#11763177	1	1	yes	characteristic
206	C₂	#14528242	1	2	yes	—
207	C₂	#14528352	1	2	yes	—
208	C₂	#14530202	1	1	yes	characteristic
209	C₂	#14530212	1	1	yes	characteristic
210	C₂	#14530229	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528912

C₂

yes

maximal, characteristic

#14528789

C₂

yes

maximal

#14526668

C₂

yes

maximal

#14526619

C₂

yes

maximal, characteristic

C₁₇×#1984

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂²

yes

—

C₂×#10587506

—

#11762679

C₂²

yes

—

#11762572

—

#11762571

—

C₂×C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#336

—

C₂²×Q₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

—

#10587506

—

C₄⋊#3622

—

C₄⋊#3622

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

Q₁₆

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

characteristic

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

#41

yes

—

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

—

C₂³×C₁₇

yes

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#1984

D₁₇

yes

—

—

—

—

C₂²×C₁₇

#397

yes

characteristic

C₂²×C₁₇

#464

yes

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

100

C₄×C₁₇

#482

yes

—

101

C₄×C₁₇

#484

yes

—

102

C₄×C₁₇

#485

yes

—

103

C₄×C₁₇

#488

yes

characteristic

104

C₄×C₁₇

#491

yes

characteristic

105

Dic₁₇

—

106

C₂×C₄×Q₈

C₂×D₁₇

yes

characteristic

107

C₂×C₄⋊C₈

—

108

C₂×C₄⋊C₈

C₂×D₁₇

yes

characteristic

109

C₂×#103

—

110

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

111

#384

—

112

#340

—

113

#340

—

114

#336

—

115

C₂×C₁₇

#1684

yes

—

116

C₂×C₁₇

#1702

yes

—

117

C₂×C₁₇

#2074

yes

characteristic

118

C₂×C₁₇

#2076

yes

characteristic

119

C₂×C₁₇

#2079

yes

characteristic

120

C₂²×Q₈

D₆₈

yes

characteristic

121

C₂²×C₈

—

122

C₂²×C₈

—

123

C₂²×C₈

C₂²⋊D₁₇

yes

—

124

C₄×Q₈

—

125

C₄×Q₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

D₆₈

yes

characteristic

131

C₂×C₄²

—

132

C₂×C₄²

C₂²×D₁₇

yes

characteristic

133

C₄⋊C₈

—

134

C₄⋊C₈

—

135

C₄⋊C₈

—

136

#103

—

137

C₄×C₈

—

138

C₁₇

#10624

yes

characteristic, complemented, Sylow

139

C₂×Q₈

—

140

C₂×Q₈

—

141

C₂×Q₈

—

142

C₂×Q₈

Dic₆₈

yes

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

C₂×D₆₈

yes

characteristic

147

C₂²×C₄

#63252

yes

characteristic

148

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

136

—

164

C₄²

—

165

C₄²

—

166

C₄²

#63254

yes

—

167

C₄²

#63254

yes

—

168

C₂³

#10587616

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

136

—

178

C₂×C₄

—

136

—

179

C₂×C₄

—

136

—

180

C₂×C₄

#10587565

yes

—

181

C₂×C₄

#10587572

yes

—

182

C₂×C₄

#10587590

yes

characteristic

183

C₂×C₄

C₂×Dic₆₈

yes

characteristic

184

C₂×C₄

#10587594

yes

—

185

C₂×C₄

#10587621

yes

characteristic

186

C₂×C₄

#10587622

yes

characteristic

187

C₂×C₄

#10587628

yes

—

188

C₂×C₄

#10587636

yes

—

189

C₈

—

190

C₈

—

191

C₈

—

136

—

192

C₂²

#11762734

yes

characteristic

193

C₂²

#11763031

yes

—

194

C₂²

#11763050

yes

characteristic

195

C₂²

#11763062

yes

characteristic

196

C₂²

#11763072

yes

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₄

—

272

—

201

C₄

#11763134

yes

—

202

C₄

#11763145

yes

—

203

C₄

#11763147

yes

—

204

C₄

#11763163

yes

characteristic

205

C₄

#11763177

yes

characteristic

206

C₂

#14528242

yes

—

207

C₂

#14528352

yes

—

208

C₂

#14530202

yes

characteristic

209

C₂

#14530212

yes

characteristic

210

C₂

#14530229

yes

characteristic

	subgroup	G/H
center	C₂³	#10587616
commutator, cosocle	C₂×C₄²×C₁₇	C₂³
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1984	C₂
socle, prime radical	C₂³×C₁₇	#121
hypercenter	#1984	D₁₇
hypocenter	C₁₇	#10624
LCS	G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#10587616

commutator, cosocle

C₂×C₄²×C₁₇

C₂³

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1984

C₂

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂×C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
138. C₁₇	#10624	semidirect

H ≅ G/N

type

138. C₁₇

#10624

semidirect

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

subgroup	index
#14528912	2
#14528789	2
#14526668	2
#14526619	2
C₁₇×#1984	2
#10624	17

subgroup

index

C₁₇×#1984

Subgroups — #16939073

Conjugacy classes of subgroups

Complements

Maximal subgroups