Group #16934186 — subgroups

Subgroups — #16934186

Order 4352.

510 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 199 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	#14529746	C₂	1	2	yes	maximal
2	#14529587	C₂	1	2	yes	maximal
3	#14528944	C₂	1	1	yes	maximal, characteristic
4	C₂×#11762530	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#498	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587489	—	2	2	no	—
7	C₂×#10587506	—	2	4	no	—
8	C₂×C₄×#3622	C₂²	1	2	yes	—
9	#11762679	C₂²	1	1	yes	characteristic
10	#11762679	C₂²	1	1	yes	characteristic
11	#11762571	—	2	4	no	—
12	#11762530	—	2	2	no	—
13	C₂×C₁₇×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₇×#498	—	2	2	no	—
15	C₁₇×#498	C₄	1	4	yes	complemented
16	C₂×C₁₇×#106	—	2	2	no	—
17	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²×Q₈×C₁₇	—	2	2	no	—
19	C₂²×C₈×C₁₇	D₄	1	2	yes	—
20	C₁₇×C₄⋊Q₈	—	2	2	no	—
21	C₁₇×C₄⋊Q₈	—	2	2	no	—
22	C₁₇×C₄⋊Q₈	—	2	2	no	—
23	C₁₇×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×C₄⋊Dic₁₇	—	2	2	no	—
29	C₂×C₄⋊Dic₁₇	—	2	1	no	—
30	C₂×C₄⋊Dic₁₇	—	2	1	no	—
31	C₂²×#3622	D₄	1	4	yes	—
32	C₁₇×#106	—	2	8	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	D₄	1	2	yes	—
35	#10587506	—	4	8	no	—
36	#10587489	—	4	4	no	—
37	C₄×#3622	—	2	2	no	—
38	C₄×#3622	—	2	2	no	—
39	C₂×Q₈×C₁₇	—	2	4	no	—
40	C₂×Q₈×C₁₇	—	4	2	no	—
41	C₂×Q₈×C₁₇	—	4	4	no	—
42	C₂×Q₈×C₁₇	—	4	4	no	—
43	C₂²×C₄×C₁₇	—	4	2	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₁₇	—	4	1	no	—
47	C₂²×Dic₁₇	—	4	1	no	—
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	4	yes	—
49	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
50	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
51	C₄⋊C₄×C₁₇	—	2	4	no	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₄⋊C₄×C₁₇	—	4	2	no	—
54	C₄⋊C₄×C₁₇	—	4	2	no	—
55	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₇	#41	1	2	yes	—
58	C₄⋊Dic₁₇	—	4	2	no	—
59	C₄⋊Dic₁₇	—	4	2	no	—
60	C₄⋊Dic₁₇	—	4	4	no	—
61	C₄⋊Dic₁₇	—	4	4	no	—
62	C₂×#3622	—	2	4	no	—
63	C₂×#3622	—	2	4	no	—
64	C₂×#3622	—	2	4	no	—
65	#9841	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	#127	1	1	yes	characteristic
67	Q₈×C₁₇	—	4	8	no	—
68	Q₈×C₁₇	—	8	4	no	—
69	C₂×C₄×C₁₇	—	4	4	no	—
70	C₂×C₄×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	—	8	2	no	—
72	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₁₇	#121	1	2	yes	—
75	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
79	C₂×C₄×C₁₇	#137	1	4	yes	—
80	C₂×Dic₁₇	—	8	1	no	—
81	C₂×Dic₁₇	—	8	1	no	—
82	C₂×Dic₁₇	—	8	2	no	—
83	C₂×Dic₁₇	—	8	2	no	—
84	C₈×C₁₇	#103	1	8	yes	complemented
85	#3622	—	4	4	no	—
86	C₂×#498	D₁₇	1	1	yes	characteristic
87	#1990	—	17	2	no	—
88	#1956	—	17	2	no	—
89	#1845	—	17	1	no	—
90	C₂×#322	—	17	1	no	—
91	C₂²×C₁₇	#390	1	1	yes	characteristic
92	C₂²×C₁₇	#492	1	1	yes	characteristic
93	C₂²×C₁₇	#494	1	1	yes	characteristic
94	C₂²×C₁₇	#496	1	2	yes	—
95	C₂²×C₁₇	#497	1	2	yes	—
96	C₄×C₁₇	—	8	4	no	—
97	C₄×C₁₇	C₂×#103	1	2	yes	—
98	C₄×C₁₇	#440	1	4	yes	—
99	C₄×C₁₇	#468	1	4	yes	—
100	C₄×C₁₇	#487	1	2	yes	—
101	Dic₁₇	—	16	1	no	—
102	Dic₁₇	—	16	1	no	—
103	C₂×C₄⋊Q₈	C₂×D₁₇	1	2	yes	—
104	#498	—	2	2	no	—
105	#498	Dic₁₇	1	4	yes	complemented
106	C₂×#107	—	34	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#103	—	34	4	no	—
109	C₂×C₄×C₈	—	17	2	no	—
110	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
111	#384	—	17	1	no	—
112	#384	—	17	1	no	—
113	#340	—	34	4	no	—
114	#322	—	34	2	no	—
115	C₂×C₁₇	#1715	1	1	yes	characteristic
116	C₂×C₁₇	#1716	1	1	yes	characteristic
117	C₂×C₁₇	#1723	1	2	yes	—
118	C₂×C₁₇	#1956	1	1	yes	characteristic
119	C₂×C₁₇	#1973	1	2	yes	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×C₈	—	17	4	no	—
122	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₂×Dic₁₇	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	34	2	no	—
130	C₂×C₄⋊C₄	—	34	2	no	—
131	C₂×C₄⋊C₄	—	34	1	no	—
132	C₂×C₄⋊C₄	—	34	1	no	—
133	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
134	#107	—	68	4	no	—
135	#106	—	2	8	no	—
136	#103	—	68	8	no	—
137	C₄×C₈	—	34	2	no	—
138	C₄×C₈	—	34	2	no	—
139	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
140	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
141	C₁₇	#9841	1	1	yes	characteristic, complemented, Sylow
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	68	1	no	—
148	C₂²×C₄	—	68	1	no	—
149	C₂²×C₄	D₄×D₁₇	1	2	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
151	C₂×C₈	—	34	4	no	—
152	C₂×C₈	—	34	4	no	—
153	C₂×C₈	—	34	4	no	—
154	C₂×C₈	#63230	1	4	yes	—
155	C₂×C₈	#63231	1	4	yes	—
156	C₂×C₈	C₂²⋊Dic₁₇	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	68	4	no	—
162	C₄⋊C₄	—	68	4	no	—
163	C₄⋊C₄	—	68	2	no	—
164	C₄⋊C₄	—	68	2	no	—
165	C₄²	#63254	1	2	yes	—
166	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
168	C₂³	#10587626	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	136	1	no	—
175	C₂×C₄	—	136	1	no	—
176	C₂×C₄	—	136	2	no	—
177	C₂×C₄	—	136	2	no	—
178	C₂×C₄	#10587600	1	4	yes	—
179	C₂×C₄	#10587601	1	4	yes	—
180	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
181	C₂×C₄	#10587622	1	1	yes	characteristic
182	C₂×C₄	#10587624	1	2	yes	—
183	C₂×C₄	C₂×#63230	1	2	yes	—
184	C₂×C₄	C₂×#63231	1	2	yes	—
185	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
186	C₈	—	68	4	no	—
187	C₈	#10587506	1	8	yes	complemented
188	C₂²	#11762712	1	1	yes	characteristic
189	C₂²	#11763183	1	1	yes	characteristic
190	C₂²	#11763188	1	1	yes	characteristic
191	C₂²	#11763195	1	2	yes	—
192	C₂²	#11763200	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	272	1	no	—
195	C₄	—	272	1	no	—
196	C₄	C₂×#10587506	1	2	yes	—
197	C₄	#11762910	1	4	yes	—
198	C₄	#11763052	1	4	yes	—
199	C₄	#11763159	1	2	yes	—
200	C₂	#14528413	1	1	yes	characteristic
201	C₂	#14528418	1	1	yes	characteristic
202	C₂	#14528458	1	2	yes	—
203	C₂	#14529587	1	1	yes	characteristic
204	C₂	#14529667	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14529746

C₂

yes

maximal

#14529587

C₂

yes

maximal

#14528944

C₂

yes

maximal, characteristic

C₂×#11762530

C₂

yes

maximal, characteristic

C₂×C₁₇×#498

C₂

yes

maximal, characteristic

C₂×#10587489

—

C₂×#10587506

—

C₂×C₄×#3622

C₂²

yes

—

#11762679

C₂²

yes

characteristic

#11762679

C₂²

yes

characteristic

#11762571

—

#11762530

—

C₂×C₁₇×C₄⋊Q₈

C₂²

yes

—

C₁₇×#498

—

C₁₇×#498

C₄

yes

complemented

C₂×C₁₇×#106

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

—

C₁₇×#106

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

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×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

yes

—

D₁₇

yes

—

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#497

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

C₄×C₁₇

#440

yes

—

C₄×C₁₇

#468

yes

—

100

C₄×C₁₇

#487

yes

—

101

Dic₁₇

—

102

Dic₁₇

—

103

C₂×C₄⋊Q₈

C₂×D₁₇

yes

—

104

#498

—

105

#498

Dic₁₇

yes

complemented

106

C₂×#107

—

107

C₂×#106

—

108

C₂×#103

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

111

#384

—

112

#384

—

113

#340

—

114

#322

—

115

C₂×C₁₇

#1715

yes

characteristic

116

C₂×C₁₇

#1716

yes

characteristic

117

C₂×C₁₇

#1723

yes

—

118

C₂×C₁₇

#1956

yes

characteristic

119

C₂×C₁₇

#1973

yes

—

120

C₂²×Q₈

—

121

C₂²×C₈

—

122

C₂²×C₈

C₂²⋊D₁₇

yes

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

C₂×Dic₁₇

yes

—

127

C₂×C₄⋊C₄

—

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²×D₁₇

yes

characteristic

134

#107

—

135

#106

—

136

#103

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₂×Dic₁₇

yes

—

140

C₄×C₈

C₂²⋊D₁₇

yes

—

141

C₁₇

#9841

yes

characteristic, complemented, Sylow

142

C₂×Q₈

—

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

D₄×D₁₇

yes

—

150

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#63230

yes

—

155

C₂×C₈

#63231

yes

—

156

C₂×C₈

C₂²⋊Dic₁₇

yes

—

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄²

#63254

yes

—

166

C₄²

C₂²×Dic₁₇

yes

characteristic

167

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

168

C₂³

#10587626

yes

characteristic

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

136

—

175

C₂×C₄

—

136

—

176

C₂×C₄

—

136

—

177

C₂×C₄

—

136

—

178

C₂×C₄

#10587600

yes

—

179

C₂×C₄

#10587601

yes

—

180

C₂×C₄

D₄×Dic₁₇

yes

—

181

C₂×C₄

#10587622

yes

characteristic

182

C₂×C₄

#10587624

yes

—

183

C₂×C₄

C₂×#63230

yes

—

184

C₂×C₄

C₂×#63231

yes

—

185

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

186

C₈

—

187

C₈

#10587506

yes

complemented

188

C₂²

#11762712

yes

characteristic

189

C₂²

#11763183

yes

characteristic

190

C₂²

#11763188

yes

characteristic

191

C₂²

#11763195

yes

—

192

C₂²

#11763200

yes

—

193

C₄

—

194

C₄

—

272

—

195

C₄

—

272

—

196

C₄

C₂×#10587506

yes

—

197

C₄

#11762910

yes

—

198

C₄

#11763052

yes

—

199

C₄

#11763159

yes

—

200

C₂

#14528413

yes

characteristic

201

C₂

#14528418

yes

characteristic

202

C₂

#14528458

yes

—

203

C₂

#14529587

yes

characteristic

204

C₂

#14529667

yes

—

	subgroup	G/H
center	C₂³	#10587626
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₂×C₁₇×#498	C₂
socle, prime radical	C₂³×C₁₇	#127
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₂×#498	D₁₇
hypocenter	C₁₇	#9841
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

subgroup

G/H

center

C₂³

#10587626

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₂×C₁₇×#498

C₂

socle, prime radical

C₂³×C₁₇

#127

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

N	H ≅ G/N	type
15. C₁₇×#498	C₄	semidirect
84. C₈×C₁₇	#103	semidirect
105. #498	Dic₁₇	semidirect
141. C₁₇	#9841	semidirect
187. C₈	#10587506	semidirect

H ≅ G/N

type

15. C₁₇×#498

C₄

semidirect

84. C₈×C₁₇

#103

semidirect

105. #498

Dic₁₇

semidirect

141. C₁₇

#9841

semidirect

187. C₈

#10587506

semidirect

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

subgroup	index
#14529746	2
#14529587	2
#14528944	2
C₂×#11762530	2
C₂×C₁₇×#498	2
#9841	17

subgroup

index

C₂×C₁₇×#498

Subgroups — #16934186

Conjugacy classes of subgroups

Complements

Maximal subgroups