Group #16934352 — subgroups

Subgroups — #16934352

Order 4352.

434 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 159 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	#14528901	C₂	1	1	yes	maximal, characteristic
2	#14528530	C₂	1	1	yes	maximal, characteristic
3	#14526625	C₂	1	4	yes	maximal
4	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	2	yes	—
6	C₂×C₄×#3622	C₂²	1	2	yes	—
7	#11762564	—	2	2	no	—
8	#11762564	—	2	2	no	—
9	C₂×C₁₇×#125	C₄	1	1	yes	characteristic
10	C₁₇×#389	C₄	1	2	yes	—
11	C₁₇×#387	—	2	2	no	—
12	C₁₇×#382	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#125	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₇	—	2	2	no	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	—	2	2	no	—
28	C₂²×#3622	—	2	2	no	—
29	C₁₇×#95	—	2	2	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₁₇×#95	—	2	2	no	—
32	C₄⋊#3622	—	2	4	no	—
33	C₄×#3622	—	2	2	no	—
34	C₄×#3622	Q₈	1	4	yes	—
35	C₂²×C₄×C₁₇	—	2	2	no	—
36	C₂²×C₄×C₁₇	—	2	2	no	—
37	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₁₇	—	2	8	no	—
43	C₄⋊C₄×C₁₇	—	2	2	no	—
44	C₄⋊C₄×C₁₇	—	2	4	no	—
45	C₄⋊C₄×C₁₇	—	4	4	no	—
46	C₄⋊C₄×C₁₇	M₁₆	1	4	yes	—
47	C₄²×C₁₇	—	2	2	no	—
48	C₄²×C₁₇	—	2	2	no	—
49	C₄²×C₁₇	—	2	4	no	—
50	C₄²×C₁₇	—	4	2	no	—
51	C₄²×C₁₇	—	4	4	no	—
52	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
53	C₄²×C₁₇	#41	1	2	yes	—
54	C₂×#3622	—	2	4	no	—
55	C₂×#3622	—	4	4	no	—
56	C₂×#3622	—	4	2	no	—
57	C₂×#3622	—	4	4	no	—
58	#9875	—	17	1	no	maximal, Sylow
59	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	4	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	4	2	no	—
69	C₂×C₄×C₁₇	—	4	2	no	—
70	C₂×C₄×C₁₇	—	4	2	no	—
71	C₂×C₄×C₁₇	—	4	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
74	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
75	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	#131	1	2	yes	—
79	C₂×C₄×C₁₇	#135	1	4	yes	—
80	#3622	—	4	4	no	—
81	#3622	—	8	2	no	—
82	C₄×#125	D₁₇	1	1	yes	characteristic
83	#1831	—	17	1	no	—
84	#1744	—	17	1	no	—
85	#1279	—	17	4	no	—
86	C₂²×C₁₇	#391	1	1	yes	characteristic
87	C₂²×C₁₇	#407	1	1	yes	characteristic
88	C₂²×C₁₇	#408	1	1	yes	characteristic
89	C₂²×C₁₇	#416	1	2	yes	—
90	C₂²×C₁₇	#421	1	2	yes	—
91	C₄×C₁₇	—	2	2	no	—
92	C₄×C₁₇	—	4	2	no	—
93	C₄×C₁₇	—	4	2	no	—
94	C₄×C₁₇	—	4	4	no	—
95	C₄×C₁₇	—	8	2	no	—
96	C₄×C₁₇	#446	1	4	yes	—
97	C₄×C₁₇	#488	1	2	yes	—
98	C₄×C₁₇	#499	1	2	yes	—
99	C₂×#125	Dic₁₇	1	1	yes	characteristic
100	C₂×C₄⋊C₈	—	17	2	no	—
101	C₂×C₄×C₈	—	17	2	no	—
102	#389	Dic₁₇	1	2	yes	—
103	#387	—	2	2	no	—
104	#382	Dic₁₇	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	2	no	—
106	C₄×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
107	C₄³	C₂×D₁₇	1	1	yes	characteristic
108	#336	—	34	2	no	—
109	#336	—	34	2	no	—
110	C₂×C₁₇	#1482	1	2	yes	—
111	C₂×C₁₇	#1488	1	2	yes	—
112	C₂×C₁₇	#1985	1	1	yes	characteristic
113	C₂×C₁₇	#1991	1	1	yes	characteristic
114	C₂×C₁₇	#1995	1	1	yes	characteristic
115	C₂²×C₈	—	34	2	no	—
116	C₂²×C₈	—	34	2	no	—
117	#125	—	2	4	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
122	C₂×C₄²	—	2	2	no	—
123	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
126	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
127	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
128	C₄⋊C₈	—	34	4	no	—
129	C₄×C₈	—	17	4	no	—
130	C₄×C₈	—	34	2	no	—
131	#95	—	2	2	no	—
132	#95	—	2	2	no	—
133	#95	—	2	2	no	—
134	C₁₇	#9875	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
138	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
139	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
142	C₂×C₈	—	34	4	no	—
143	C₂×C₈	—	68	4	no	—
144	C₂×C₈	—	68	2	no	—
145	C₂×C₈	—	68	4	no	—
146	C₄⋊C₄	—	2	8	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	4	no	—
150	C₄⋊C₄	#63223	1	4	yes	—
151	C₄²	—	2	2	no	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	—	4	4	no	—
156	C₄²	Q₈×D₁₇	1	2	yes	—
157	C₄²	#63256	1	2	yes	—
158	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	C₂×#63223	1	2	yes	—
173	C₂×C₄	#10587630	1	1	yes	characteristic
174	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
175	C₂×C₄	#10587632	1	2	yes	—
176	C₂×C₄	#10587633	1	1	yes	characteristic
177	C₂×C₄	#10587634	1	2	yes	—
178	C₂×C₄	#10587636	1	4	yes	—
179	C₈	—	68	4	no	—
180	C₈	—	136	2	no	—
181	C₂²	#11762716	1	1	yes	characteristic
182	C₂²	#11762773	1	1	yes	characteristic
183	C₂²	#11762776	1	1	yes	characteristic
184	C₂²	#11762805	1	2	yes	—
185	C₂²	#11762826	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	#11762940	1	4	yes	—
192	C₄	#11763166	1	2	yes	—
193	C₄	#11763208	1	2	yes	—
194	C₂	#14527192	1	2	yes	—
195	C₂	#14527218	1	2	yes	—
196	C₂	#14529725	1	1	yes	characteristic
197	C₂	#14529750	1	1	yes	characteristic
198	C₂	#14529766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528901

C₂

yes

maximal, characteristic

#14528530

C₂

yes

maximal, characteristic

#14526625

C₂

yes

maximal

C₄×C₁₇×#125

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

—

#11762564

—

#11762564

—

C₂×C₁₇×#125

C₄

yes

characteristic

C₁₇×#389

C₄

yes

—

C₁₇×#387

—

C₁₇×#382

C₄

yes

characteristic

C₄×C₄⋊C₄×C₁₇

—

C₄×C₄⋊C₄×C₁₇

C₂²

yes

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₂²×#3622

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

—

C₄⋊#3622

—

C₄×#3622

—

C₄×#3622

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

M₁₆

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×Q₈

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

—

—

C₂³×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#135

yes

—

#3622

—

#3622

—

C₄×#125

D₁₇

yes

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#421

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#446

yes

—

C₄×C₁₇

#488

yes

—

C₄×C₁₇

#499

yes

—

C₂×#125

Dic₁₇

yes

characteristic

100

C₂×C₄⋊C₈

—

101

C₂×C₄×C₈

—

102

#389

Dic₁₇

yes

—

103

#387

—

104

#382

Dic₁₇

yes

characteristic

105

C₄×C₄⋊C₄

—

106

C₄×C₄⋊C₄

C₂×D₁₇

yes

—

107

C₄³

C₂×D₁₇

yes

characteristic

108

#336

—

109

#336

—

110

C₂×C₁₇

#1482

yes

—

111

C₂×C₁₇

#1488

yes

—

112

C₂×C₁₇

#1985

yes

characteristic

113

C₂×C₁₇

#1991

yes

characteristic

114

C₂×C₁₇

#1995

yes

characteristic

115

C₂²×C₈

—

116

C₂²×C₈

—

117

#125

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

—

120

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

121

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

122

C₂×C₄²

—

123

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

124

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

125

C₂×C₄²

C₂²⋊D₁₇

yes

—

126

C₂×C₄²

C₂²⋊D₁₇

yes

—

127

C₂×C₄²

C₂²×D₁₇

yes

characteristic

128

C₄⋊C₈

—

129

C₄×C₈

—

130

C₄×C₈

—

131

#95

—

132

#95

—

133

#95

—

134

C₁₇

#9875

yes

characteristic, complemented, Sylow

135

C₂²×C₄

—

136

C₂²×C₄

—

137

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

138

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

139

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

140

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

141

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄⋊C₄

—

150

C₄⋊C₄

#63223

yes

—

151

C₄²

—

152

C₄²

—

153

C₄²

—

154

C₄²

—

155

C₄²

—

156

C₄²

Q₈×D₁₇

yes

—

157

C₄²

#63256

yes

—

158

C₂³

C₂×C₂²⋊Dic₁₇

yes

characteristic

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

C₂×#63223

yes

—

173

C₂×C₄

#10587630

yes

characteristic

174

C₂×C₄

Q₈×Dic₁₇

yes

—

175

C₂×C₄

#10587632

yes

—

176

C₂×C₄

#10587633

yes

characteristic

177

C₂×C₄

#10587634

yes

—

178

C₂×C₄

#10587636

yes

—

179

C₈

—

180

C₈

—

136

—

181

C₂²

#11762716

yes

characteristic

182

C₂²

#11762773

yes

characteristic

183

C₂²

#11762776

yes

characteristic

184

C₂²

#11762805

yes

—

185

C₂²

#11762826

yes

—

186

C₄

—

187

C₄

—

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

#11762940

yes

—

192

C₄

#11763166

yes

—

193

C₄

#11763208

yes

—

194

C₂

#14527192

yes

—

195

C₂

#14527218

yes

—

196

C₂

#14529725

yes

characteristic

197

C₂

#14529750

yes

characteristic

198

C₂

#14529766

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#9875
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₇

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₄×C₁₇×#125

C₂

socle, prime radical

C₂³×C₁₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#125

N	H ≅ G/N	type
134. C₁₇	#9875	semidirect

H ≅ G/N

type

134. C₁₇

#9875

semidirect

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

subgroup	index
#14528901	2
#14528530	2
#14526625	2
C₄×C₁₇×#125	2
#9875	17

subgroup

index

C₄×C₁₇×#125

Subgroups — #16934352

Conjugacy classes of subgroups

Complements

Maximal subgroups