Group #16950683 — subgroups

Subgroups — #16950683

Order 4352.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	#14530539	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14526866	C₂	1	1	yes	maximal, characteristic
4	#14526865	C₂	1	2	yes	maximal
5	C₁₇×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	2	yes	—
7	C₂×C₄×#3622	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
12	C₄²×Dic₁₇	C₄	1	1	yes	characteristic
13	#11762627	—	2	4	no	—
14	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₇	—	2	2	no	—
20	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂²×#3622	C₂×C₄	1	1	yes	characteristic
24	C₁₇×M₃₂	—	2	2	no	—
25	C₁₇×M₃₂	—	2	2	no	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₄×C₈×C₁₇	—	2	2	no	—
28	C₈×Dic₁₇	—	2	2	no	—
29	C₈×Dic₁₇	—	2	2	no	—
30	C₈×Dic₁₇	C₈	1	2	yes	—
31	C₈×Dic₁₇	C₈	1	2	yes	—
32	C₈×Dic₁₇	D₄	1	2	yes	—
33	C₈×Dic₁₇	Q₈	1	2	yes	—
34	#10587483	—	2	2	no	—
35	#10587483	—	2	2	no	—
36	C₂×#63214	—	2	2	no	—
37	C₄×#3622	—	2	2	no	—
38	C₂²×C₄×C₁₇	—	2	1	no	—
39	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇	—	2	1	no	—
41	C₂²×Dic₁₇	M₁₆	1	2	yes	—
42	C₂×C₈×C₁₇	—	2	1	no	—
43	C₂×C₈×C₁₇	—	2	1	no	—
44	C₂×C₈×C₁₇	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₇	#41	1	2	yes	—
48	C₂×C₈×C₁₇	#41	1	2	yes	—
49	C₄²×C₁₇	—	2	1	no	—
50	C₄²×C₁₇	M₁₆	1	2	yes	—
51	C₄×Dic₁₇	—	2	8	no	—
52	C₄×Dic₁₇	—	2	1	no	—
53	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
54	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
55	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
56	C₄×Dic₁₇	M₁₆	1	2	yes	—
57	C₂×#3622	—	2	1	no	—
58	C₂×#3622	—	2	1	no	—
59	C₂×#3622	—	2	1	no	—
60	C₂×#3622	—	2	1	no	—
61	C₂×#3622	C₄⋊C₄	1	2	yes	—
62	C₂×#3622	C₂×C₈	1	2	yes	—
63	C₁₆×C₁₇	—	4	2	no	—
64	#63214	—	4	2	no	—
65	#12883	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₇	—	2	4	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
71	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₁₇	—	2	4	no	—
76	C₂×Dic₁₇	—	2	2	no	—
77	C₂×Dic₁₇	—	2	4	no	—
78	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
79	C₂×Dic₁₇	C₄⋊C₈	1	2	yes	—
80	C₈×C₁₇	—	2	2	no	—
81	C₈×C₁₇	#124	1	2	yes	—
82	C₈×C₁₇	#125	1	2	yes	—
83	#3622	—	2	2	no	—
84	#3622	—	2	2	no	—
85	#2150	—	17	2	no	—
86	C₄²×C₈	—	17	1	no	—
87	#1376	—	17	2	no	—
88	#1376	—	17	1	no	—
89	#1376	D₁₇	1	1	yes	characteristic
90	C₂²×C₁₇	—	2	2	no	—
91	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₁₇	#432	1	1	yes	characteristic
93	C₂²×C₁₇	#432	1	1	yes	characteristic
94	C₄×C₁₇	—	2	4	no	—
95	C₄×C₁₇	#382	1	1	yes	characteristic
96	C₄×C₁₇	#423	1	1	yes	characteristic
97	C₄×C₁₇	#431	1	1	yes	characteristic
98	C₄×C₁₇	#431	1	1	yes	characteristic
99	Dic₁₇	—	2	4	no	—
100	Dic₁₇	—	2	2	no	—
101	Dic₁₇	#364	1	4	yes	—
102	C₂×M₃₂	—	17	2	no	—
103	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
104	C₂×C₄×C₈	—	17	1	no	—
105	C₂×C₄×C₈	—	17	1	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	17	1	no	—
108	C₂×C₄×C₈	—	17	1	no	—
109	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
110	C₄³	—	17	1	no	—
111	C₄⋊C₁₆	—	34	4	no	—
112	C₂×C₁₇	—	2	1	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	#1831	1	1	yes	characteristic
115	C₂×C₁₇	C₂×#364	1	2	yes	—
116	C₂²×C₈	—	17	1	no	—
117	C₂²×C₈	—	17	1	no	—
118	C₂²×C₈	C₄×D₁₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
120	C₂×C₄²	—	17	1	no	—
121	C₂×C₄²	—	17	1	no	—
122	C₂×C₄²	—	34	2	no	—
123	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
124	M₃₂	—	2	2	no	—
125	M₃₂	—	2	2	no	—
126	M₃₂	—	34	2	no	—
127	M₃₂	—	34	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	34	2	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	17	2	no	—
132	C₄×C₈	—	17	2	no	—
133	C₄×C₈	—	17	2	no	—
134	C₄×C₈	—	17	2	no	—
135	C₄×C₈	—	34	2	no	—
136	C₄×C₈	—	34	2	no	—
137	C₄×C₈	—	34	2	no	—
138	C₁₇	#12883	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	17	2	no	—
141	C₂²×C₄	—	34	1	no	—
142	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	17	2	no	—
146	C₂×C₈	—	17	2	no	—
147	C₂×C₈	—	34	1	no	—
148	C₂×C₈	—	34	1	no	—
149	C₂×C₈	—	34	1	no	—
150	C₂×C₈	—	34	1	no	—
151	C₂×C₈	C₈×D₁₇	1	2	yes	—
152	C₂×C₈	#63252	1	2	yes	—
153	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
154	C₂×C₈	#63254	1	2	yes	—
155	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
156	C₄²	—	2	1	no	—
157	C₄²	—	17	2	no	—
158	C₄²	—	17	1	no	—
159	C₄²	—	17	1	no	—
160	C₄²	—	17	2	no	—
161	C₄²	—	34	8	no	—
162	C₄²	—	34	1	no	—
163	C₄²	#63218	1	2	yes	—
164	C₁₆	—	4	2	no	—
165	C₁₆	—	68	2	no	—
166	C₂³	M₁₆×D₁₇	1	1	yes	characteristic
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	17	2	no	—
170	C₂×C₄	—	17	2	no	—
171	C₂×C₄	—	34	4	no	—
172	C₂×C₄	—	34	4	no	—
173	C₂×C₄	—	34	2	no	—
174	C₂×C₄	#10587558	1	1	yes	characteristic
175	C₂×C₄	#10587563	1	1	yes	characteristic
176	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
177	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
178	C₂×C₄	C₂×#63218	1	1	yes	characteristic
179	C₂×C₄	M₁₆×D₁₇	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	34	2	no	—
182	C₈	—	34	2	no	—
183	C₈	#10587571	1	2	yes	—
184	C₈	#10587575	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₄⋊C₈×D₁₇	1	1	yes	characteristic
187	C₂²	#11762863	1	1	yes	characteristic
188	C₂²	#11762864	1	1	yes	characteristic
189	C₄	—	2	4	no	—
190	C₄	—	17	4	no	—
191	C₄	—	34	4	no	—
192	C₄	—	34	2	no	—
193	C₄	#11762673	1	1	yes	characteristic
194	C₄	#11762831	1	1	yes	characteristic
195	C₄	#11762859	1	1	yes	characteristic
196	C₄	#11762860	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#14528899	1	1	yes	characteristic
200	C₂	D₁₇×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14530539

C₂

yes

maximal

C₄×C₈×Dic₁₇

C₂

yes

maximal, characteristic

#14526866

C₂

yes

maximal, characteristic

#14526865

C₂

yes

maximal

C₁₇×#1376

C₂

yes

maximal, characteristic

C₂×#10587483

C₂²

yes

—

C₂×C₄×#3622

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₄

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

C₄²×Dic₁₇

C₄

yes

characteristic

#11762627

—

C₂×C₁₇×M₃₂

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₂²×#3622

C₂×C₄

yes

characteristic

C₁₇×M₃₂

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

—

C₈×Dic₁₇

—

C₈×Dic₁₇

—

C₈×Dic₁₇

C₈

yes

—

C₈×Dic₁₇

C₈

yes

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

Q₈

yes

—

—

—

—

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

M₁₆

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#3622

C₂×C₈

yes

—

C₁₆×C₁₇

—

#63214

—

#12883

—

maximal, Sylow

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄⋊C₈

yes

—

C₂×Dic₁₇

C₄⋊C₈

yes

—

C₈×C₁₇

—

C₈×C₁₇

#124

yes

—

C₈×C₁₇

#125

yes

—

#3622

—

#3622

—

#2150

—

C₄²×C₈

—

#1376

—

#1376

—

#1376

D₁₇

yes

characteristic

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

100

Dic₁₇

—

101

Dic₁₇

#364

yes

—

102

C₂×M₃₂

—

103

C₂×M₃₂

C₂×D₁₇

yes

—

104

C₂×C₄×C₈

—

105

C₂×C₄×C₈

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

C₂×C₄×C₈

—

109

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

110

C₄³

—

111

C₄⋊C₁₆

—

112

C₂×C₁₇

—

113

C₂×C₁₇

—

114

C₂×C₁₇

#1831

yes

characteristic

115

C₂×C₁₇

C₂×#364

yes

—

116

C₂²×C₈

—

117

C₂²×C₈

—

118

C₂²×C₈

C₄×D₁₇

yes

characteristic

119

C₂²×C₈

C₂²×D₁₇

yes

characteristic

120

C₂×C₄²

—

121

C₂×C₄²

—

122

C₂×C₄²

—

123

C₂×C₄²

C₄×D₁₇

yes

characteristic

124

M₃₂

—

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

C₄×C₈

—

131

C₄×C₈

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₁₇

#12883

yes

characteristic, complemented, Sylow

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

—

142

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

C₈×D₁₇

yes

—

152

C₂×C₈

#63252

yes

—

153

C₂×C₈

D₄×D₁₇

yes

characteristic

154

C₂×C₈

#63254

yes

—

155

C₂×C₈

Q₈×D₁₇

yes

characteristic

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

—

161

C₄²

—

162

C₄²

—

163

C₄²

#63218

yes

—

164

C₁₆

—

165

C₁₆

—

166

C₂³

M₁₆×D₁₇

yes

characteristic

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

#10587558

yes

characteristic

175

C₂×C₄

#10587563

yes

characteristic

176

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

177

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

178

C₂×C₄

C₂×#63218

yes

characteristic

179

C₂×C₄

M₁₆×D₁₇

yes

characteristic

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

#10587571

yes

—

184

C₈

#10587575

yes

—

185

C₂²

—

186

C₂²

C₄⋊C₈×D₁₇

yes

characteristic

187

C₂²

#11762863

yes

characteristic

188

C₂²

#11762864

yes

characteristic

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

—

193

C₄

#11762673

yes

characteristic

194

C₄

#11762831

yes

characteristic

195

C₄

#11762859

yes

characteristic

196

C₄

#11762860

yes

characteristic

197

C₂

—

198

C₂

—

199

C₂

#14528899

yes

characteristic

200

C₂

D₁₇×#364

yes

—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1376	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₇	C₂×M₁₆
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1376	D₁₇
hypocenter	C₁₇	#12883
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

subgroup

G/H

center

C₂×C₈

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1376

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₈

prime radical

C₂³×C₁₇

C₂×M₁₆

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

N	H ≅ G/N	type
138. C₁₇	#12883	semidirect

H ≅ G/N

type

138. C₁₇

#12883

semidirect

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

subgroup	index
#14530539	2
C₄×C₈×Dic₁₇	2
#14526866	2
#14526865	2
C₁₇×#1376	2
#12883	17

subgroup

index

C₄×C₈×Dic₁₇

C₁₇×#1376

Subgroups — #16950683

Conjugacy classes of subgroups

Complements

Maximal subgroups