Group #16954187 — subgroups

Subgroups — #16954187

Order 4352.

342 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#11762635	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11762588	C₂	1	1	yes	maximal, characteristic
3	#14528810	C₂	1	2	yes	maximal
4	#14526875	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂×#10587489	C₂²	1	2	yes	—
9	#11762666	—	2	2	no	—
10	#11762635	—	2	2	no	—
11	#11762588	—	2	2	no	—
12	#11762571	—	2	2	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×#63227	—	4	2	no	—
19	C₂×C₄⋊Dic₁₇	—	2	2	no	—
20	C₂²×#3622	D₄	1	1	yes	characteristic
21	C₂×C₈×D₁₇	—	2	1	no	—
22	C₂×C₈×D₁₇	—	2	2	no	—
23	C₂×C₈×D₁₇	—	2	2	no	—
24	C₂×C₈×D₁₇	—	2	1	no	—
25	C₂×C₁₆×C₁₇	—	2	2	no	—
26	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
27	C₁₇×#107	—	2	2	no	—
28	C₁₇×#107	—	2	2	no	—
29	#10587502	—	4	2	no	—
30	#10587489	—	2	2	no	—
31	#10587489	—	2	2	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂×#63214	—	2	2	no	—
34	C₂³×D₁₇	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₇	—	4	2	no	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
39	C₂×C₄×D₁₇	—	2	1	no	—
40	C₂×C₄×D₁₇	—	2	2	no	—
41	C₂×C₄×D₁₇	—	2	1	no	—
42	C₂×C₄×D₁₇	—	2	2	no	—
43	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	2	yes	—
45	C₂×C₈×C₁₇	#41	1	2	yes	—
46	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
47	C₄⋊C₄×C₁₇	—	4	2	no	—
48	C₄⋊C₄×C₁₇	—	4	2	no	—
49	#63227	—	8	4	no	—
50	C₄⋊Dic₁₇	—	4	2	no	—
51	C₄⋊Dic₁₇	—	4	2	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	—	2	2	no	—
54	C₈×D₁₇	—	4	1	no	—
55	C₈×D₁₇	—	4	1	no	—
56	C₈×D₁₇	—	4	2	no	—
57	C₁₆×C₁₇	—	2	2	no	—
58	#63214	—	4	1	no	—
59	#13493	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
61	C₂²×D₁₇	—	2	1	no	—
62	C₂²×D₁₇	—	2	2	no	—
63	C₂²×D₁₇	—	2	1	no	—
64	C₂²×D₁₇	—	2	2	no	—
65	C₂²×D₁₇	Q₃₂	1	2	yes	—
66	C₂×C₄×C₁₇	—	8	2	no	—
67	C₂×C₄×C₁₇	—	8	2	no	—
68	C₂×C₄×C₁₇	—	8	2	no	—
69	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#123	1	2	yes	—
71	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#137	1	2	yes	—
73	C₂×Dic₁₇	—	2	1	no	—
74	C₂×Dic₁₇	—	2	2	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₂×Dic₁₇	—	8	2	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	C₄×D₁₇	—	4	1	no	—
79	C₄×D₁₇	—	4	1	no	—
80	C₄×D₁₇	—	4	2	no	—
81	C₈×C₁₇	#123	1	1	yes	characteristic
82	C₈×C₁₇	#124	1	1	yes	characteristic
83	C₈×C₁₇	#126	1	2	yes	—
84	#3622	—	4	1	no	—
85	C₂×#366	—	17	1	no	—
86	C₂×C₂²⋊C₁₆	—	17	1	no	—
87	#1808	—	17	2	no	—
88	#1380	—	17	2	no	—
89	#1380	D₁₇	1	1	yes	characteristic, complemented
90	C₂²×C₁₇	#466	1	1	yes	characteristic
91	C₂²×C₁₇	#480	1	2	yes	—
92	C₂²×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
93	C₂²×C₁₇	#508	1	1	yes	characteristic
94	C₂²×C₁₇	#509	1	2	yes	—
95	C₂×D₁₇	—	2	2	no	—
96	C₂×D₁₇	—	2	4	no	—
97	C₂×D₁₇	—	4	1	no	—
98	C₂×D₁₇	—	4	1	no	—
99	C₂×D₁₇	—	4	2	no	—
100	C₄×C₁₇	—	16	2	no	—
101	C₄×C₁₇	#397	1	1	yes	characteristic
102	C₄×C₁₇	#470	1	1	yes	characteristic
103	C₄×C₁₇	#485	1	2	yes	—
104	Dic₁₇	—	4	1	no	—
105	Dic₁₇	—	16	2	no	—
106	C₂³×C₈	—	17	1	no	—
107	C₂²×C₁₆	—	17	1	no	—
108	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
109	C₂×#107	—	17	2	no	—
110	C₂×#107	C₂×D₁₇	1	2	yes	—
111	#380	—	34	2	no	—
112	#366	—	34	2	no	—
113	C₂²⋊C₁₆	—	34	2	no	—
114	#340	—	34	2	no	—
115	C₂×C₁₇	#2075	1	1	yes	characteristic
116	C₂×C₁₇	#2216	1	1	yes	characteristic
117	C₂×C₁₇	#2218	1	1	yes	characteristic
118	C₂×C₁₇	#2234	1	2	yes	—
119	C₂×C₁₇	#2236	1	2	yes	—
120	D₁₇	—	4	2	no	—
121	C₂³×C₄	—	17	1	no	—
122	C₂²×C₈	—	17	1	no	—
123	C₂²×C₈	—	34	1	no	—
124	C₂²×C₈	—	34	2	no	—
125	C₂²×C₈	—	34	2	no	—
126	C₂²×C₈	—	34	1	no	—
127	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	34	2	no	—
130	C₂×C₂²⋊C₄	—	68	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	2	no	—
134	C₂×C₁₆	D₆₈	1	2	yes	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	34	2	no	—
138	#107	—	34	2	no	—
139	#95	—	68	2	no	—
140	C₁₇	#13493	1	1	yes	characteristic, complemented, Sylow
141	C₂⁴	—	17	1	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	17	1	no	—
144	C₂²×C₄	—	34	2	no	—
145	C₂²×C₄	—	34	1	no	—
146	C₂²×C₄	—	34	2	no	—
147	C₂²×C₄	—	34	1	no	—
148	C₂²×C₄	—	68	2	no	—
149	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
150	C₂×C₈	—	34	2	no	—
151	C₂×C₈	—	34	1	no	—
152	C₂×C₈	—	68	1	no	—
153	C₂×C₈	—	68	1	no	—
154	C₂×C₈	—	68	2	no	—
155	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
156	C₂×C₈	#63252	1	2	yes	—
157	C₂×C₈	#63254	1	2	yes	—
158	C₂×C₈	#63256	1	1	yes	characteristic
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	68	2	no	—
162	C₄⋊C₄	—	68	2	no	—
163	C₂²⋊C₄	—	136	4	no	—
164	C₁₆	—	2	2	no	—
165	C₁₆	—	68	1	no	—
166	C₂³	—	17	2	no	—
167	C₂³	—	34	1	no	—
168	C₂³	—	34	2	no	—
169	C₂³	—	34	1	no	—
170	C₂³	—	34	2	no	—
171	C₂³	D₈×D₁₇	1	1	yes	characteristic
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	34	1	no	—
176	C₂×C₄	—	34	2	no	—
177	C₂×C₄	—	68	1	no	—
178	C₂×C₄	—	68	1	no	—
179	C₂×C₄	—	68	2	no	—
180	C₂×C₄	—	136	2	no	—
181	C₂×C₄	—	136	2	no	—
182	C₂×C₄	—	136	2	no	—
183	C₂×C₄	#10587569	1	2	yes	—
184	C₂×C₄	#10587581	1	1	yes	characteristic
185	C₂×C₄	#10587601	1	2	yes	—
186	C₂×C₄	#10587605	1	1	yes	characteristic
187	C₈	—	68	1	no	—
188	C₈	#10587561	1	1	yes	characteristic
189	C₈	#10587572	1	1	yes	characteristic
190	C₈	#10587584	1	2	yes	—
191	C₂²	—	34	2	no	—
192	C₂²	—	34	4	no	—
193	C₂²	—	68	1	no	—
194	C₂²	—	68	2	no	—
195	C₂²	—	68	1	no	—
196	C₂²	#11763046	1	1	yes	characteristic
197	C₂²	#11763121	1	2	yes	—
198	C₂²	Q₃₂×D₁₇	1	1	yes	characteristic
199	C₂²	#11763238	1	1	yes	characteristic
200	C₂²	#11763246	1	2	yes	—
201	C₄	—	16	2	no	—
202	C₄	—	272	2	no	—
203	C₄	—	68	1	no	—
204	C₄	#11762736	1	1	yes	characteristic
205	C₄	#11763068	1	1	yes	characteristic
206	C₄	#11763149	1	2	yes	—
207	C₂	—	68	2	no	—
208	C₂	#14530209	1	1	yes	characteristic
209	C₂	#14530876	1	1	yes	characteristic
210	C₂	#14530888	1	1	yes	characteristic
211	C₂	#14530973	1	2	yes	—
212	C₂	#14530985	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762635

C₂

yes

maximal, characteristic, complemented

C₂×#11762588

C₂

yes

maximal, characteristic

#14528810

C₂

yes

maximal

#14526875

C₂

yes

maximal, complemented

C₁₇×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₁₇

C₂²

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

C₂×#10587489

C₂²

yes

—

—

—

—

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#107

C₂²

yes

—

C₂²×C₄×D₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×#63227

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

characteristic

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₁₇×#107

—

C₁₇×#107

—

—

—

—

—

—

C₂³×D₁₇

D₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

—

#13493

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

Q₃₂

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

#135

yes

characteristic

C₂×C₄×C₁₇

#137

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

#3622

—

C₂×#366

—

C₂×C₂²⋊C₁₆

—

#1808

—

#1380

—

#1380

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

#466

yes

characteristic

C₂²×C₁₇

#480

yes

—

C₂²×C₁₇

C₂×Q₃₂

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

100

C₄×C₁₇

—

101

C₄×C₁₇

#397

yes

characteristic

102

C₄×C₁₇

#470

yes

characteristic

103

C₄×C₁₇

#485

yes

—

104

Dic₁₇

—

105

Dic₁₇

—

106

C₂³×C₈

—

107

C₂²×C₁₆

—

108

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

109

C₂×#107

—

110

C₂×#107

C₂×D₁₇

yes

—

111

#380

—

112

#366

—

113

C₂²⋊C₁₆

—

114

#340

—

115

C₂×C₁₇

#2075

yes

characteristic

116

C₂×C₁₇

#2216

yes

characteristic

117

C₂×C₁₇

#2218

yes

characteristic

118

C₂×C₁₇

#2234

yes

—

119

C₂×C₁₇

#2236

yes

—

120

D₁₇

—

121

C₂³×C₄

—

122

C₂²×C₈

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

C₂²×D₁₇

yes

characteristic

128

C₂×C₄⋊C₄

—

129

C₂×C₄⋊C₄

—

130

C₂×C₂²⋊C₄

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

C₂×C₁₆

D₆₈

yes

—

135

#107

—

136

#107

—

137

#107

—

138

#107

—

139

#95

—

140

C₁₇

#13493

yes

characteristic, complemented, Sylow

141

C₂⁴

—

142

C₂²×C₄

—

143

C₂²×C₄

—

144

C₂²×C₄

—

145

C₂²×C₄

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

D₄×D₁₇

yes

characteristic

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂×D₆₈

yes

characteristic

156

C₂×C₈

#63252

yes

—

157

C₂×C₈

#63254

yes

—

158

C₂×C₈

#63256

yes

characteristic

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₂²⋊C₄

—

136

—

164

C₁₆

—

165

C₁₆

—

166

C₂³

—

167

C₂³

—

168

C₂³

—

169

C₂³

—

170

C₂³

—

171

C₂³

D₈×D₁₇

yes

characteristic

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

136

—

181

C₂×C₄

—

136

—

182

C₂×C₄

—

136

—

183

C₂×C₄

#10587569

yes

—

184

C₂×C₄

#10587581

yes

characteristic

185

C₂×C₄

#10587601

yes

—

186

C₂×C₄

#10587605

yes

characteristic

187

C₈

—

188

C₈

#10587561

yes

characteristic

189

C₈

#10587572

yes

characteristic

190

C₈

#10587584

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

—

194

C₂²

—

195

C₂²

—

196

C₂²

#11763046

yes

characteristic

197

C₂²

#11763121

yes

—

198

C₂²

Q₃₂×D₁₇

yes

characteristic

199

C₂²

#11763238

yes

characteristic

200

C₂²

#11763246

yes

—

201

C₄

—

202

C₄

—

272

—

203

C₄

—

204

C₄

#11762736

yes

characteristic

205

C₄

#11763068

yes

characteristic

206

C₄

#11763149

yes

—

207

C₂

—

208

C₂

#14530209

yes

characteristic

209

C₂

#14530876

yes

characteristic

210

C₂

#14530888

yes

characteristic

211

C₂

#14530973

yes

—

212

C₂

#14530985

yes

—

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator, cosocle	C₂²×C₈×C₁₇	C₂³
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1380	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂³×D₁₇	D₈
hypercenter	#1380	D₁₇
hypocenter	C₁₇	#13493
LCS	G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₁₇

commutator, cosocle

C₂²×C₈×C₁₇

C₂³

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1380

C₂

socle

C₂³×C₁₇

C₂×D₈

prime radical

C₂³×D₁₇

D₈

D₁₇

C₁₇

G ▷ C₂²×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
1. C₂×#11762635	C₂	semidirect
4. #14526875	C₂	semidirect
5. C₁₇×#1380	C₂	semidirect
89. #1380	D₁₇	semidirect
140. C₁₇	#13493	semidirect

H ≅ G/N

type

1. C₂×#11762635

C₂

semidirect

4. #14526875

C₂

semidirect

5. C₁₇×#1380

C₂

semidirect

89. #1380

D₁₇

semidirect

140. C₁₇

#13493

semidirect

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

subgroup	index
C₂×#11762635	2
C₂×#11762588	2
#14528810	2
#14526875	2
C₁₇×#1380	2
#13493	17

subgroup

index

C₁₇×#1380

Subgroups — #16954187

Conjugacy classes of subgroups

Complements

Maximal subgroups