Group #16950481 — subgroups

Subgroups — #16950481

Order 4352.

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

normal subgroups only

This group has 358 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	C₂×#11763226	C₂	1	1	yes	maximal, characteristic
2	#14530558	C₂	1	2	yes	maximal, complemented
3	#14528830	C₂	1	1	yes	maximal, characteristic
4	#14526881	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1813	C₂	1	1	yes	maximal, characteristic
6	#11763226	—	2	4	no	—
7	C₂×#10587483	C₄	1	2	yes	complemented
8	C₂×#10587483	C₂²	1	1	yes	characteristic
9	C₂²×#63214	C₄	1	2	yes	complemented
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	#11762847	C₂²	1	2	yes	—
12	#11762783	—	2	1	no	—
13	#11762633	—	2	4	no	—
14	#11762566	—	2	1	no	—
15	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
16	C₂×C₁₇×#108	C₂²	1	2	yes	—
17	C₁₇×#413	—	2	1	no	—
18	C₁₇×#338	—	2	1	no	—
19	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
20	C₁₇×#131	—	2	2	no	—
21	C₁₇×#131	—	2	2	no	—
22	C₂×M₁₆×C₁₇	—	2	2	no	—
23	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
24	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	#10587612	—	2	2	no	—
28	C₁₇×#108	—	2	2	no	—
29	C₁₇×#108	—	2	2	no	—
30	C₁₇×#101	—	4	1	no	—
31	C₁₇×#100	—	4	1	no	—
32	#10587505	—	4	2	no	—
33	#10587489	—	2	2	no	—
34	#10587488	—	2	2	no	—
35	#10587483	—	2	4	no	—
36	#10587483	D₄	1	4	yes	—
37	C₂×#63214	—	2	4	no	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	C₂×C₄	1	1	yes	characteristic
40	C₂×#63214	C₂×C₄	1	1	yes	characteristic
41	C₂×#63214	C₂×C₄	1	2	yes	—
42	#41×C₁₇	—	2	2	no	—
43	#41×C₁₇	—	2	2	no	—
44	C₂×Q₈×C₁₇	SD₁₆	1	1	yes	characteristic
45	C₂×D₄×C₁₇	SD₁₆	1	1	yes	characteristic
46	C₂×D₄×C₁₇	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₁₇	—	2	2	no	—
50	M₁₆×C₁₇	—	2	4	no	—
51	M₁₆×C₁₇	—	4	2	no	—
52	M₁₆×C₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	—	4	2	no	—
56	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₇	#41	1	2	yes	—
61	C₂²⋊Dic₁₇	—	4	2	no	—
62	C₄⋊Dic₁₇	—	4	2	no	—
63	C₄×Dic₁₇	—	4	2	no	—
64	#63214	—	2	1	no	—
65	#63214	—	2	1	no	—
66	#63214	C₂²⋊C₄	1	4	yes	—
67	#12832	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#106	1	2	yes	—
69	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
70	Q₈×C₁₇	—	2	2	no	—
71	D₄×C₁₇	—	2	4	no	—
72	D₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	—	2	2	no	—
75	C₂×C₄×C₁₇	#106	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	#121	1	2	yes	—
78	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#135	1	2	yes	—
80	C₂×Dic₁₇	—	8	2	no	—
81	C₈×C₁₇	—	2	2	no	—
82	C₈×C₁₇	—	8	2	no	—
83	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₁₇	#117	1	1	yes	characteristic
85	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×#504	—	17	1	no	—
87	#2155	—	17	2	no	—
88	#1813	D₁₇	1	1	yes	characteristic
89	#1812	—	17	1	no	—
90	#1382	—	17	2	no	—
91	C₂²×C₁₇	—	2	2	no	—
92	C₂²×C₁₇	—	4	1	no	—
93	C₂²×C₁₇	—	4	2	no	—
94	C₂²×C₁₇	C₂×#106	1	1	yes	characteristic
95	C₂²×C₁₇	#465	1	2	yes	—
96	C₄×C₁₇	—	2	2	no	—
97	C₄×C₁₇	#380	1	1	yes	characteristic
98	C₄×C₁₇	#427	1	1	yes	characteristic
99	C₄×C₁₇	#465	1	2	yes	—
100	Dic₁₇	—	16	2	no	—
101	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
102	#504	—	34	4	no	—
103	C₂×M₃₂	—	17	1	no	—
104	C₂×M₃₂	—	17	2	no	—
105	C₂²×C₁₆	—	17	2	no	—
106	C₂²×C₁₆	—	17	1	no	—
107	C₂×#108	C₂×D₁₇	1	2	yes	—
108	#427	—	17	2	no	—
109	#413	—	2	1	no	—
110	#410	—	34	1	no	—
111	#365	—	34	4	no	—
112	#338	—	2	1	no	—
113	H(3,ℤ/4)	—	34	1	no	—
114	C₂×C₁₇	—	2	2	no	—
115	C₂×C₁₇	—	2	1	no	—
116	C₂×C₁₇	—	4	2	no	—
117	C₂×C₁₇	#1807	1	1	yes	characteristic
118	C₂×#41	C₂²⋊D₁₇	1	1	yes	characteristic
119	#131	—	2	2	no	—
120	#131	—	2	2	no	—
121	C₂×M₁₆	—	2	2	no	—
122	C₂×M₁₆	Dic₃₄	1	2	yes	—
123	C₂×M₁₆	C₂²⋊D₁₇	1	1	yes	characteristic
124	C₂²×C₈	C₄×D₁₇	1	2	yes	—
125	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
126	#117	—	34	2	no	—
127	M₃₂	—	17	4	no	—
128	M₃₂	—	34	4	no	—
129	C₂×C₁₆	—	17	2	no	—
130	C₂×C₁₆	—	17	1	no	—
131	C₂×C₁₆	—	17	1	no	—
132	C₂×C₁₆	—	34	4	no	—
133	C₂×C₁₆	—	34	2	no	—
134	#108	—	2	2	no	—
135	#108	—	2	2	no	—
136	#107	—	34	2	no	—
137	#106	—	34	2	no	—
138	#101	—	4	1	no	—
139	#100	—	4	1	no	—
140	#99	—	68	2	no	—
141	C₁₇	#12832	1	1	yes	characteristic, complemented, Sylow
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#63229	1	1	yes	characteristic
145	C₂×D₄	#63229	1	1	yes	characteristic
146	C₂×D₄	#63230	1	2	yes	—
147	C₂²×C₄	#63225	1	2	yes	—
148	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	4	2	no	—
152	M₁₆	—	4	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	#63225	1	2	yes	—
157	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
159	C₂×C₈	D₄×D₁₇	1	2	yes	—
160	C₂×C₈	#63254	1	2	yes	—
161	C₄⋊C₄	—	68	2	no	—
162	C₂²⋊C₄	—	68	2	no	—
163	C₄²	—	68	2	no	—
164	C₁₆	—	17	4	no	—
165	C₁₆	—	34	1	no	—
166	C₁₆	—	34	1	no	—
167	C₂³	#10587479	1	2	yes	—
168	C₂³	C₂×#63230	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	136	2	no	—
175	C₂×C₄	#10587479	1	2	yes	—
176	C₂×C₄	C₂×#63225	1	1	yes	characteristic
177	C₂×C₄	C₂×#63229	1	1	yes	characteristic
178	C₂×C₄	#10587625	1	2	yes	—
179	C₂×C₄	#10587636	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	8	2	no	—
182	C₈	#10587563	1	1	yes	characteristic
183	C₈	#10587564	1	2	yes	—
184	C₈	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#10587479	1	1	yes	characteristic
189	C₂²	#11763040	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	272	2	no	—
192	C₄	#11762667	1	1	yes	characteristic
193	C₄	#11762845	1	1	yes	characteristic
194	C₄	#11763036	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#14528807	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763226

C₂

yes

maximal, characteristic

#14530558

C₂

yes

maximal, complemented

#14528830

C₂

yes

maximal, characteristic

#14526881

C₂

yes

maximal, complemented

C₁₇×#1813

C₂

yes

maximal, characteristic

#11763226

—

C₂×#10587483

C₄

yes

complemented

C₂×#10587483

C₂²

yes

characteristic

C₂²×#63214

C₄

yes

complemented

C₂²×#63214

C₂²

yes

characteristic

#11762847

C₂²

yes

—

#11762783

—

#11762633

—

#11762566

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₂×C₁₇×#108

C₂²

yes

—

C₁₇×#413

—

C₁₇×#338

—

C₂×#41×C₁₇

D₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂×M₁₆×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

#10587612

—

C₁₇×#108

—

C₁₇×#108

—

C₁₇×#101

—

C₁₇×#100

—

—

—

—

—

D₄

yes

—

C₂×#63214

—

C₂×#63214

—

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

C₂×C₄

yes

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

SD₁₆

yes

characteristic

C₂×D₄×C₁₇

SD₁₆

yes

characteristic

C₂×D₄×C₁₇

SD₁₆

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

#63214

—

#63214

—

#63214

C₂²⋊C₄

yes

—

#12832

—

maximal, Sylow

C₂³×C₁₇

#106

yes

—

C₂³×C₁₇

C₂×SD₁₆

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#106

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

#117

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₂×#504

—

#2155

—

#1813

D₁₇

yes

characteristic

#1812

—

#1382

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#106

yes

characteristic

C₂²×C₁₇

#465

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

100

Dic₁₇

—

101

C₂×#131

C₂×D₁₇

yes

characteristic

102

#504

—

103

C₂×M₃₂

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

—

106

C₂²×C₁₆

—

107

C₂×#108

C₂×D₁₇

yes

—

108

#427

—

109

#413

—

110

#410

—

111

#365

—

112

#338

—

113

H(3,ℤ/4)

—

114

C₂×C₁₇

—

115

C₂×C₁₇

—

116

C₂×C₁₇

—

117

C₂×C₁₇

#1807

yes

characteristic

118

C₂×#41

C₂²⋊D₁₇

yes

characteristic

119

#131

—

120

#131

—

121

C₂×M₁₆

—

122

C₂×M₁₆

Dic₃₄

yes

—

123

C₂×M₁₆

C₂²⋊D₁₇

yes

characteristic

124

C₂²×C₈

C₄×D₁₇

yes

—

125

C₂²×C₈

C₂²×D₁₇

yes

characteristic

126

#117

—

127

M₃₂

—

128

M₃₂

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#108

—

135

#108

—

136

#107

—

137

#106

—

138

#101

—

139

#100

—

140

#99

—

141

C₁₇

#12832

yes

characteristic, complemented, Sylow

142

#41

—

143

#41

—

144

C₂×Q₈

#63229

yes

characteristic

145

C₂×D₄

#63229

yes

characteristic

146

C₂×D₄

#63230

yes

—

147

C₂²×C₄

#63225

yes

—

148

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

152

M₁₆

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

#63225

yes

—

157

C₂×C₈

C₂×Dic₃₄

yes

characteristic

158

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

159

C₂×C₈

D₄×D₁₇

yes

—

160

C₂×C₈

#63254

yes

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

—

167

C₂³

#10587479

yes

—

168

C₂³

C₂×#63230

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

136

—

175

C₂×C₄

#10587479

yes

—

176

C₂×C₄

C₂×#63225

yes

characteristic

177

C₂×C₄

C₂×#63229

yes

characteristic

178

C₂×C₄

#10587625

yes

—

179

C₂×C₄

#10587636

yes

—

180

C₈

—

181

C₈

—

182

C₈

#10587563

yes

characteristic

183

C₈

#10587564

yes

—

184

C₈

C₂²⋊C₄×D₁₇

yes

characteristic

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#10587479

yes

characteristic

189

C₂²

#11763040

yes

—

190

C₄

—

191

C₄

—

272

—

192

C₄

#11762667

yes

characteristic

193

C₄

#11762845

yes

characteristic

194

C₄

#11763036

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#14528807

yes

characteristic

	subgroup	G/H
center	C₄	#11762845
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1813	C₂
socle	C₂×C₁₇	#1807
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1813	D₁₇
hypocenter	C₁₇	#12832
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

subgroup

G/H

center

C₄

#11762845

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1813

C₂

socle

C₂×C₁₇

#1807

prime radical

C₂×#41×C₁₇

D₄

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ #1813

N	H ≅ G/N	type
2. #14530558	C₂	semidirect
4. #14526881	C₂	semidirect
7. C₂×#10587483	C₄	semidirect
9. C₂²×#63214	C₄	semidirect
141. C₁₇	#12832	semidirect

H ≅ G/N

type

2. #14530558

C₂

semidirect

4. #14526881

C₂

semidirect

7. C₂×#10587483

C₄

semidirect

9. C₂²×#63214

C₄

semidirect

141. C₁₇

#12832

semidirect

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

subgroup	index
C₂×#11763226	2
#14530558	2
#14528830	2
#14526881	2
C₁₇×#1813	2
#12832	17

subgroup

index

C₁₇×#1813

Subgroups — #16950481

Conjugacy classes of subgroups

Complements

Maximal subgroups