Group #16950663 — subgroups

Subgroups — #16950663

Order 4352.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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	#14537395	C₂	1	1	yes	maximal, characteristic
2	#14528830	C₂	1	1	yes	maximal, characteristic
3	#14526899	C₂	1	2	yes	maximal, complemented
4	#14526880	C₂	1	2	yes	maximal, complemented
5	C₁₇×#2113	C₂	1	1	yes	maximal, characteristic
6	#11763576	—	2	1	no	—
7	#11762858	—	2	1	no	—
8	#11762858	C₄	1	4	yes	complemented
9	C₂×#10587490	—	2	2	no	—
10	C₂×#10587490	C₂²	1	2	yes	—
11	#11762847	C₂²	1	2	yes	—
12	#11762783	—	2	1	no	—
13	#11762566	—	2	1	no	—
14	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₁₇×C₂²⋊C₁₆	—	2	2	no	—
18	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
19	C₁₇×#131	—	2	2	no	—
20	C₁₇×#131	—	2	2	no	—
21	C₂×M₁₆×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₇	D₄	1	2	yes	—
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587634	—	4	4	no	—
26	#10587612	—	2	2	no	—
27	C₂×#63223	—	2	4	no	—
28	C₂×#63223	—	2	2	no	—
29	C₁₇×M₃₂	—	2	2	no	—
30	C₂×C₁₆×C₁₇	—	2	1	no	—
31	C₂×C₁₆×C₁₇	—	2	1	no	—
32	C₂×C₁₆×C₁₇	—	2	2	no	—
33	#10587505	—	4	2	no	—
34	#10587490	—	2	4	no	—
35	#10587490	—	2	2	no	—
36	#10587490	C₂×C₄	1	4	yes	—
37	#10587489	—	2	2	no	—
38	#10587488	—	2	2	no	—
39	#41×C₁₇	—	2	2	no	—
40	#41×C₁₇	—	2	2	no	—
41	C₂×Q₈×C₁₇	Q₁₆	1	1	yes	characteristic
42	C₂×D₄×C₁₇	SD₁₆	1	2	yes	—
43	C₂×D₄×C₁₇	Q₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₁₇	—	2	2	no	—
47	M₁₆×C₁₇	—	2	4	no	—
48	C₂×C₈×C₁₇	—	2	2	no	—
49	C₂×C₈×C₁₇	—	2	2	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	#41	1	2	yes	—
54	C₂×C₈×C₁₇	#41	1	2	yes	—
55	C₂²⋊Dic₁₇	—	4	2	no	—
56	C₄⋊Dic₁₇	—	4	2	no	—
57	C₄×Dic₁₇	—	4	2	no	—
58	#63223	—	2	4	no	—
59	#63223	—	4	4	no	—
60	#63223	—	4	2	no	—
61	C₂×#3622	—	4	4	no	—
62	C₂×#3622	—	4	2	no	—
63	C₁₆×C₁₇	—	4	1	no	—
64	C₁₆×C₁₇	—	4	1	no	—
65	#12877	—	17	1	no	maximal, Sylow
66	C₂³×C₁₇	#103	1	2	yes	—
67	C₂³×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
68	Q₈×C₁₇	—	2	2	no	—
69	D₄×C₁₇	—	2	4	no	—
70	D₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	#103	1	2	yes	—
74	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#123	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
78	C₂×Dic₁₇	—	8	2	no	—
79	C₈×C₁₇	—	2	2	no	—
80	C₈×C₁₇	#117	1	2	yes	—
81	C₈×C₁₇	#123	1	1	yes	characteristic
82	C₈×C₁₇	#123	1	1	yes	characteristic
83	#3622	—	4	4	no	—
84	#2916	—	17	1	no	—
85	#2113	D₁₇	1	1	yes	characteristic
86	#1812	—	17	1	no	—
87	#1389	—	17	2	no	—
88	#1382	—	17	2	no	—
89	C₂²×C₁₇	—	2	2	no	—
90	C₂²×C₁₇	—	4	1	no	—
91	C₂²×C₁₇	—	4	2	no	—
92	C₂²×C₁₇	C₂×#103	1	1	yes	characteristic
93	C₂²×C₁₇	#481	1	2	yes	—
94	C₄×C₁₇	—	2	2	no	—
95	C₄×C₁₇	#381	1	1	yes	characteristic
96	C₄×C₁₇	#418	1	1	yes	characteristic
97	C₄×C₁₇	#484	1	2	yes	—
98	Dic₁₇	—	16	2	no	—
99	#568	—	34	1	no	—
100	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
101	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
103	#430	—	17	4	no	—
104	#430	—	34	1	no	—
105	C₂×#108	—	17	2	no	—
106	C₂×#108	—	34	2	no	—
107	#427	—	17	2	no	—
108	#410	—	34	1	no	—
109	C₂²⋊C₁₆	—	2	2	no	—
110	H(3,ℤ/4)	—	34	1	no	—
111	C₂×C₁₇	—	2	2	no	—
112	C₂×C₁₇	—	2	1	no	—
113	C₂×C₁₇	—	4	2	no	—
114	C₂×C₁₇	#1824	1	1	yes	characteristic
115	C₂×#41	D₆₈	1	1	yes	characteristic
116	#131	—	2	2	no	—
117	#131	—	2	2	no	—
118	#131	—	68	4	no	—
119	C₂×M₁₆	—	34	2	no	—
120	C₂×M₁₆	—	34	4	no	—
121	C₂×M₁₆	C₄×D₁₇	1	2	yes	—
122	C₂×M₁₆	D₆₈	1	1	yes	characteristic
123	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
124	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
125	#117	—	34	2	no	—
126	M₃₂	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	#108	—	17	4	no	—
131	#108	—	34	2	no	—
132	#108	—	34	4	no	—
133	#107	—	34	2	no	—
134	#106	—	34	2	no	—
135	#99	—	68	2	no	—
136	C₁₇	#12877	1	1	yes	characteristic, complemented, Sylow
137	#41	—	2	2	no	—
138	#41	—	2	2	no	—
139	C₂×Q₈	Dic₆₈	1	1	yes	characteristic
140	C₂×D₄	#63219	1	2	yes	—
141	C₂×D₄	Dic₆₈	1	1	yes	characteristic
142	C₂²×C₄	#63227	1	2	yes	—
143	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
144	M₁₆	—	2	2	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	34	4	no	—
147	M₁₆	—	68	4	no	—
148	M₁₆	—	68	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	68	4	no	—
152	C₂×C₈	—	68	2	no	—
153	C₂×C₈	#63227	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
155	C₂×C₈	#63254	1	2	yes	—
156	C₂×C₈	#63254	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
158	C₄⋊C₄	—	68	2	no	—
159	C₂²⋊C₄	—	68	2	no	—
160	C₄²	—	68	2	no	—
161	C₁₆	—	4	1	no	—
162	C₁₆	—	4	1	no	—
163	C₂³	#10587487	1	2	yes	—
164	C₂³	C₂×#63219	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	D₄	—	2	4	no	—
167	D₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	136	2	no	—
171	C₂×C₄	#10587487	1	2	yes	—
172	C₂×C₄	#10587572	1	2	yes	—
173	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
174	C₂×C₄	#10587594	1	2	yes	—
175	C₂×C₄	C₂×#63227	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	68	4	no	—
178	C₈	#10587563	1	2	yes	—
179	C₈	#10587621	1	1	yes	characteristic
180	C₈	#10587621	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	4	1	no	—
183	C₂²	—	4	2	no	—
184	C₂²	C₂×#10587487	1	1	yes	characteristic
185	C₂²	#11763128	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	272	2	no	—
188	C₄	#11762670	1	1	yes	characteristic
189	C₄	#11762815	1	1	yes	characteristic
190	C₄	#11763145	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	—	2	1	no	—
193	C₂	—	4	2	no	—
194	C₂	#14528875	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14537395

C₂

yes

maximal, characteristic

#14528830

C₂

yes

maximal, characteristic

#14526899

C₂

yes

maximal, complemented

#14526880

C₂

yes

maximal, complemented

C₁₇×#2113

C₂

yes

maximal, characteristic

#11763576

—

#11762858

—

#11762858

C₄

yes

complemented

C₂×#10587490

—

C₂×#10587490

C₂²

yes

—

#11762847

C₂²

yes

—

#11762783

—

#11762566

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×C₂²⋊C₁₆

—

C₂×#41×C₁₇

D₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

—

—

—

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

—

—

—

C₂×C₄

yes

—

#10587489

—

#10587488

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

Q₁₆

yes

characteristic

C₂×D₄×C₁₇

SD₁₆

yes

—

C₂×D₄×C₁₇

Q₁₆

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂×C₈×C₁₇

#41

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

—

—

—

—

—

C₁₆×C₁₇

—

C₁₆×C₁₇

—

#12877

—

maximal, Sylow

C₂³×C₁₇

#103

yes

—

C₂³×C₁₇

C₂×SD₁₆

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#103

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#123

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#117

yes

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

—

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#103

yes

characteristic

C₂²×C₁₇

#481

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

#568

—

100

C₂×#131

C₂×D₁₇

yes

characteristic

101

C₂×M₃₂

C₂×D₁₇

yes

characteristic

102

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

103

#430

—

104

#430

—

105

C₂×#108

—

106

C₂×#108

—

107

#427

—

108

#410

—

109

C₂²⋊C₁₆

—

110

H(3,ℤ/4)

—

111

C₂×C₁₇

—

112

C₂×C₁₇

—

113

C₂×C₁₇

—

114

C₂×C₁₇

#1824

yes

characteristic

115

C₂×#41

D₆₈

yes

characteristic

116

#131

—

117

#131

—

118

#131

—

119

C₂×M₁₆

—

120

C₂×M₁₆

—

121

C₂×M₁₆

C₄×D₁₇

yes

—

122

C₂×M₁₆

D₆₈

yes

characteristic

123

C₂²×C₈

C₂²⋊D₁₇

yes

—

124

C₂²×C₈

C₂²×D₁₇

yes

characteristic

125

#117

—

126

M₃₂

—

127

C₂×C₁₆

—

128

C₂×C₁₆

—

129

C₂×C₁₆

—

130

#108

—

131

#108

—

132

#108

—

133

#107

—

134

#106

—

135

#99

—

136

C₁₇

#12877

yes

characteristic, complemented, Sylow

137

#41

—

138

#41

—

139

C₂×Q₈

Dic₆₈

yes

characteristic

140

C₂×D₄

#63219

yes

—

141

C₂×D₄

Dic₆₈

yes

characteristic

142

C₂²×C₄

#63227

yes

—

143

C₂²×C₄

C₂×D₆₈

yes

characteristic

144

M₁₆

—

145

M₁₆

—

146

M₁₆

—

147

M₁₆

—

148

M₁₆

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

#63227

yes

—

154

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

155

C₂×C₈

#63254

yes

—

156

C₂×C₈

#63254

yes

—

157

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

—

161

C₁₆

—

162

C₁₆

—

163

C₂³

#10587487

yes

—

164

C₂³

C₂×#63219

yes

characteristic

165

Q₈

—

166

D₄

—

167

D₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

136

—

171

C₂×C₄

#10587487

yes

—

172

C₂×C₄

#10587572

yes

—

173

C₂×C₄

C₂×Dic₆₈

yes

characteristic

174

C₂×C₄

#10587594

yes

—

175

C₂×C₄

C₂×#63227

yes

characteristic

176

C₈

—

177

C₈

—

178

C₈

#10587563

yes

—

179

C₈

#10587621

yes

characteristic

180

C₈

#10587621

yes

characteristic

181

C₂²

—

182

C₂²

—

183

C₂²

—

184

C₂²

C₂×#10587487

yes

characteristic

185

C₂²

#11763128

yes

—

186

C₄

—

187

C₄

—

272

—

188

C₄

#11762670

yes

characteristic

189

C₄

#11762815

yes

characteristic

190

C₄

#11763145

yes

—

191

C₂

—

192

C₂

—

193

C₂

—

194

C₂

#14528875

yes

characteristic

	subgroup	G/H
center	C₄	#11762815
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#2113	C₂
socle	C₂×C₁₇	#1824
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#2113	D₁₇
hypocenter	C₁₇	#12877
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

subgroup

G/H

center

C₄

#11762815

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#2113

C₂

socle

C₂×C₁₇

#1824

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 ◁ C₂²×C₄ ◁ C₂×#131 ◁ #2113

N	H ≅ G/N	type
3. #14526899	C₂	semidirect
4. #14526880	C₂	semidirect
8. #11762858	C₄	semidirect
136. C₁₇	#12877	semidirect

H ≅ G/N

type

3. #14526899

C₂

semidirect

4. #14526880

C₂

semidirect

8. #11762858

C₄

semidirect

136. C₁₇

#12877

semidirect

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

subgroup	index
#14537395	2
#14528830	2
#14526899	2
#14526880	2
C₁₇×#2113	2
#12877	17

subgroup

index

C₁₇×#2113

Subgroups — #16950663

Conjugacy classes of subgroups

Complements

Maximal subgroups