Group #16933721 — subgroups

Subgroups — #16933721

Order 4352.

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

normal subgroups only

This group has 478 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	#14528915	C₂	1	1	yes	maximal, characteristic, complemented
2	#14527344	C₂	1	4	yes	maximal, complemented
3	#14527076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	2	yes	—
6	#11762675	C₂²	1	2	yes	—
7	#11762572	—	2	2	no	—
8	#11762543	—	2	8	no	—
9	#11762531	—	2	2	no	—
10	C₂×C₁₇×#135	—	2	2	no	—
11	C₁₇×#532	C₂²	1	2	yes	—
12	C₁₇×#495	—	2	2	no	—
13	C₁₇×#494	C₄	1	1	yes	characteristic, complemented
14	C₁₇×#493	C₄	1	1	yes	characteristic, complemented
15	C₁₇×#492	C₄	1	2	yes	complemented
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#41×C₁₇	—	2	2	no	—
18	C₁₇×#135	—	4	8	no	—
19	C₂×Q₁₆×C₁₇	—	2	2	no	—
20	C₂×SD₁₆×C₁₇	—	2	4	no	—
21	C₂×D₈×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	D₄	1	2	yes	—
23	C₁₇×C₄⋊Q₈	C₂×C₄	1	2	yes	—
24	C₁₇×#127	C₂×C₄	1	2	yes	—
25	C₁₇×#124	—	2	2	no	—
26	C₁₇×#121	—	2	4	no	—
27	C₄×D₄×C₁₇	—	2	4	no	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊Dic₁₇	—	2	2	no	—
30	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₁₇	D₄	1	2	yes	—
33	#10587502	—	2	4	no	—
34	C₂²⋊#3622	D₄	1	4	yes	—
35	C₄⋊#3622	—	2	2	no	—
36	C₄×#3622	—	2	2	no	—
37	#41×C₁₇	—	4	8	no	—
38	C₂×Q₈×C₁₇	—	2	2	no	—
39	C₂×D₄×C₁₇	—	2	2	no	—
40	C₂×D₄×C₁₇	—	2	4	no	—
41	C₂×D₄×C₁₇	—	4	2	no	—
42	C₂²×C₄×C₁₇	—	4	2	no	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₇	—	4	2	no	—
46	Q₁₆×C₁₇	—	4	4	no	—
47	SD₁₆×C₁₇	—	4	8	no	—
48	D₈×C₁₇	—	4	4	no	—
49	C₂×C₈×C₁₇	—	2	4	no	—
50	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
52	C₄⋊C₄×C₁₇	—	2	4	no	—
53	C₂²⋊C₄×C₁₇	—	4	4	no	—
54	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₇	#41	1	2	yes	—
57	C₄⋊Dic₁₇	—	4	4	no	—
58	C₂×#3622	—	2	4	no	—
59	#9760	—	17	1	no	maximal, Sylow
60	C₂³×C₁₇	—	4	2	no	—
61	C₂³×C₁₇	#127	1	1	yes	characteristic
62	Q₈×C₁₇	—	4	4	no	—
63	D₄×C₁₇	—	4	4	no	—
64	D₄×C₁₇	—	8	4	no	—
65	D₄×C₁₇	—	8	4	no	—
66	C₂×C₄×C₁₇	—	4	2	no	—
67	C₂×C₄×C₁₇	—	4	2	no	—
68	C₂×C₄×C₁₇	—	8	4	no	—
69	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₁₇	#121	1	2	yes	—
72	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
73	C₂×C₄×C₁₇	AGL(1,ℤ/8)	1	4	yes	—
74	C₂×Dic₁₇	—	8	4	no	—
75	C₂×Dic₁₇	—	8	2	no	—
76	C₈×C₁₇	—	2	2	no	—
77	C₈×C₁₇	—	2	2	no	—
78	#3622	—	4	4	no	—
79	#3147	D₁₇	1	1	yes	characteristic
80	#1835	—	17	1	no	—
81	#1517	—	17	4	no	—
82	#1452	—	17	1	no	—
83	C₂²×C₁₇	—	2	2	no	—
84	C₂²×C₁₇	—	8	2	no	—
85	C₂²×C₁₇	—	8	4	no	—
86	C₂²×C₁₇	#390	1	1	yes	characteristic
87	C₂²×C₁₇	#496	1	2	yes	—
88	C₄×C₁₇	—	2	2	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	8	2	no	—
91	C₄×C₁₇	#418	1	2	yes	—
92	C₄×C₁₇	#490	1	2	yes	—
93	Dic₁₇	—	16	2	no	—
94	C₂×#135	—	2	2	no	—
95	#532	C₂×D₁₇	1	2	yes	—
96	#495	—	2	2	no	—
97	#494	Dic₁₇	1	1	yes	characteristic, complemented
98	#493	Dic₁₇	1	1	yes	characteristic, complemented
99	#492	Dic₁₇	1	2	yes	complemented
100	#431	—	17	2	no	—
101	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
102	#382	—	17	2	no	—
103	#340	—	34	2	no	—
104	#327	—	34	8	no	—
105	#322	—	34	2	no	—
106	C₂×C₁₇	—	2	2	no	—
107	C₂×C₁₇	—	16	2	no	—
108	C₂×C₁₇	#1961	1	1	yes	characteristic
109	C₂×C₁₇	#1976	1	2	yes	—
110	C₂×#41	—	2	2	no	—
111	#135	—	4	8	no	—
112	C₂×Q₁₆	—	2	2	no	—
113	C₂×SD₁₆	—	2	4	no	—
114	C₂×D₈	—	2	2	no	—
115	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
116	C₄⋊Q₈	C₂×Dic₁₇	1	2	yes	—
117	#127	C₂×Dic₁₇	1	2	yes	—
118	#124	—	2	2	no	—
119	#121	—	2	4	no	—
120	C₄×D₄	—	2	4	no	—
121	C₂×C₄⋊C₄	—	34	2	no	—
122	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
123	C₄⋊C₈	—	34	2	no	—
124	C₂²⋊C₈	—	17	4	no	—
125	C₄×C₈	—	34	2	no	—
126	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₁₇	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
129	#95	—	34	4	no	—
130	C₁₇	#9760	1	1	yes	characteristic, complemented, Sylow
131	#41	—	4	8	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂×D₄	—	4	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	68	2	no	—
138	C₂²×C₄	D₄×D₁₇	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
140	Q₁₆	—	4	4	no	—
141	SD₁₆	—	4	8	no	—
142	D₈	—	4	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	34	4	no	—
145	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
146	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	68	4	no	—
149	C₂²⋊C₄	—	4	4	no	—
150	C₄²	#63254	1	2	yes	—
151	C₄²	C₂²×Dic₁₇	1	1	yes	characteristic
152	C₄²	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
153	C₂³	—	4	2	no	—
154	C₂³	#10587626	1	1	yes	characteristic
155	Q₈	—	4	4	no	—
156	D₄	—	4	4	no	—
157	D₄	—	8	4	no	—
158	D₄	—	8	4	no	—
159	C₂×C₄	—	4	2	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	136	4	no	—
163	C₂×C₄	—	136	2	no	—
164	C₂×C₄	#10587618	1	4	yes	—
165	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
166	C₂×C₄	#10587622	1	1	yes	characteristic
167	C₂×C₄	#10587624	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	68	4	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	8	2	no	—
174	C₂²	—	8	4	no	—
175	C₂²	#11762712	1	1	yes	characteristic
176	C₂²	#11763196	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	2	2	no	—
179	C₄	—	8	2	no	—
180	C₄	—	272	2	no	—
181	C₄	#11762814	1	2	yes	—
182	C₄	#11763173	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	16	2	no	—
185	C₂	#14529609	1	1	yes	characteristic
186	C₂	#14529683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528915

C₂

yes

maximal, characteristic, complemented

#14527344

C₂

yes

maximal, complemented

#14527076

C₂

yes

maximal, characteristic, complemented

C₁₇×#3147

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762675

C₂²

yes

—

#11762572

—

#11762543

—

#11762531

—

C₂×C₁₇×#135

—

C₁₇×#532

C₂²

yes

—

C₁₇×#495

—

C₁₇×#494

C₄

yes

characteristic, complemented

C₁₇×#493

C₄

yes

characteristic, complemented

C₁₇×#492

C₄

yes

complemented

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×#41×C₁₇

—

C₁₇×#135

—

C₂×Q₁₆×C₁₇

—

C₂×SD₁₆×C₁₇

—

C₂×D₈×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₁₇×C₄⋊Q₈

C₂×C₄

yes

—

C₁₇×#127

C₂×C₄

yes

—

C₁₇×#124

—

C₁₇×#121

—

C₄×D₄×C₁₇

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

—

#10587502

—

C₂²⋊#3622

D₄

yes

—

C₄⋊#3622

—

C₄×#3622

—

#41×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

Q₁₆×C₁₇

—

SD₁₆×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

#41

yes

—

C₄⋊Dic₁₇

—

C₂×#3622

—

#9760

—

maximal, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

#127

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#3147

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#496

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#418

yes

—

C₄×C₁₇

#490

yes

—

Dic₁₇

—

C₂×#135

—

#532

C₂×D₁₇

yes

—

#495

—

#494

Dic₁₇

yes

characteristic, complemented

#493

Dic₁₇

yes

characteristic, complemented

#492

Dic₁₇

yes

complemented

100

#431

—

101

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

102

#382

—

103

#340

—

104

#327

—

105

#322

—

106

C₂×C₁₇

—

107

C₂×C₁₇

—

108

C₂×C₁₇

#1961

yes

characteristic

109

C₂×C₁₇

#1976

yes

—

110

C₂×#41

—

111

#135

—

112

C₂×Q₁₆

—

113

C₂×SD₁₆

—

114

C₂×D₈

—

115

C₂²×C₈

C₂²⋊D₁₇

yes

—

116

C₄⋊Q₈

C₂×Dic₁₇

yes

—

117

#127

C₂×Dic₁₇

yes

—

118

#124

—

119

#121

—

120

C₄×D₄

—

121

C₂×C₄⋊C₄

—

122

C₂×C₄²

C₂²×D₁₇

yes

characteristic

123

C₄⋊C₈

—

124

C₂²⋊C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×Dic₁₇

yes

characteristic

127

C₄×C₈

C₂×Dic₁₇

yes

characteristic

128

C₄×C₈

C₂²⋊D₁₇

yes

—

129

#95

—

130

C₁₇

#9760

yes

characteristic, complemented, Sylow

131

#41

—

132

C₂×Q₈

—

133

C₂×D₄

—

134

C₂×D₄

—

135

C₂×D₄

—

136

C₂²×C₄

—

137

C₂²×C₄

—

138

C₂²×C₄

D₄×D₁₇

yes

—

139

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

140

Q₁₆

—

141

SD₁₆

—

142

D₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

C₂²⋊Dic₁₇

yes

—

146

C₂×C₈

C₂²⋊Dic₁₇

yes

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₂²⋊C₄

—

150

C₄²

#63254

yes

—

151

C₄²

C₂²×Dic₁₇

yes

characteristic

152

C₄²

C₂×C₂²⋊D₁₇

yes

characteristic

153

C₂³

—

154

C₂³

#10587626

yes

characteristic

155

Q₈

—

156

D₄

—

157

D₄

—

158

D₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

136

—

163

C₂×C₄

—

136

—

164

C₂×C₄

#10587618

yes

—

165

C₂×C₄

D₄×Dic₁₇

yes

—

166

C₂×C₄

#10587622

yes

characteristic

167

C₂×C₄

#10587624

yes

—

168

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

169

C₈

—

170

C₈

—

171

C₈

—

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

#11762712

yes

characteristic

176

C₂²

#11763196

yes

—

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

272

—

181

C₄

#11762814

yes

—

182

C₄

#11763173

yes

—

183

C₂

—

184

C₂

—

185

C₂

#14529609

yes

characteristic

186

C₂

#14529683

yes

—

	subgroup	G/H
center	C₂²	#11762712
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting, prime radical	C₁₇×#3147	C₂
socle	C₂²×C₁₇	#390
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#3147	D₁₇
hypocenter	C₁₇	#9760
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

subgroup

G/H

center

C₂²

#11762712

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting, prime radical

C₁₇×#3147

C₂

socle

C₂²×C₁₇

#390

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3147

N	H ≅ G/N	type
1. #14528915	C₂	semidirect
2. #14527344	C₂	semidirect
3. #14527076	C₂	semidirect
13. C₁₇×#494	C₄	semidirect
14. C₁₇×#493	C₄	semidirect
15. C₁₇×#492	C₄	semidirect
97. #494	Dic₁₇	semidirect
98. #493	Dic₁₇	semidirect
99. #492	Dic₁₇	semidirect
130. C₁₇	#9760	semidirect

H ≅ G/N

type

1. #14528915

C₂

semidirect

2. #14527344

C₂

semidirect

3. #14527076

C₂

semidirect

13. C₁₇×#494

C₄

semidirect

14. C₁₇×#493

C₄

semidirect

15. C₁₇×#492

C₄

semidirect

97. #494

Dic₁₇

semidirect

98. #493

Dic₁₇

semidirect

99. #492

Dic₁₇

semidirect

130. C₁₇

#9760

semidirect

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

subgroup	index
#14528915	2
#14527344	2
#14527076	2
C₁₇×#3147	2
#9760	17

subgroup

index

C₁₇×#3147

Subgroups — #16933721

Conjugacy classes of subgroups

Complements

Maximal subgroups