Group #17948541 — subgroups

Subgroups — #17948541

Order 4352.

242 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 98 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	C₄×#10587529	C₂	1	1	yes	maximal, characteristic
2	#14544329	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587529	C₄	1	2	yes	—
5	#11763810	—	2	1	no	—
6	#11763695	—	2	1	no	—
7	C₄×#63216	—	2	1	no	—
8	#11763673	—	2	1	no	—
9	#11763672	C₄	1	2	yes	—
10	C₄⋊C₈×D₁₇	—	2	1	no	—
11	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
12	#11762562	C₄	1	1	yes	characteristic
13	#11762561	—	2	1	no	—
14	C₁₇×#335	C₄	1	1	yes	characteristic
15	C₂×#63242	—	4	1	no	—
16	C₂×C₈×D₁₇	—	4	1	no	—
17	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×D₁₇	Q₈	1	1	yes	characteristic
19	C₄²×D₁₇	D₄	1	1	yes	characteristic
20	#10587544	—	4	1	no	—
21	C₄⋊#3624	—	2	2	no	—
22	C₂×#63216	—	2	2	no	—
23	#10587529	—	2	2	no	—
24	#10587529	C₈	1	4	yes	complemented
25	C₁₇×C₄⋊C₈	—	2	1	no	—
26	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
27	#10587485	—	4	1	no	—
28	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₁₇	D₄	1	1	yes	characteristic
30	C₄⋊#3622	—	2	1	no	—
31	C₄×#3622	D₄	1	1	yes	characteristic
32	C₂×C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₇	D₈	1	1	yes	characteristic
34	C₂×C₄×D₁₇	Q₁₆	1	1	yes	characteristic
35	C₂×#3624	—	4	2	no	—
36	#63242	—	8	1	no	—
37	#63242	—	8	1	no	—
38	C₂×C₈×C₁₇	—	4	1	no	—
39	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
41	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
44	C₄×Dic₁₇	SD₁₆	1	1	yes	characteristic
45	C₂×#3622	—	4	1	no	—
46	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
48	C₈×D₁₇	—	2	2	no	—
49	C₈×D₁₇	—	8	1	no	—
50	C₈×D₁₇	—	8	1	no	—
51	C₈×D₁₇	C₂×C₈	1	2	yes	—
52	C₈×D₁₇	M₁₆	1	2	yes	—
53	#63216	—	2	4	no	—
54	#7154	—	17	1	no	maximal, Sylow
55	C₂²×D₁₇	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
57	C₂×C₄×C₁₇	#102	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#103	1	1	yes	characteristic
59	C₂×Dic₁₇	#102	1	1	yes	characteristic
60	C₂×Dic₁₇	#103	1	1	yes	characteristic
61	C₂×Dic₁₇	#106	1	1	yes	characteristic
62	C₄×D₁₇	—	2	2	no	—
63	C₄×D₁₇	C₄⋊C₈	1	2	yes	—
64	C₄×D₁₇	#108	1	2	yes	—
65	C₄×D₁₇	D₁₆	1	1	yes	characteristic
66	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
67	C₄×D₁₇	SD₃₂	1	1	yes	characteristic
68	C₄×D₁₇	Q₃₂	1	1	yes	characteristic
69	#3624	—	8	2	no	—
70	C₈×C₁₇	—	2	1	no	—
71	C₈×C₁₇	—	8	1	no	—
72	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
73	C₈×C₁₇	#97	1	1	yes	characteristic
74	#3622	—	2	1	no	—
75	#3622	—	8	1	no	—
76	#3622	C₂²⋊C₈	1	2	yes	—
77	C₂×C₄×C₁₆	—	17	1	no	—
78	C₂×#335	—	17	1	no	—
79	C₂×#335	—	17	1	no	—
80	C₂²×C₁₇	#340	1	1	yes	characteristic
81	C₂×D₁₇	#335	1	2	yes	—
82	C₂×D₁₇	#366	1	1	yes	characteristic
83	C₂×D₁₇	#367	1	1	yes	characteristic
84	C₂×D₁₇	#368	1	2	yes	—
85	C₄×C₁₇	—	2	1	no	—
86	C₄×C₁₇	#336	1	1	yes	characteristic
87	C₄×C₁₇	#341	1	1	yes	characteristic
88	C₄×C₁₇	#357	1	1	yes	characteristic
89	C₄×C₁₇	#358	1	1	yes	characteristic
90	Dic₁₇	—	2	1	no	—
91	Dic₁₇	#334	1	2	yes	—
92	Dic₁₇	#357	1	1	yes	characteristic
93	Dic₁₇	#358	1	1	yes	characteristic
94	C₂²×C₁₆	—	17	2	no	—
95	C₂×C₄⋊C₈	—	34	1	no	—
96	C₂×C₄⋊C₈	—	34	1	no	—
97	C₂×C₄×C₈	—	17	1	no	—
98	C₄×C₁₆	—	34	1	no	—
99	C₄×C₁₆	—	34	1	no	—
100	#335	—	17	2	no	—
101	#335	—	17	1	no	—
102	#335	—	34	1	no	—
103	#335	—	34	1	no	—
104	#335	#595	1	1	yes	characteristic
105	C₂×C₁₇	#1277	1	1	yes	characteristic
106	C₂×C₁₇	#1380	1	1	yes	characteristic
107	C₂×C₁₇	#1381	1	1	yes	characteristic
108	D₁₇	#1371	1	2	yes	—
109	D₁₇	#1372	1	2	yes	—
110	C₂²×C₈	—	17	1	no	—
111	C₂²×C₈	—	17	1	no	—
112	C₂²×C₈	—	68	1	no	—
113	C₂²×C₈	—	68	1	no	—
114	C₂×C₄²	—	17	1	no	—
115	C₂×C₁₆	—	17	4	no	—
116	C₂×C₁₆	—	34	2	no	—
117	C₂×C₁₆	—	34	2	no	—
118	C₄⋊C₈	—	2	1	no	—
119	C₄⋊C₈	—	34	2	no	—
120	C₄⋊C₈	—	34	1	no	—
121	C₄⋊C₈	—	68	1	no	—
122	C₄⋊C₈	—	68	1	no	—
123	C₄×C₈	—	17	1	no	—
124	C₄×C₈	—	17	1	no	—
125	C₄×C₈	—	17	1	no	—
126	C₄×C₈	C₂×#595	1	1	yes	characteristic
127	C₁₇	#7154	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₄	—	17	1	no	—
129	C₂²×C₄	—	17	1	no	—
130	C₂²×C₄	—	17	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	17	1	no	—
133	C₂×C₈	—	17	2	no	—
134	C₂×C₈	—	17	2	no	—
135	C₂×C₈	—	17	1	no	—
136	C₂×C₈	—	34	2	no	—
137	C₂×C₈	—	68	2	no	—
138	C₂×C₈	—	68	1	no	—
139	C₂×C₈	—	136	1	no	—
140	C₂×C₈	—	136	1	no	—
141	C₂×C₈	—	136	1	no	—
142	C₂×C₈	—	136	1	no	—
143	C₂×C₈	C₄×#595	1	1	yes	characteristic
144	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
145	C₄²	—	17	1	no	—
146	C₄²	—	17	1	no	—
147	C₄²	—	17	1	no	—
148	C₄²	C₂²⋊#595	1	1	yes	characteristic
149	C₁₆	—	34	4	no	—
150	C₂³	—	17	1	no	—
151	C₂×C₄	—	17	1	no	—
152	C₂×C₄	—	17	1	no	—
153	C₂×C₄	—	17	1	no	—
154	C₂×C₄	—	17	1	no	—
155	C₂×C₄	—	17	1	no	—
156	C₂×C₄	—	17	1	no	—
157	C₂×C₄	—	17	2	no	—
158	C₂×C₄	—	17	2	no	—
159	C₂×C₄	—	17	1	no	—
160	C₂×C₄	—	34	2	no	—
161	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
162	C₂×C₄	#10587547	1	1	yes	characteristic
163	C₂×C₄	#10587549	1	1	yes	characteristic
164	C₈	—	2	1	no	—
165	C₈	—	8	1	no	—
166	C₈	—	17	2	no	—
167	C₈	—	34	1	no	—
168	C₈	—	136	2	no	—
169	C₈	—	136	1	no	—
170	C₈	C₈×#595	1	1	yes	characteristic
171	C₈	#10587532	1	1	yes	characteristic
172	C₂²	—	17	1	no	—
173	C₂²	—	17	1	no	—
174	C₂²	—	17	2	no	—
175	C₂²	—	17	2	no	—
176	C₂²	#11763682	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	17	1	no	—
179	C₄	—	17	1	no	—
180	C₄	—	17	2	no	—
181	C₄	—	34	1	no	—
182	C₄	#11763674	1	1	yes	characteristic
183	C₄	#11763684	1	1	yes	characteristic
184	C₄	#11763721	1	1	yes	characteristic
185	C₄	#11763722	1	1	yes	characteristic
186	C₂	—	17	2	no	—
187	C₂	—	17	2	no	—
188	C₂	#14543613	1	1	yes	characteristic
189	C₂	#14543841	1	1	yes	characteristic
190	C₂	#14543843	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587529

C₂

yes

maximal, characteristic

#14544329

C₂

yes

maximal, characteristic

D₁₇×#335

C₂

yes

maximal, characteristic

C₂×#10587529

C₄

yes

—

—

—

—

—

C₄

yes

—

C₄⋊C₈×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762562

C₄

yes

characteristic

#11762561

—

C₁₇×#335

C₄

yes

characteristic

C₂×#63242

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

Q₈

yes

characteristic

C₄²×D₁₇

D₄

yes

—

—

—

—

C₈

yes

complemented

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

#10587485

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₄⋊#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×D₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

Q₁₆

yes

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₄×Dic₁₇

SD₁₆

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

—

C₈×D₁₇

M₁₆

yes

—

—

—

C₂²×D₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₂×Dic₁₇

yes

C₄×D₁₇

—

C₄×D₁₇

C₄⋊C₈

yes

—

C₄×D₁₇

#108

yes

—

C₄×D₁₇

D₁₆

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

C₄×D₁₇

SD₃₂

yes

characteristic

C₄×D₁₇

Q₃₂

yes

characteristic

#3624

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

—

—

C₂²⋊C₈

yes

—

C₂×C₄×C₁₆

—

C₂×#335

—

C₂×#335

—

C₂²×C₁₇

#340

yes

characteristic

C₂×D₁₇

#335

yes

—

C₂×D₁₇

yes

C₂×D₁₇

yes

C₂×D₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

#334

yes

—

Dic₁₇

yes

Dic₁₇

yes

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

100

#335

—

101

#335

—

102

#335

—

103

#335

—

104

#335

#595

yes

characteristic

105

C₂×C₁₇

#1277

yes

characteristic

106

C₂×C₁₇

#1380

yes

characteristic

107

C₂×C₁₇

#1381

yes

characteristic

108

D₁₇

#1371

yes

—

109

D₁₇

#1372

yes

—

110

C₂²×C₈

—

111

C₂²×C₈

—

112

C₂²×C₈

—

113

C₂²×C₈

—

114

C₂×C₄²

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₄⋊C₈

—

122

C₄⋊C₈

—

123

C₄×C₈

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

C₂×#595

yes

characteristic

127

C₁₇

#7154

yes

characteristic, complemented, Sylow

128

C₂²×C₄

—

129

C₂²×C₄

—

130

C₂²×C₄

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

136

—

140

C₂×C₈

—

136

—

141

C₂×C₈

—

136

—

142

C₂×C₈

—

136

—

143

C₂×C₈

C₄×#595

yes

characteristic

144

C₂×C₈

C₄⋊#595

yes

characteristic

145

C₄²

—

146

C₄²

—

147

C₄²

—

148

C₄²

C₂²⋊#595

yes

characteristic

149

C₁₆

—

150

C₂³

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

162

C₂×C₄

#10587547

yes

characteristic

163

C₂×C₄

#10587549

yes

characteristic

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

136

—

169

C₈

—

136

—

170

C₈

C₈×#595

yes

characteristic

171

C₈

#10587532

yes

characteristic

172

C₂²

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

176

C₂²

#11763682

yes

characteristic

177

C₄

—

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

#11763674

yes

characteristic

183

C₄

#11763684

yes

characteristic

184

C₄

#11763721

yes

characteristic

185

C₄

#11763722

yes

characteristic

186

C₂

—

187

C₂

—

188

C₂

#14543613

yes

characteristic

189

C₂

#14543841

yes

characteristic

190

C₂

#14543843

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587547
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#335	C₄
socle	C₂²×C₁₇	#340
prime radical	C₂²×D₁₇	#107
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#335	#595
hypocenter	C₁₇	#7154
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

subgroup

G/H

center

C₂×C₄

#10587547

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#335

C₄

C₂²×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

N	H ≅ G/N	type
24. #10587529	C₈	semidirect
127. C₁₇	#7154	semidirect

H ≅ G/N

type

24. #10587529

C₈

semidirect

127. C₁₇

#7154

semidirect

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

subgroup	index
C₄×#10587529	2
#14544329	2
D₁₇×#335	2
#7154	17

subgroup

index

C₄×#10587529

#14544329

D₁₇×#335

#7154

Subgroups — #17948541

Conjugacy classes of subgroups

Complements

Maximal subgroups