Group #17948634 — subgroups

Subgroups — #17948634

Order 4352.

258 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14545123	C₂	1	1	yes	maximal, characteristic
2	#14544586	C₂	1	1	yes	maximal, characteristic
3	D₁₇×#322	C₂	1	1	yes	maximal, characteristic
4	#11763865	—	2	1	no	—
5	D₁₇⋊#108	C₄	1	1	yes	characteristic
6	#11763819	C₄	1	1	yes	characteristic
7	#11763772	—	2	1	no	—
8	#11763732	—	2	1	no	—
9	#11763731	C₄	1	2	yes	complemented
10	#11763680	—	2	1	no	—
11	C₄×C₈×D₁₇	—	2	1	no	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762531	—	2	1	no	—
14	#11762530	C₄	1	1	yes	characteristic, complemented
15	C₁₇×#322	C₄	1	1	yes	characteristic, complemented
16	#10587677	—	2	1	no	—
17	D₁₇⋊M₁₆	—	2	1	no	—
18	C₂×C₈×D₁₇	—	2	2	no	—
19	C₂×C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	C₄²×D₁₇	D₄	1	1	yes	characteristic
22	#10587538	—	2	2	no	—
23	#10587536	C₈	1	2	yes	complemented
24	#10587535	—	2	1	no	—
25	#10587534	C₈	1	2	yes	complemented
26	#10587533	—	2	1	no	—
27	#10587532	—	4	1	no	—
28	C₈×#595	—	4	1	no	—
29	#10587530	—	4	1	no	—
30	#10587529	—	4	1	no	—
31	C₄×C₈×C₁₇	—	2	1	no	—
32	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₁₇	—	2	2	no	—
34	C₈×Dic₁₇	C₂×C₄	1	1	yes	characteristic
35	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
36	C₄×#3622	—	2	1	no	—
37	C₄×#3622	D₄	1	1	yes	characteristic
38	C₂×C₄×D₁₇	—	2	1	no	—
39	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²⋊#595	—	4	1	no	—
41	#63247	—	4	1	no	—
42	C₄⋊#595	—	2	2	no	—
43	C₄×#595	—	4	1	no	—
44	#63243	—	2	2	no	—
45	#63242	—	4	1	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×#3622	—	2	2	no	—
53	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
55	C₈×D₁₇	—	4	4	no	—
56	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
57	C₈×D₁₇	C₂×C₈	1	1	yes	characteristic
58	C₈×D₁₇	M₁₆	1	1	yes	characteristic
59	C₈×D₁₇	M₁₆	1	1	yes	characteristic
60	C₈×D₁₇	D₈	1	1	yes	characteristic
61	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
62	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
63	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
64	#63216	—	4	2	no	—
65	#7182	—	17	1	no	maximal, Sylow
66	C₂²×D₁₇	#104	1	1	yes	characteristic
67	C₂×#595	—	4	2	no	—
68	C₂×C₄×C₁₇	—	2	1	no	—
69	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
70	C₂×Dic₁₇	—	2	1	no	—
71	C₂×Dic₁₇	#104	1	1	yes	characteristic
72	C₄×D₁₇	—	2	4	no	—
73	C₄×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
75	C₄×D₁₇	#102	1	1	yes	characteristic
76	C₄×D₁₇	#103	1	1	yes	characteristic
77	#3624	—	4	2	no	—
78	C₈×C₁₇	—	4	2	no	—
79	C₈×C₁₇	C₄×C₈	1	1	yes	characteristic
80	C₈×C₁₇	#97	1	1	yes	characteristic
81	C₈×C₁₇	#106	1	1	yes	characteristic
82	C₈×C₁₇	#107	1	1	yes	characteristic
83	#3622	—	4	2	no	—
84	#3622	#102	1	1	yes	characteristic
85	#3622	#103	1	1	yes	characteristic
86	#3622	C₄⋊C₈	1	1	yes	characteristic
87	#3622	C₄⋊C₈	1	1	yes	characteristic
88	C₂×#364	—	17	1	no	—
89	#1772	—	17	1	no	—
90	C₂×#322	—	17	1	no	—
91	C₂²×C₁₇	#339	1	1	yes	characteristic
92	C₂×D₁₇	—	2	2	no	—
93	C₂×D₁₇	D₄⋊C₈	1	1	yes	characteristic
94	C₂×D₁₇	Q₈⋊C₈	1	1	yes	characteristic
95	#595	—	8	2	no	—
96	C₄×C₁₇	—	2	2	no	—
97	C₄×C₁₇	#336	1	1	yes	characteristic
98	C₄×C₁₇	#340	1	1	yes	characteristic
99	Dic₁₇	—	2	2	no	—
100	Dic₁₇	D₄⋊C₈	1	1	yes	characteristic
101	Dic₁₇	Q₈⋊C₈	1	1	yes	characteristic
102	C₂×M₃₂	—	34	1	no	—
103	C₂×#108	—	17	1	no	—
104	#427	—	17	1	no	—
105	#405	—	34	1	no	—
106	C₂×C₄×C₈	—	17	1	no	—
107	C₂×C₄×C₈	—	34	1	no	—
108	#364	—	17	2	no	—
109	#364	—	34	1	no	—
110	#339	—	34	1	no	—
111	#322	—	17	1	no	—
112	#322	—	34	1	no	—
113	#322	#595	1	1	yes	characteristic, complemented
114	C₂×C₁₇	—	2	1	no	—
115	C₂×C₁₇	#1279	1	1	yes	characteristic
116	D₁₇	—	2	1	no	—
117	D₁₇	#1336	1	2	yes	—
118	C₂×M₁₆	—	34	1	no	—
119	C₂²×C₈	—	17	1	no	—
120	C₂²×C₈	—	17	1	no	—
121	C₂²×C₈	—	34	2	no	—
122	#117	—	34	1	no	—
123	C₂×C₄²	—	17	1	no	—
124	M₃₂	—	34	2	no	—
125	M₃₂	—	68	1	no	—
126	C₂×C₁₆	—	68	1	no	—
127	#108	—	17	2	no	—
128	#108	—	34	1	no	—
129	#107	—	17	2	no	—
130	#106	—	34	1	no	—
131	#97	—	68	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	17	1	no	—
134	C₄×C₈	—	17	1	no	—
135	C₄×C₈	—	17	1	no	—
136	C₄×C₈	—	34	2	no	—
137	C₄×C₈	—	34	1	no	—
138	C₄×C₈	—	68	1	no	—
139	C₄×C₈	C₂×#595	1	1	yes	characteristic
140	C₁₇	#7182	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₄	—	17	1	no	—
142	C₂²×C₄	—	34	1	no	—
143	M₁₆	—	34	2	no	—
144	M₁₆	—	68	1	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	17	1	no	—
147	C₂×C₈	—	17	1	no	—
148	C₂×C₈	—	17	1	no	—
149	C₂×C₈	—	17	1	no	—
150	C₂×C₈	—	17	1	no	—
151	C₂×C₈	—	17	1	no	—
152	C₂×C₈	—	17	1	no	—
153	C₂×C₈	—	17	1	no	—
154	C₂×C₈	—	17	1	no	—
155	C₂×C₈	—	17	1	no	—
156	C₂×C₈	—	34	2	no	—
157	C₂×C₈	—	68	4	no	—
158	C₂×C₈	—	68	1	no	—
159	C₂×C₈	C₄×#595	1	1	yes	characteristic
160	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
161	C₄⋊C₄	—	34	2	no	—
162	C₂²⋊C₄	—	68	1	no	—
163	C₄²	—	17	1	no	—
164	C₄²	—	34	1	no	—
165	C₄²	—	68	1	no	—
166	C₄²	C₂²⋊#595	1	1	yes	characteristic
167	C₁₆	—	68	2	no	—
168	C₂³	—	17	1	no	—
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	17	1	no	—
171	C₂×C₄	—	17	1	no	—
172	C₂×C₄	—	17	1	no	—
173	C₂×C₄	—	17	1	no	—
174	C₂×C₄	—	17	1	no	—
175	C₂×C₄	—	34	4	no	—
176	C₂×C₄	—	34	1	no	—
177	C₂×C₄	—	68	2	no	—
178	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	—	17	1	no	—
181	C₈	—	17	1	no	—
182	C₈	—	17	1	no	—
183	C₈	—	17	1	no	—
184	C₈	—	68	2	no	—
185	C₈	—	68	2	no	—
186	C₈	C₈×#595	1	1	yes	characteristic
187	C₈	#10587532	1	1	yes	characteristic
188	C₈	#10587533	1	1	yes	characteristic
189	C₈	#10587534	1	1	yes	characteristic
190	C₂²	—	17	1	no	—
191	C₂²	—	17	1	no	—
192	C₂²	—	34	2	no	—
193	C₂²	#11763680	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	17	1	no	—
196	C₄	—	17	1	no	—
197	C₄	—	34	2	no	—
198	C₄	—	136	2	no	—
199	C₄	#11763674	1	1	yes	characteristic
200	C₄	#11763683	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	17	2	no	—
203	C₂	—	34	1	no	—
204	C₂	#14543619	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545123

C₂

yes

maximal, characteristic

#14544586

C₂

yes

maximal, characteristic

D₁₇×#322

C₂

yes

maximal, characteristic

#11763865

—

D₁₇⋊#108

C₄

yes

characteristic

#11763819

C₄

yes

—

—

C₄

yes

complemented

#11763680

—

C₄×C₈×D₁₇

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762531

—

#11762530

C₄

yes

characteristic, complemented

C₁₇×#322

C₄

yes

characteristic, complemented

#10587677

—

D₁₇⋊M₁₆

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

D₄

yes

characteristic

#10587538

—

#10587536

C₈

yes

complemented

#10587535

—

#10587534

C₈

yes

—

—

—

—

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

—

C₈×Dic₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₄×#3622

—

C₄×#3622

D₄

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂²⋊#595

—

—

—

—

—

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₂²⋊C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₂×#3622

—

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

C₂×C₈

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

M₁₆

yes

characteristic

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

—

—

C₂²×D₁₇

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

#104

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₂²⋊C₈

yes

characteristic

C₄×D₁₇

C₂²⋊C₈

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

C₈×C₁₇

—

C₈×C₁₇

C₄×C₈

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

yes

C₄⋊C₈

yes

characteristic

#3622

C₄⋊C₈

yes

—

—

—

C₂²×C₁₇

#339

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

D₄⋊C₈

yes

characteristic

C₂×D₁₇

Q₈⋊C₈

yes

characteristic

#595

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

100

Dic₁₇

D₄⋊C₈

yes

characteristic

101

Dic₁₇

Q₈⋊C₈

yes

characteristic

102

C₂×M₃₂

—

103

C₂×#108

—

104

#427

—

105

#405

—

106

C₂×C₄×C₈

—

107

C₂×C₄×C₈

—

108

#364

—

109

#364

—

110

#339

—

111

#322

—

112

#322

—

113

#322

#595

yes

characteristic, complemented

114

C₂×C₁₇

—

115

C₂×C₁₇

#1279

yes

characteristic

116

D₁₇

—

117

D₁₇

#1336

yes

—

118

C₂×M₁₆

—

119

C₂²×C₈

—

120

C₂²×C₈

—

121

C₂²×C₈

—

122

#117

—

123

C₂×C₄²

—

124

M₃₂

—

125

M₃₂

—

126

C₂×C₁₆

—

127

#108

—

128

#108

—

129

#107

—

130

#106

—

131

#97

—

132

C₄×C₈

—

133

C₄×C₈

—

134

C₄×C₈

—

135

C₄×C₈

—

136

C₄×C₈

—

137

C₄×C₈

—

138

C₄×C₈

—

139

C₄×C₈

C₂×#595

yes

characteristic

140

C₁₇

#7182

yes

characteristic, complemented, Sylow

141

C₂²×C₄

—

142

C₂²×C₄

—

143

M₁₆

—

144

M₁₆

—

145

C₂×C₈

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₂×C₈

—

150

C₂×C₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₄×#595

yes

characteristic

160

C₂×C₈

C₄⋊#595

yes

characteristic

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₄²

—

164

C₄²

—

165

C₄²

—

166

C₄²

C₂²⋊#595

yes

characteristic

167

C₁₆

—

168

C₂³

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

179

C₈

—

180

C₈

—

181

C₈

—

182

C₈

—

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

C₈×#595

yes

characteristic

187

C₈

#10587532

yes

characteristic

188

C₈

#10587533

yes

characteristic

189

C₈

#10587534

yes

characteristic

190

C₂²

—

191

C₂²

—

192

C₂²

—

193

C₂²

#11763680

yes

characteristic

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

136

—

199

C₄

#11763674

yes

characteristic

200

C₄

#11763683

yes

characteristic

201

C₂

—

202

C₂

—

203

C₂

—

204

C₂

#14543619

yes

characteristic

	subgroup	G/H
center	C₄	#11763683
commutator	C₈×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#322	C₄
socle	C₂×C₁₇	#1279
prime radical	C₂²×D₁₇	#104
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#322	#595
hypocenter	C₁₇	#7182
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₄

#11763683

commutator

C₈×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#322

C₄

C₂×C₁₇

C₂²×D₁₇

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
9. #11763731	C₄	semidirect
14. #11762530	C₄	semidirect
15. C₁₇×#322	C₄	semidirect
23. #10587536	C₈	semidirect
25. #10587534	C₈	semidirect
113. #322	#595	semidirect
140. C₁₇	#7182	semidirect

H ≅ G/N

type

9. #11763731

C₄

semidirect

14. #11762530

C₄

semidirect

15. C₁₇×#322

C₄

semidirect

23. #10587536

C₈

semidirect

25. #10587534

C₈

semidirect

113. #322

#595

semidirect

140. C₁₇

#7182

semidirect

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

subgroup	index
#14545123	2
#14544586	2
D₁₇×#322	2
#7182	17

subgroup

index

#14545123

#14544586

D₁₇×#322

#7182

Subgroups — #17948634

Conjugacy classes of subgroups

Complements

Maximal subgroups