Group #17948590 — subgroups

Subgroups — #17948590

Order 4352.

266 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 82 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	#14545117	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊#595	C₂	1	1	yes	maximal, characteristic
3	#14530377	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587530	—	2	1	no	—
5	#11763810	C₄	1	1	yes	characteristic
6	C₂×C₈×#595	—	2	1	no	—
7	C₄×C₄⋊#595	C₄	1	1	yes	characteristic
8	#11763730	—	2	1	no	—
9	C₄⋊#63216	C₄	1	2	yes	complemented
10	#11763674	—	2	1	no	—
11	C₂×#10587469	—	2	1	no	—
12	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
13	#11762585	C₄	1	1	yes	characteristic, complemented
14	C₁₆×Dic₁₇	—	2	1	no	—
15	C₁₇×#346	C₄	1	1	yes	characteristic, complemented
16	C₂×C₄⋊#595	—	2	1	no	—
17	C₂×C₄×#595	—	2	1	no	—
18	C₂×#63242	—	2	1	no	—
19	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
20	C₂×C₈×D₁₇	D₄	1	1	yes	characteristic
21	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
22	(C₂×#595)⋊C₄	C₈	1	2	yes	—
23	#10587544	C₈	1	2	yes	—
24	C₄⋊#3624	—	2	1	no	—
25	C₂×#63216	—	2	2	no	—
26	C₈×#595	—	4	1	no	—
27	C₈×#595	—	4	1	no	—
28	#10587530	—	4	1	no	—
29	#10587530	—	4	1	no	—
30	C₂×C₁₆×C₁₇	—	2	1	no	—
31	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₁₇	D₄	1	1	yes	characteristic
33	C₈×Dic₁₇	Q₈	1	1	yes	characteristic
34	C₂×#63214	—	2	1	no	—
35	C₄×#3622	C₂×C₄	1	1	yes	characteristic
36	#10587469	—	2	2	no	—
37	#10587469	—	2	2	no	—
38	C₂²×#595	—	2	2	no	—
39	C₂×C₄×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×D₁₇	M₁₆	1	1	yes	characteristic
42	C₂×#3624	—	2	2	no	—
43	C₄⋊#595	—	2	4	no	—
44	C₄×#595	—	4	1	no	—
45	C₄×#595	—	4	1	no	—
46	#63242	—	4	1	no	—
47	#63242	—	4	1	no	—
48	C₂×C₈×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄²×C₁₇	C₄²	1	1	yes	characteristic
51	C₄×Dic₁₇	C₄⋊C₄	1	1	yes	characteristic
52	C₄×Dic₁₇	C₂×C₈	1	1	yes	characteristic
53	C₄×Dic₁₇	M₁₆	1	1	yes	characteristic
54	C₂×#3622	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×#3622	C₄⋊C₄	1	1	yes	characteristic
56	C₈×D₁₇	—	2	1	no	—
57	C₈×D₁₇	—	2	1	no	—
58	C₈×D₁₇	D₈	1	1	yes	characteristic
59	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
60	C₈×D₁₇	SD₁₆	1	1	yes	characteristic
61	C₈×D₁₇	Q₁₆	1	1	yes	characteristic
62	#63216	—	4	2	no	—
63	C₁₆×C₁₇	—	2	2	no	—
64	#63214	—	2	2	no	—
65	#7169	—	17	1	no	maximal, Sylow
66	C₂²×D₁₇	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×#595	—	2	4	no	—
68	C₂×#595	—	4	2	no	—
69	C₂×#595	—	4	2	no	—
70	C₂×C₄×C₁₇	#95	1	1	yes	characteristic
71	C₂×C₄×C₁₇	C₄×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#97	1	1	yes	characteristic
73	C₂×Dic₁₇	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
76	C₄×D₁₇	—	2	2	no	—
77	C₄×D₁₇	—	4	1	no	—
78	C₄×D₁₇	#102	1	1	yes	characteristic
79	C₄×D₁₇	#103	1	1	yes	characteristic
80	C₄×D₁₇	#104	1	1	yes	characteristic
81	C₄×D₁₇	#104	1	1	yes	characteristic
82	#3624	—	4	2	no	—
83	C₈×C₁₇	—	2	1	no	—
84	C₈×C₁₇	#106	1	1	yes	characteristic
85	C₈×C₁₇	#107	1	1	yes	characteristic
86	#3622	—	2	1	no	—
87	#3622	#102	1	1	yes	characteristic
88	#3622	#103	1	1	yes	characteristic
89	#2150	—	17	1	no	—
90	C₄×M₃₂	—	17	1	no	—
91	C₈×C₄⋊C₄	—	17	1	no	—
92	C₂²×C₁₇	#336	1	1	yes	characteristic
93	C₂×D₁₇	—	2	1	no	—
94	C₂×D₁₇	—	2	1	no	—
95	C₂×D₁₇	D₄⋊C₈	1	1	yes	characteristic
96	C₂×D₁₇	Q₈⋊C₈	1	1	yes	characteristic
97	#595	—	4	4	no	—
98	C₄×C₁₇	—	2	1	no	—
99	C₄×C₁₇	—	2	1	no	—
100	C₄×C₁₇	#339	1	1	yes	characteristic
101	C₄×C₁₇	#340	1	1	yes	characteristic
102	Dic₁₇	—	2	1	no	—
103	Dic₁₇	D₄⋊C₈	1	1	yes	characteristic
104	Dic₁₇	Q₈⋊C₈	1	1	yes	characteristic
105	Dic₁₇	#364	1	2	yes	—
106	C₂×M₃₂	—	34	1	no	—
107	C₂×M₃₂	—	34	1	no	—
108	C₂×C₄⋊C₈	—	17	1	no	—
109	C₂×C₄×C₈	—	17	1	no	—
110	C₂×C₄×C₈	—	34	1	no	—
111	C₄×C₄⋊C₄	—	17	1	no	—
112	C₄⋊C₁₆	—	17	2	no	—
113	C₄⋊C₁₆	—	34	1	no	—
114	#346	—	17	1	no	—
115	#346	#595	1	1	yes	characteristic, complemented
116	C₄×C₁₆	—	34	1	no	—
117	#336	—	34	1	no	—
118	C₂×C₁₇	#1279	1	1	yes	characteristic
119	C₂×C₁₇	#1376	1	1	yes	characteristic
120	C₂×C₁₇	#1378	1	1	yes	characteristic
121	D₁₇	—	2	1	no	—
122	D₁₇	—	2	1	no	—
123	C₂²×C₈	—	17	1	no	—
124	C₂²×C₈	—	17	1	no	—
125	C₂²×C₈	—	34	1	no	—
126	C₂×C₄⋊C₄	—	34	1	no	—
127	C₂×C₄²	—	17	1	no	—
128	C₂×C₄²	—	34	1	no	—
129	M₃₂	—	34	2	no	—
130	M₃₂	—	34	2	no	—
131	M₃₂	—	68	1	no	—
132	M₃₂	—	68	1	no	—
133	C₂×C₁₆	—	2	1	no	—
134	C₂×C₁₆	—	34	1	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₄⋊C₈	—	17	2	no	—
137	C₄⋊C₈	—	34	1	no	—
138	C₄×C₈	—	17	1	no	—
139	C₄×C₈	—	17	1	no	—
140	C₄×C₈	—	17	1	no	—
141	C₄×C₈	—	68	1	no	—
142	C₄×C₈	—	68	1	no	—
143	C₄×C₈	C₂×#595	1	1	yes	characteristic
144	#95	—	17	2	no	—
145	C₁₇	#7169	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₄	—	17	1	no	—
147	C₂²×C₄	—	17	1	no	—
148	C₂²×C₄	—	17	1	no	—
149	C₂²×C₄	—	34	2	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	1	no	—
157	C₂×C₈	—	34	1	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	—	68	1	no	—
160	C₂×C₈	—	68	1	no	—
161	C₂×C₈	C₄⋊#595	1	1	yes	characteristic
162	C₂×C₈	C₂²⋊#595	1	1	yes	characteristic
163	C₄⋊C₄	—	34	4	no	—
164	C₄²	—	17	1	no	—
165	C₄²	—	17	1	no	—
166	C₄²	—	17	1	no	—
167	C₄²	—	68	1	no	—
168	C₄²	—	68	1	no	—
169	C₄²	C₄×#595	1	1	yes	characteristic
170	C₁₆	—	2	2	no	—
171	C₁₆	—	34	2	no	—
172	C₁₆	—	68	2	no	—
173	C₂³	—	17	1	no	—
174	C₂×C₄	—	17	1	no	—
175	C₂×C₄	—	17	1	no	—
176	C₂×C₄	—	17	1	no	—
177	C₂×C₄	—	17	1	no	—
178	C₂×C₄	—	17	1	no	—
179	C₂×C₄	—	17	1	no	—
180	C₂×C₄	—	17	1	no	—
181	C₂×C₄	—	34	2	no	—
182	C₂×C₄	—	34	4	no	—
183	C₂×C₄	—	68	1	no	—
184	C₂×C₄	—	68	2	no	—
185	C₂×C₄	—	68	2	no	—
186	C₂×C₄	C₈×#595	1	1	yes	characteristic
187	C₂×C₄	#10587532	1	1	yes	characteristic
188	C₂×C₄	(C₂×#595)⋊C₄	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	17	1	no	—
191	C₈	—	17	1	no	—
192	C₈	—	34	1	no	—
193	C₈	—	68	2	no	—
194	C₈	#10587533	1	1	yes	characteristic
195	C₈	#10587534	1	1	yes	characteristic
196	C₂²	—	17	1	no	—
197	C₂²	—	17	1	no	—
198	C₂²	—	34	1	no	—
199	C₂²	—	34	1	no	—
200	C₂²	#11763674	1	1	yes	characteristic
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	17	1	no	—
204	C₄	—	17	1	no	—
205	C₄	—	17	2	no	—
206	C₄	—	34	1	no	—
207	C₄	—	68	4	no	—
208	C₄	#11763680	1	1	yes	characteristic
209	C₄	#11763683	1	1	yes	characteristic
210	C₂	—	34	1	no	—
211	C₂	—	34	1	no	—
212	C₂	#14543619	1	1	yes	characteristic
213	C₂	#14543827	1	1	yes	characteristic
214	C₂	#14543836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545117

C₂

yes

maximal, characteristic

C₈×C₄⋊#595

C₂

yes

maximal, characteristic

#14530377

C₂

yes

maximal, characteristic

C₂×#10587530

—

#11763810

C₄

yes

characteristic

C₂×C₈×#595

—

C₄×C₄⋊#595

C₄

yes

characteristic

#11763730

—

C₄⋊#63216

C₄

yes

complemented

#11763674

—

C₂×#10587469

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

#11762585

C₄

yes

characteristic, complemented

C₁₆×Dic₁₇

—

C₁₇×#346

C₄

yes

characteristic, complemented

C₂×C₄⋊#595

—

C₂×C₄×#595

—

C₂×#63242

—

C₂×C₈×D₁₇

D₄

yes

characteristic

C₂×C₈×D₁₇

D₄

yes

characteristic

C₄²×D₁₇

C₂×C₄

yes

characteristic

(C₂×#595)⋊C₄

C₈

yes

—

#10587544

C₈

yes

—

—

—

—

—

—

—

C₂×C₁₆×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₈×Dic₁₇

D₄

yes

characteristic

C₈×Dic₁₇

Q₈

yes

characteristic

C₂×#63214

—

C₄×#3622

C₂×C₄

yes

characteristic

#10587469

—

#10587469

—

C₂²×#595

—

C₂×C₄×D₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

characteristic

C₂×C₄×D₁₇

M₁₆

yes

—

—

—

—

—

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₄²×C₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₄⋊C₄

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

characteristic

C₄×Dic₁₇

M₁₆

yes

characteristic

C₂×#3622

C₂²⋊C₄

yes

characteristic

C₂×#3622

C₄⋊C₄

yes

characteristic

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

D₈

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

SD₁₆

yes

characteristic

C₈×D₁₇

Q₁₆

yes

characteristic

#63216

—

C₁₆×C₁₇

—

#63214

—

#7169

—

maximal, Sylow

C₂²×D₁₇

C₂²⋊C₈

yes

—

—

—

C₂×C₄×C₁₇

#95

yes

characteristic

C₂×C₄×C₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

#97

yes

characteristic

C₂×Dic₁₇

C₂²⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₂×Dic₁₇

C₄⋊C₈

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

C₄×D₁₇

yes

—

C₈×C₁₇

—

C₈×C₁₇

yes

C₈×C₁₇

yes

—

yes

yes

—

C₄×M₃₂

—

C₈×C₄⋊C₄

—

C₂²×C₁₇

#336

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

D₄⋊C₈

yes

characteristic

C₂×D₁₇

Q₈⋊C₈

yes

characteristic

#595

—

C₄×C₁₇

—

C₄×C₁₇

—

100

C₄×C₁₇

#339

yes

characteristic

101

C₄×C₁₇

#340

yes

characteristic

102

Dic₁₇

—

103

Dic₁₇

D₄⋊C₈

yes

characteristic

104

Dic₁₇

Q₈⋊C₈

yes

characteristic

105

Dic₁₇

#364

yes

—

106

C₂×M₃₂

—

107

C₂×M₃₂

—

108

C₂×C₄⋊C₈

—

109

C₂×C₄×C₈

—

110

C₂×C₄×C₈

—

111

C₄×C₄⋊C₄

—

112

C₄⋊C₁₆

—

113

C₄⋊C₁₆

—

114

#346

—

115

#346

#595

yes

characteristic, complemented

116

C₄×C₁₆

—

117

#336

—

118

C₂×C₁₇

#1279

yes

characteristic

119

C₂×C₁₇

#1376

yes

characteristic

120

C₂×C₁₇

#1378

yes

characteristic

121

D₁₇

—

122

D₁₇

—

123

C₂²×C₈

—

124

C₂²×C₈

—

125

C₂²×C₈

—

126

C₂×C₄⋊C₄

—

127

C₂×C₄²

—

128

C₂×C₄²

—

129

M₃₂

—

130

M₃₂

—

131

M₃₂

—

132

M₃₂

—

133

C₂×C₁₆

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄×C₈

—

139

C₄×C₈

—

140

C₄×C₈

—

141

C₄×C₈

—

142

C₄×C₈

—

143

C₄×C₈

C₂×#595

yes

characteristic

144

#95

—

145

C₁₇

#7169

yes

characteristic, complemented, Sylow

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

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

—

160

C₂×C₈

—

161

C₂×C₈

C₄⋊#595

yes

characteristic

162

C₂×C₈

C₂²⋊#595

yes

characteristic

163

C₄⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₄²

—

169

C₄²

C₄×#595

yes

characteristic

170

C₁₆

—

171

C₁₆

—

172

C₁₆

—

173

C₂³

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

185

C₂×C₄

—

186

C₂×C₄

C₈×#595

yes

characteristic

187

C₂×C₄

#10587532

yes

characteristic

188

C₂×C₄

(C₂×#595)⋊C₄

yes

characteristic

189

C₈

—

190

C₈

—

191

C₈

—

192

C₈

—

193

C₈

—

194

C₈

#10587533

yes

characteristic

195

C₈

#10587534

yes

characteristic

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#11763674

yes

characteristic

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

—

207

C₄

—

208

C₄

#11763680

yes

characteristic

209

C₄

#11763683

yes

characteristic

210

C₂

—

211

C₂

—

212

C₂

#14543619

yes

characteristic

213

C₂

#14543827

yes

characteristic

214

C₂

#14543836

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×#595)⋊C₄
commutator	C₂×C₄×C₁₇	C₄×C₈
Frattini	C₄×C₈	C₂×#595
Fitting	C₁₇×#346	C₄
socle	C₂²×C₁₇	#336
prime radical	C₂²×D₁₇	C₂²⋊C₈
cosocle	C₄×C₈×D₁₇	C₂²
hypercenter	#346	#595
hypocenter	C₁₇	#7169
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₂×C₄

(C₂×#595)⋊C₄

commutator

C₂×C₄×C₁₇

C₄×C₈

Frattini

C₄×C₈

C₂×#595

Fitting

C₁₇×#346

C₄

socle

C₂²×C₁₇

#336

prime radical

C₂²×D₁₇

C₂²⋊C₈

cosocle

C₄×C₈×D₁₇

C₂²

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
9. C₄⋊#63216	C₄	semidirect
13. #11762585	C₄	semidirect
15. C₁₇×#346	C₄	semidirect
115. #346	#595	semidirect
145. C₁₇	#7169	semidirect

H ≅ G/N

type

9. C₄⋊#63216

C₄

semidirect

13. #11762585

C₄

semidirect

15. C₁₇×#346

C₄

semidirect

115. #346

#595

semidirect

145. C₁₇

#7169

semidirect

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

subgroup	index
#14545117	2
C₈×C₄⋊#595	2
#14530377	2
#7169	17

subgroup

index

#14545117

C₈×C₄⋊#595

#14530377

#7169

Subgroups — #17948590

Conjugacy classes of subgroups

Complements

Maximal subgroups