Group #17189825 — subgroups

Subgroups — #17189825

Order 4352.

482 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 187 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	#14539296	C₂	1	2	yes	maximal
2	#14531069	C₂	1	4	yes	maximal, complemented
3	#14531068	C₂	1	4	yes	maximal, complemented
4	#14530568	C₂	1	1	yes	maximal, characteristic
5	#14530566	C₂	1	2	yes	maximal
6	C₂×#11762584	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#2158	C₂	1	1	yes	maximal, characteristic
8	#11763362	—	2	2	no	—
9	C₂²×#63214	C₂²	1	1	yes	characteristic
10	#11763213	C₂²	1	4	yes	—
11	#11763210	C₂²	1	4	yes	—
12	#11763209	—	2	2	no	—
13	#11763206	—	2	2	no	—
14	#11762847	C₂²	1	2	yes	—
15	#11762845	—	2	2	no	—
16	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
17	#11762637	C₂²	1	2	yes	—
18	#11762636	C₂²	1	4	yes	—
19	#11762635	C₂²	1	2	yes	—
20	#11762634	C₂²	1	4	yes	—
21	#11762584	C₂²	1	2	yes	—
22	#11762584	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
24	C₁₇×#427	C₂²	1	2	yes	—
25	C₁₇×#367	C₂²	1	2	yes	—
26	C₁₇×#366	C₂²	1	2	yes	—
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#117	—	2	2	no	—
29	#10587612	—	2	2	no	—
30	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
31	C₂²×#3622	D₄	1	1	yes	characteristic
32	#10587576	—	2	4	no	—
33	#10587574	—	2	4	no	—
34	#10587564	—	4	4	no	—
35	C₄×Dic₃₄	—	4	4	no	—
36	C₂×C₁₆×C₁₇	Q₈	1	4	yes	complemented
37	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
38	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
39	C₁₇×#107	C₂³	1	4	yes	—
40	C₁₇×#106	—	2	2	no	—
41	#10587489	C₂³	1	4	yes	—
42	#10587488	—	2	2	no	—
43	C₈×Dic₁₇	D₄	1	2	yes	—
44	C₈×Dic₁₇	C₂³	1	2	yes	—
45	C₂×#63214	Q₈	1	4	yes	complemented
46	C₂×#63214	C₂³	1	2	yes	—
47	#10587479	—	2	4	no	—
48	#10587478	—	2	4	no	—
49	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₁₇	D₈	1	2	yes	—
51	C₂×Dic₃₄	—	4	4	no	—
52	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₇	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
56	C₄⋊C₄×C₁₇	—	2	4	no	—
57	C₂²⋊C₄×C₁₇	—	4	2	no	—
58	C₄²×C₁₇	—	4	2	no	—
59	C₂²⋊Dic₁₇	—	4	2	no	—
60	C₄⋊Dic₁₇	—	2	4	no	—
61	C₄⋊Dic₁₇	—	4	4	no	—
62	#63225	—	4	4	no	—
63	#63225	—	4	4	no	—
64	C₄×Dic₁₇	—	4	2	no	—
65	C₄×Dic₁₇	D₈	1	2	yes	—
66	C₄×Dic₁₇	C₂×D₄	1	2	yes	—
67	C₂×#3622	Q₁₆	1	4	yes	—
68	C₂×#3622	C₂×D₄	1	2	yes	—
69	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
70	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
71	#63214	C₂×Q₈	1	4	yes	—
72	#33728	—	17	1	no	maximal, Sylow
73	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	—	4	4	no	—
75	C₂×C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×Dic₁₇	—	4	2	no	—
80	C₂×Dic₁₇	—	4	4	no	—
81	C₂×Dic₁₇	C₂×D₈	1	4	yes	—
82	Dic₃₄	—	8	8	no	—
83	C₈×C₁₇	C₄⋊Q₈	1	2	yes	—
84	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₁₇	C₂²×Q₈	1	1	yes	characteristic
86	#3622	C₂×Q₁₆	1	4	yes	—
87	#3160	—	17	2	no	—
88	#2255	—	17	4	no	—
89	#2255	—	17	4	no	—
90	#2158	—	17	1	no	—
91	#2158	—	17	2	no	—
92	#2158	D₁₇	1	1	yes	characteristic
93	C₂×#346	—	17	1	no	—
94	C₂²×C₁₇	—	2	2	no	—
95	C₂²×C₁₇	#500	1	2	yes	—
96	C₂²×C₁₇	C₂²×D₈	1	1	yes	characteristic
97	C₄×C₁₇	—	8	4	no	—
98	C₄×C₁₇	#500	1	2	yes	—
99	C₄×C₁₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₁₇	C₂²×Q₁₆	1	1	yes	characteristic
101	Dic₁₇	—	2	4	no	—
102	Dic₁₇	—	8	4	no	—
103	#533	—	34	2	no	—
104	C₂²×C₁₆	—	17	1	no	—
105	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
106	#500	—	17	4	no	—
107	#499	—	17	4	no	—
108	#499	—	34	2	no	—
109	#498	—	34	2	no	—
110	#427	—	17	2	no	—
111	#427	—	34	2	no	—
112	#427	C₂×D₁₇	1	2	yes	—
113	C₂×C₄×C₈	—	17	1	no	—
114	#367	—	17	2	no	—
115	#367	—	17	4	no	—
116	#367	C₂×D₁₇	1	2	yes	—
117	#366	—	17	2	no	—
118	#366	—	17	4	no	—
119	#366	C₂×D₁₇	1	2	yes	—
120	#346	—	17	2	no	—
121	#346	—	17	2	no	—
122	C₂×C₁₇	—	2	2	no	—
123	C₂×C₁₇	C₂×#500	1	1	yes	characteristic
124	C₂×C₁₇	#3414	1	2	yes	—
125	C₂²×C₈	—	17	1	no	—
126	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
127	C₄⋊Q₈	—	34	4	no	—
128	#125	—	34	4	no	—
129	C₂²⋊Q₈	—	68	4	no	—
130	C₄×Q₈	—	68	4	no	—
131	#117	—	2	2	no	—
132	#117	—	34	2	no	—
133	C₂×C₄²	—	17	1	no	—
134	C₂×C₁₆	—	17	4	no	—
135	C₂×C₁₆	—	17	2	no	—
136	C₂×C₁₆	Dic₃₄	1	4	yes	complemented
137	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
138	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
139	#107	—	17	4	no	—
140	#107	—	34	4	no	—
141	#107	C₂²×D₁₇	1	4	yes	—
142	#106	—	2	2	no	—
143	#106	—	34	4	no	—
144	#106	—	34	2	no	—
145	C₄×C₈	—	17	2	no	—
146	C₄×C₈	—	17	2	no	—
147	C₁₇	#33728	1	1	yes	characteristic, complemented, Sylow
148	C₂×Q₈	—	68	4	no	—
149	C₂²×C₄	—	17	2	no	—
150	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
151	C₂×C₈	—	17	2	no	—
152	C₂×C₈	—	17	4	no	—
153	C₂×C₈	C₂×Dic₃₄	1	2	yes	—
154	C₂×C₈	D₄×D₁₇	1	1	yes	characteristic
155	C₂×C₈	Q₈×D₁₇	1	2	yes	—
156	C₂×C₈	C₂³×D₁₇	1	1	yes	characteristic
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	34	4	no	—
159	C₄⋊C₄	—	68	4	no	—
160	C₄⋊C₄	—	68	4	no	—
161	C₄⋊C₄	—	68	4	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	68	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	17	2	no	—
166	C₄²	—	17	2	no	—
167	C₄²	—	68	2	no	—
168	C₁₆	—	17	4	no	—
169	C₁₆	C₂×Dic₃₄	1	2	yes	—
170	C₁₆	C₂×Dic₃₄	1	2	yes	—
171	C₂³	D₈×D₁₇	1	1	yes	characteristic
172	Q₈	—	136	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	17	4	no	—
175	C₂×C₄	—	68	2	no	—
176	C₂×C₄	—	68	4	no	—
177	C₂×C₄	#10587574	1	2	yes	—
178	C₂×C₄	D₈×D₁₇	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₁₇	1	2	yes	—
180	C₂×C₄	C₂×D₄×D₁₇	1	1	yes	characteristic
181	C₈	—	17	4	no	—
182	C₈	#10587574	1	2	yes	—
183	C₈	C₂²×Dic₃₄	1	1	yes	characteristic
184	C₈	C₂×Q₈×D₁₇	1	1	yes	characteristic
185	C₂²	—	2	2	no	—
186	C₂²	#11763213	1	2	yes	—
187	C₂²	C₂×D₈×D₁₇	1	1	yes	characteristic
188	C₄	—	8	4	no	—
189	C₄	—	34	4	no	—
190	C₄	—	136	4	no	—
191	C₄	#11763213	1	2	yes	—
192	C₄	C₂×#10587574	1	1	yes	characteristic
193	C₄	C₂×Q₁₆×D₁₇	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₂×#11763213	1	1	yes	characteristic
196	C₂	#14541917	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14539296

C₂

yes

maximal

#14531069

C₂

yes

maximal, complemented

#14531068

C₂

yes

maximal, complemented

#14530568

C₂

yes

maximal, characteristic

#14530566

C₂

yes

maximal

C₂×#11762584

C₂

yes

maximal, characteristic

C₁₇×#2158

C₂

yes

maximal, characteristic

#11763362

—

C₂²×#63214

C₂²

yes

characteristic

#11763213

C₂²

yes

—

#11763210

C₂²

yes

—

#11763209

—

#11763206

—

#11762847

C₂²

yes

—

#11762845

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762637

C₂²

yes

—

#11762636

C₂²

yes

—

#11762635

C₂²

yes

—

#11762634

C₂²

yes

—

#11762584

C₂²

yes

—

#11762584

C₂²

yes

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₁₇×#427

C₂²

yes

—

C₁₇×#367

C₂²

yes

—

C₁₇×#366

C₂²

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

—

#10587612

—

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

D₄

yes

—

—

—

C₄×Dic₃₄

—

C₂×C₁₆×C₁₇

Q₈

yes

complemented

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₂×C₁₆×C₁₇

C₂³

yes

characteristic

C₁₇×#107

C₂³

yes

—

C₁₇×#106

—

#10587489

C₂³

yes

—

#10587488

—

C₈×Dic₁₇

D₄

yes

—

C₈×Dic₁₇

C₂³

yes

—

C₂×#63214

Q₈

yes

complemented

C₂×#63214

C₂³

yes

—

#10587479

—

#10587478

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

D₈

yes

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

—

C₂×C₈×C₁₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

D₈

yes

—

C₄×Dic₁₇

C₂×D₄

yes

—

C₂×#3622

Q₁₆

yes

—

C₂×#3622

C₂×D₄

yes

—

C₁₆×C₁₇

C₂×Q₈

yes

—

C₁₆×C₁₇

C₂×Q₈

yes

—

#63214

C₂×Q₈

yes

—

#33728

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₈

yes

—

Dic₃₄

—

C₈×C₁₇

C₄⋊Q₈

yes

—

C₈×C₁₇

C₂²×Q₈

yes

characteristic

C₈×C₁₇

C₂²×Q₈

yes

characteristic

#3622

C₂×Q₁₆

yes

—

—

—

—

—

—

D₁₇

yes

characteristic

C₂×#346

—

C₂²×C₁₇

—

C₂²×C₁₇

#500

yes

—

C₂²×C₁₇

C₂²×D₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

#500

yes

—

C₄×C₁₇

C₂×C₄⋊Q₈

yes

characteristic

100

C₄×C₁₇

C₂²×Q₁₆

yes

characteristic

101

Dic₁₇

—

102

Dic₁₇

—

103

#533

—

104

C₂²×C₁₆

—

105

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

106

#500

—

107

#499

—

108

#499

—

109

#498

—

110

#427

—

111

#427

—

112

#427

C₂×D₁₇

yes

—

113

C₂×C₄×C₈

—

114

#367

—

115

#367

—

116

#367

C₂×D₁₇

yes

—

117

#366

—

118

#366

—

119

#366

C₂×D₁₇

yes

—

120

#346

—

121

#346

—

122

C₂×C₁₇

—

123

C₂×C₁₇

C₂×#500

yes

characteristic

124

C₂×C₁₇

#3414

yes

—

125

C₂²×C₈

—

126

C₂²×C₈

C₂²×D₁₇

yes

characteristic

127

C₄⋊Q₈

—

128

#125

—

129

C₂²⋊Q₈

—

130

C₄×Q₈

—

131

#117

—

132

#117

—

133

C₂×C₄²

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

Dic₃₄

yes

complemented

137

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

138

C₂×C₁₆

C₂²×D₁₇

yes

characteristic

139

#107

—

140

#107

—

141

#107

C₂²×D₁₇

yes

—

142

#106

—

143

#106

—

144

#106

—

145

C₄×C₈

—

146

C₄×C₈

—

147

C₁₇

#33728

yes

characteristic, complemented, Sylow

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

D₄×D₁₇

yes

characteristic

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

C₂×Dic₃₄

yes

—

154

C₂×C₈

D₄×D₁₇

yes

characteristic

155

C₂×C₈

Q₈×D₁₇

yes

—

156

C₂×C₈

C₂³×D₁₇

yes

characteristic

157

C₄⋊C₄

—

158

C₄⋊C₄

—

159

C₄⋊C₄

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₂²⋊C₄

—

163

C₂²⋊C₄

—

164

C₄²

—

165

C₄²

—

166

C₄²

—

167

C₄²

—

168

C₁₆

—

169

C₁₆

C₂×Dic₃₄

yes

—

170

C₁₆

C₂×Dic₃₄

yes

—

171

C₂³

D₈×D₁₇

yes

characteristic

172

Q₈

—

136

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

#10587574

yes

—

178

C₂×C₄

D₈×D₁₇

yes

characteristic

179

C₂×C₄

Q₁₆×D₁₇

yes

—

180

C₂×C₄

C₂×D₄×D₁₇

yes

characteristic

181

C₈

—

182

C₈

#10587574

yes

—

183

C₈

C₂²×Dic₃₄

yes

characteristic

184

C₈

C₂×Q₈×D₁₇

yes

characteristic

185

C₂²

—

186

C₂²

#11763213

yes

—

187

C₂²

C₂×D₈×D₁₇

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

136

—

191

C₄

#11763213

yes

—

192

C₄

C₂×#10587574

yes

characteristic

193

C₄

C₂×Q₁₆×D₁₇

yes

characteristic

194

C₂

—

195

C₂

C₂×#11763213

yes

characteristic

196

C₂

#14541917

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₇
Fitting	C₁₇×#2158	C₂
socle	C₂²×C₁₇	C₂²×D₈
prime radical	C₂³×C₁₇	C₂×D₈
hypercenter	#2158	D₁₇
hypocenter	C₁₇	#33728
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

subgroup

G/H

center

C₂×C₄

D₈×D₁₇

commutator, cosocle

C₂×C₈×C₁₇

C₂⁴

Frattini

C₂×C₈

C₂³×D₁₇

Fitting

C₁₇×#2158

C₂

socle

C₂²×C₁₇

C₂²×D₈

prime radical

C₂³×C₁₇

C₂×D₈

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

N	H ≅ G/N	type
2. #14531069	C₂	semidirect
3. #14531068	C₂	semidirect
36. C₂×C₁₆×C₁₇	Q₈	semidirect
45. C₂×#63214	Q₈	semidirect
136. C₂×C₁₆	Dic₃₄	semidirect
147. C₁₇	#33728	semidirect

H ≅ G/N

type

2. #14531069

C₂

semidirect

3. #14531068

C₂

semidirect

36. C₂×C₁₆×C₁₇

Q₈

semidirect

45. C₂×#63214

Q₈

semidirect

136. C₂×C₁₆

Dic₃₄

semidirect

147. C₁₇

#33728

semidirect

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

subgroup	index
#14539296	2
#14531069	2
#14531068	2
#14530568	2
#14530566	2
C₂×#11762584	2
C₁₇×#2158	2
#33728	17

subgroup

index

C₁₇×#2158

Subgroups — #17189825

Conjugacy classes of subgroups

Complements

Maximal subgroups