Group #16933781 — subgroups

Subgroups — #16933781

Order 4352.

414 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×#11763214	C₂	1	1	yes	maximal, characteristic
2	#14527470	C₂	1	2	yes	maximal
3	#14527466	C₂	1	2	yes	maximal
4	#14527067	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587555	C₂²	1	2	yes	—
7	#11763214	—	2	1	no	—
8	#11763214	—	2	1	no	—
9	#11763214	C₄	1	4	yes	—
10	#11762862	C₂²	1	1	yes	characteristic
11	#11762862	C₂²	1	1	yes	characteristic
12	C₂×#10587489	—	2	2	no	—
13	#11762557	—	2	4	no	—
14	#11762554	—	2	4	no	—
15	#11762530	—	2	1	no	—
16	C₈×#3622	—	2	1	no	—
17	C₂×C₁₇×#108	—	2	2	no	—
18	C₁₇×#423	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₃₄	—	2	2	no	—
21	C₂×M₁₆×C₁₇	—	2	2	no	—
22	C₂²×C₈×C₁₇	D₄	1	2	yes	—
23	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₇	—	2	2	no	—
25	C₂×C₄⋊Dic₁₇	—	2	2	no	—
26	#10587555	—	2	2	no	—
27	#10587555	—	2	2	no	—
28	#10587555	—	2	2	no	—
29	#10587555	C₂×C₄	1	4	yes	—
30	C₁₇×#108	—	4	4	no	—
31	C₁₇×C₄⋊C₈	—	2	2	no	—
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
36	C₂²⋊#3622	—	2	1	no	—
37	C₂²⋊#3622	—	2	1	no	—
38	#10587489	—	2	4	no	—
39	#10587489	—	2	4	no	—
40	C₄⋊#3622	D₄	1	2	yes	—
41	C₄⋊#3622	D₄	1	2	yes	—
42	C₄×#3622	—	2	1	no	—
43	C₄×#3622	—	2	1	no	—
44	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
46	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
47	C₂²×Dic₁₇	—	4	2	no	—
48	C₂×Dic₃₄	—	2	4	no	—
49	C₂×Dic₃₄	—	4	2	no	—
50	C₂×Dic₃₄	—	8	4	no	—
51	M₁₆×C₁₇	—	4	4	no	—
52	C₂×C₈×C₁₇	—	4	2	no	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₇	SD₁₆	1	4	yes	—
56	C₂×C₈×C₁₇	Q₁₆	1	4	yes	—
57	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊Dic₁₇	—	2	4	no	—
62	C₄⋊Dic₁₇	—	2	4	no	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	—	2	2	no	—
67	#9771	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	#124	1	1	yes	characteristic
69	C₂×C₄×C₁₇	—	2	2	no	—
70	C₂×C₄×C₁₇	—	2	2	no	—
71	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
79	C₂×Dic₁₇	—	4	4	no	—
80	C₂×Dic₁₇	—	8	4	no	—
81	Dic₃₄	—	8	4	no	—
82	Dic₃₄	—	8	4	no	—
83	C₈×C₁₇	—	8	2	no	—
84	C₈×C₁₇	#103	1	4	yes	—
85	C₈×C₁₇	#103	1	4	yes	—
86	#3622	—	4	2	no	—
87	#3622	—	4	2	no	—
88	C₂×#500	—	17	1	no	—
89	#1848	D₁₇	1	1	yes	characteristic
90	#1549	—	17	2	no	—
91	#1548	—	17	2	no	—
92	#1450	—	17	1	no	—
93	C₂²×C₁₇	—	2	1	no	—
94	C₂²×C₁₇	—	2	1	no	—
95	C₂²×C₁₇	#390	1	1	yes	characteristic
96	C₂²×C₁₇	#487	1	2	yes	—
97	C₄×C₁₇	—	2	1	no	—
98	C₄×C₁₇	—	2	1	no	—
99	C₄×C₁₇	C₂×#103	1	2	yes	—
100	C₄×C₁₇	#472	1	2	yes	—
101	C₄×C₁₇	#473	1	2	yes	—
102	C₄×C₁₇	#492	1	1	yes	characteristic
103	C₄×C₁₇	#494	1	1	yes	characteristic
104	Dic₁₇	—	8	4	no	—
105	C₂×C₄⋊Q₈	—	17	2	no	—
106	#500	—	17	4	no	—
107	#500	—	34	1	no	—
108	#500	—	34	1	no	—
109	#431	—	17	1	no	—
110	#431	—	17	1	no	—
111	C₂×#108	—	2	2	no	—
112	C₂×#107	—	34	2	no	—
113	#423	C₂×D₁₇	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
115	#333	—	34	4	no	—
116	#332	—	34	4	no	—
117	#322	—	34	1	no	—
118	C₈²	—	34	1	no	—
119	C₂×C₁₇	—	2	2	no	—
120	C₂×C₁₇	#1956	1	1	yes	characteristic
121	C₂×C₁₇	#1978	1	1	yes	characteristic
122	C₂×C₁₇	#1979	1	1	yes	characteristic
123	C₂²×Q₈	—	34	2	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
126	C₄⋊Q₈	—	17	4	no	—
127	C₄⋊Q₈	—	34	2	no	—
128	C₄⋊Q₈	—	34	2	no	—
129	C₄⋊Q₈	—	34	2	no	—
130	C₂×C₄⋊C₄	—	34	2	no	—
131	C₂×C₄⋊C₄	—	34	2	no	—
132	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
133	#108	—	4	4	no	—
134	#107	—	34	4	no	—
135	#107	—	34	4	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	17	2	no	—
139	C₄⋊C₈	—	17	2	no	—
140	C₂²⋊C₈	—	34	1	no	—
141	C₂²⋊C₈	—	34	1	no	—
142	C₄×C₈	—	34	1	no	—
143	C₄×C₈	—	34	1	no	—
144	C₄×C₈	C₄×D₁₇	1	2	yes	—
145	C₄×C₈	D₆₈	1	1	yes	characteristic
146	C₄×C₈	D₆₈	1	1	yes	characteristic
147	C₁₇	#9771	1	1	yes	characteristic, complemented, Sylow
148	C₂×Q₈	—	34	4	no	—
149	C₂×Q₈	—	68	2	no	—
150	C₂×Q₈	—	136	4	no	—
151	C₂²×C₄	—	68	2	no	—
152	C₂²×C₄	#63256	1	1	yes	characteristic
153	C₂²×C₄	#63256	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
155	M₁₆	—	4	4	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	2	no	—
159	C₂×C₈	#63227	1	2	yes	—
160	C₂×C₈	#63227	1	2	yes	—
161	C₂×C₈	#63229	1	4	yes	—
162	C₂×C₈	#63231	1	4	yes	—
163	C₄⋊C₄	—	34	4	no	—
164	C₄⋊C₄	—	34	4	no	—
165	C₄⋊C₄	—	68	2	no	—
166	C₄⋊C₄	—	68	2	no	—
167	C₄²	C₂×C₄×D₁₇	1	1	yes	characteristic
168	C₄²	C₂×D₆₈	1	1	yes	characteristic
169	C₄²	D₄×D₁₇	1	1	yes	characteristic
170	C₄²	D₄×D₁₇	1	1	yes	characteristic
171	C₂³	#10587633	1	1	yes	characteristic
172	Q₈	—	136	4	no	—
173	Q₈	—	136	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	68	4	no	—
177	C₂×C₄	—	136	4	no	—
178	C₂×C₄	#10587579	1	1	yes	characteristic
179	C₂×C₄	#10587579	1	1	yes	characteristic
180	C₂×C₄	#10587581	1	1	yes	characteristic
181	C₂×C₄	#10587581	1	1	yes	characteristic
182	C₂×C₄	C₂×#63227	1	1	yes	characteristic
183	C₂×C₄	C₂×#63229	1	2	yes	—
184	C₂×C₄	#10587626	1	1	yes	characteristic
185	C₂×C₄	C₂×#63231	1	2	yes	—
186	C₈	—	8	2	no	—
187	C₈	—	68	2	no	—
188	C₈	—	68	2	no	—
189	C₈	#10587481	1	4	yes	—
190	C₈	#10587481	1	4	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	#11762710	1	1	yes	characteristic
194	C₂²	#11763161	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	136	4	no	—
198	C₄	C₂×#10587481	1	2	yes	—
199	C₄	#11763080	1	2	yes	—
200	C₄	#11763085	1	2	yes	—
201	C₄	#11763181	1	1	yes	characteristic
202	C₄	#11763188	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#14529585	1	1	yes	characteristic
205	C₂	#14529692	1	1	yes	characteristic
206	C₂	#14529697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763214

C₂

yes

maximal, characteristic

#14527470

C₂

yes

maximal

#14527466

C₂

yes

maximal

#14527067

C₂

yes

maximal, characteristic

C₁₇×#1848

C₂

yes

maximal, characteristic

C₂×#10587555

C₂²

yes

—

#11763214

—

#11763214

—

#11763214

C₄

yes

—

#11762862

C₂²

yes

characteristic

#11762862

C₂²

yes

—

—

—

—

—

C₂×C₁₇×#108

—

C₁₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Dic₃₄

—

C₂×M₁₆×C₁₇

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

—

—

—

—

C₂×C₄

yes

—

C₁₇×#108

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

characteristic

C₄×C₈×C₁₇

D₄

yes

characteristic

C₂²⋊#3622

—

C₂²⋊#3622

—

#10587489

—

#10587489

—

C₄⋊#3622

D₄

yes

—

C₄⋊#3622

D₄

yes

—

C₄×#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×D₄

yes

C₂²×C₄×C₁₇

yes

C₂²×C₄×C₁₇

yes

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

Q₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

#103

yes

—

C₈×C₁₇

#103

yes

—

—

—

—

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#390

yes

characteristic

C₂²×C₁₇

#487

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#103

yes

—

100

C₄×C₁₇

#472

yes

—

101

C₄×C₁₇

#473

yes

—

102

C₄×C₁₇

#492

yes

characteristic

103

C₄×C₁₇

#494

yes

characteristic

104

Dic₁₇

—

105

C₂×C₄⋊Q₈

—

106

#500

—

107

#500

—

108

#500

—

109

#431

—

110

#431

—

111

C₂×#108

—

112

C₂×#107

—

113

#423

C₂×D₁₇

yes

—

114

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

115

#333

—

116

#332

—

117

#322

—

118

C₈²

—

119

C₂×C₁₇

—

120

C₂×C₁₇

#1956

yes

characteristic

121

C₂×C₁₇

#1978

yes

characteristic

122

C₂×C₁₇

#1979

yes

characteristic

123

C₂²×Q₈

—

124

C₂×M₁₆

—

125

C₂²×C₈

C₂²⋊D₁₇

yes

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

—

128

C₄⋊Q₈

—

129

C₄⋊Q₈

—

130

C₂×C₄⋊C₄

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄²

C₂²×D₁₇

yes

characteristic

133

#108

—

134

#107

—

135

#107

—

136

C₄⋊C₈

—

137

C₄⋊C₈

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₂²⋊C₈

—

141

C₂²⋊C₈

—

142

C₄×C₈

—

143

C₄×C₈

—

144

C₄×C₈

C₄×D₁₇

yes

—

145

C₄×C₈

D₆₈

yes

characteristic

146

C₄×C₈

D₆₈

yes

characteristic

147

C₁₇

#9771

yes

characteristic, complemented, Sylow

148

C₂×Q₈

—

149

C₂×Q₈

—

150

C₂×Q₈

—

136

—

151

C₂²×C₄

—

152

C₂²×C₄

#63256

yes

characteristic

153

C₂²×C₄

#63256

yes

characteristic

154

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

155

M₁₆

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

#63227

yes

—

160

C₂×C₈

#63227

yes

—

161

C₂×C₈

#63229

yes

—

162

C₂×C₈

#63231

yes

—

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₄²

C₂×C₄×D₁₇

yes

characteristic

168

C₄²

C₂×D₆₈

yes

characteristic

169

C₄²

D₄×D₁₇

yes

characteristic

170

C₄²

D₄×D₁₇

yes

characteristic

171

C₂³

#10587633

yes

characteristic

172

Q₈

—

136

—

173

Q₈

—

136

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

136

—

178

C₂×C₄

#10587579

yes

characteristic

179

C₂×C₄

#10587579

yes

characteristic

180

C₂×C₄

#10587581

yes

characteristic

181

C₂×C₄

#10587581

yes

characteristic

182

C₂×C₄

C₂×#63227

yes

characteristic

183

C₂×C₄

C₂×#63229

yes

—

184

C₂×C₄

#10587626

yes

characteristic

185

C₂×C₄

C₂×#63231

yes

—

186

C₈

—

187

C₈

—

188

C₈

—

189

C₈

#10587481

yes

—

190

C₈

#10587481

yes

—

191

C₂²

—

192

C₂²

—

193

C₂²

#11762710

yes

characteristic

194

C₂²

#11763161

yes

—

195

C₄

—

196

C₄

—

197

C₄

—

136

—

198

C₄

C₂×#10587481

yes

—

199

C₄

#11763080

yes

—

200

C₄

#11763085

yes

—

201

C₄

#11763181

yes

characteristic

202

C₄

#11763188

yes

characteristic

203

C₂

—

204

C₂

#14529585

yes

characteristic

205

C₂

#14529692

yes

characteristic

206

C₂

#14529697

yes

characteristic

	subgroup	G/H
center	C₂²	#11762710
commutator	C₄²×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1848	C₂
socle	C₂²×C₁₇	#390
prime radical	C₂³×C₁₇	#124
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1848	D₁₇
hypocenter	C₁₇	#9771
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂²

#11762710

commutator

C₄²×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1848

C₂

C₂²×C₁₇

C₂³×C₁₇

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
147. C₁₇	#9771	semidirect

H ≅ G/N

type

147. C₁₇

#9771

semidirect

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

subgroup	index
C₂×#11763214	2
#14527470	2
#14527466	2
#14527067	2
C₁₇×#1848	2
#9771	17

subgroup

index

C₁₇×#1848

Subgroups — #16933781

Conjugacy classes of subgroups

Complements

Maximal subgroups