Group #16934454 — subgroups

Subgroups — #16934454

Order 4352.

430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 167 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	#14528901	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	#14526667	C₂	1	4	yes	maximal
4	C₁₇×#2859	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	2	yes	—
6	C₂×C₄⋊#3622	C₂²	1	2	yes	—
7	C₂×C₄×#3622	—	2	1	no	—
8	#11762564	—	2	1	no	—
9	#11762564	—	2	2	no	—
10	C₂×C₁₇×#125	—	2	1	no	—
11	C₁₇×#389	—	2	2	no	—
12	C₁₇×#389	C₄	1	4	yes	—
13	C₁₇×#384	C₂²	1	2	yes	—
14	C₁₇×#382	—	2	1	no	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#125	—	4	2	no	—
17	C₁₇×#125	—	4	2	no	—
18	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₇	D₄	1	2	yes	—
25	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂²×#3622	—	2	2	no	—
28	C₂²×#3622	—	2	2	no	—
29	C₁₇×#95	—	2	4	no	—
30	C₄⋊#3622	—	2	2	no	—
31	C₄⋊#3622	—	2	4	no	—
32	C₄⋊#3622	Q₈	1	4	yes	—
33	C₄×#3622	—	4	2	no	—
34	C₂²×C₄×C₁₇	—	2	4	no	—
35	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₁₇	—	4	4	no	—
41	C₄⋊C₄×C₁₇	—	4	8	no	—
42	C₄⋊C₄×C₁₇	—	4	2	no	—
43	C₄⋊C₄×C₁₇	—	4	2	no	—
44	C₄⋊C₄×C₁₇	—	4	4	no	—
45	C₄²×C₁₇	—	2	4	no	—
46	C₄²×C₁₇	—	2	2	no	—
47	C₄²×C₁₇	—	4	2	no	—
48	C₄²×C₁₇	D₈	1	2	yes	—
49	C₄²×C₁₇	SD₁₆	1	2	yes	—
50	C₄²×C₁₇	C₂×Q₈	1	2	yes	—
51	C₄²×C₁₇	#41	1	2	yes	—
52	C₂×#3622	—	2	4	no	—
53	C₂×#3622	—	4	2	no	—
54	C₂×#3622	—	4	2	no	—
55	C₂×#3622	—	4	2	no	—
56	C₂×#3622	—	4	4	no	—
57	#9899	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	2	2	no	—
60	C₂×C₄×C₁₇	—	2	2	no	—
61	C₂×C₄×C₁₇	—	2	2	no	—
62	C₂×C₄×C₁₇	—	2	2	no	—
63	C₂×C₄×C₁₇	—	2	2	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	4	4	no	—
66	C₂×C₄×C₁₇	—	4	4	no	—
67	C₂×C₄×C₁₇	—	4	4	no	—
68	C₂×C₄×C₁₇	#102	1	4	yes	—
69	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
70	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
71	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₇	#135	1	4	yes	—
76	C₂×C₄×C₁₇	#137	1	2	yes	—
77	#3622	—	4	4	no	—
78	#3622	—	8	2	no	—
79	#2859	D₁₇	1	1	yes	characteristic
80	#1831	—	17	1	no	—
81	C₄×C₄⋊C₈	—	17	1	no	—
82	#1296	—	17	4	no	—
83	C₂²×C₁₇	#391	1	1	yes	characteristic
84	C₂²×C₁₇	#413	1	2	yes	—
85	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
86	C₂²×C₁₇	#416	1	2	yes	—
87	C₂²×C₁₇	#419	1	1	yes	characteristic
88	C₄×C₁₇	—	2	4	no	—
89	C₄×C₁₇	—	2	2	no	—
90	C₄×C₁₇	—	4	2	no	—
91	C₄×C₁₇	—	8	4	no	—
92	C₄×C₁₇	#474	1	2	yes	—
93	C₄×C₁₇	#475	1	2	yes	—
94	C₄×C₁₇	#488	1	2	yes	—
95	C₄×C₁₇	#499	1	2	yes	—
96	C₂×#125	—	2	1	no	—
97	C₂×C₄⋊C₈	—	17	2	no	—
98	C₂×C₄⋊C₈	—	17	2	no	—
99	C₂×C₄×C₈	—	34	1	no	—
100	#389	—	2	2	no	—
101	#389	Dic₁₇	1	4	yes	—
102	#384	C₂×D₁₇	1	2	yes	—
103	#382	—	2	1	no	—
104	C₄³	C₂×D₁₇	1	1	yes	characteristic
105	#336	—	34	2	no	—
106	#336	—	34	1	no	—
107	C₂×C₁₇	#1540	1	2	yes	—
108	C₂×C₁₇	#1554	1	2	yes	—
109	C₂×C₁₇	#1988	1	1	yes	characteristic
110	C₂×C₁₇	#1991	1	1	yes	characteristic
111	C₂×C₁₇	#1998	1	1	yes	characteristic
112	C₂²×C₈	—	34	2	no	—
113	C₂²×C₈	—	34	2	no	—
114	#125	—	4	2	no	—
115	#125	—	4	2	no	—
116	C₂×C₄⋊C₄	—	2	4	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
120	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
121	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
122	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
123	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
124	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
125	C₄⋊C₈	—	17	4	no	—
126	C₄⋊C₈	—	34	2	no	—
127	C₄⋊C₈	—	34	4	no	—
128	C₄×C₈	—	68	2	no	—
129	#95	—	2	4	no	—
130	C₁₇	#9899	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
133	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
134	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
137	C₂×C₈	—	34	4	no	—
138	C₂×C₈	—	68	2	no	—
139	C₂×C₈	—	68	2	no	—
140	C₂×C₈	—	68	2	no	—
141	C₂×C₈	—	68	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	#63228	1	2	yes	—
151	C₄²	#63229	1	2	yes	—
152	C₄²	Q₈×D₁₇	1	2	yes	—
153	C₄²	#63256	1	2	yes	—
154	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#10587503	1	4	yes	—
165	C₂×C₄	C₂×#63228	1	1	yes	characteristic
166	C₂×C₄	C₂×#63229	1	1	yes	characteristic
167	C₂×C₄	#10587628	1	2	yes	—
168	C₂×C₄	#10587630	1	1	yes	characteristic
169	C₂×C₄	Q₈×Dic₁₇	1	2	yes	—
170	C₂×C₄	#10587632	1	2	yes	—
171	C₂×C₄	#10587633	1	1	yes	characteristic
172	C₂×C₄	#10587636	1	4	yes	—
173	C₈	—	68	4	no	—
174	C₈	—	136	2	no	—
175	C₂²	#11762716	1	1	yes	characteristic
176	C₂²	#11762793	1	2	yes	—
177	C₂²	C₂×#10587503	1	1	yes	characteristic
178	C₂²	#11762805	1	2	yes	—
179	C₂²	#11762818	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	8	4	no	—
184	C₄	#11763086	1	2	yes	—
185	C₄	#11763091	1	2	yes	—
186	C₄	#11763166	1	2	yes	—
187	C₄	#11763208	1	2	yes	—
188	C₂	#14527433	1	2	yes	—
189	C₂	#14527494	1	2	yes	—
190	C₂	#14529737	1	1	yes	characteristic
191	C₂	#14529750	1	1	yes	characteristic
192	C₂	#14529778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528901

C₂

yes

maximal, characteristic

C₄×C₄⋊#3622

C₂

yes

maximal, characteristic

#14526667

C₂

yes

maximal

C₁₇×#2859

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂²

yes

—

C₂×C₄⋊#3622

C₂²

yes

—

C₂×C₄×#3622

—

#11762564

—

#11762564

—

C₂×C₁₇×#125

—

C₁₇×#389

—

C₁₇×#389

C₄

yes

—

C₁₇×#384

C₂²

yes

—

C₁₇×#382

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#125

—

C₁₇×#125

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₂²×#3622

—

C₁₇×#95

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

Q₈

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

D₈

yes

—

C₄²×C₁₇

SD₁₆

yes

—

C₄²×C₁₇

C₂×Q₈

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

—

—

—

C₂³×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#102

yes

—

C₂×C₄×C₁₇

C₄×Q₈

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#124

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×C₄×C₁₇

#137

yes

—

#3622

—

#3622

—

#2859

D₁₇

yes

characteristic

#1831

—

C₄×C₄⋊C₈

—

#1296

—

C₂²×C₁₇

#391

yes

characteristic

C₂²×C₁₇

#413

yes

—

C₂²×C₁₇

C₂×#102

yes

characteristic

C₂²×C₁₇

#416

yes

—

C₂²×C₁₇

#419

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#474

yes

—

C₄×C₁₇

#475

yes

—

C₄×C₁₇

#488

yes

—

C₄×C₁₇

#499

yes

—

C₂×#125

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

100

#389

—

101

#389

Dic₁₇

yes

—

102

#384

C₂×D₁₇

yes

—

103

#382

—

104

C₄³

C₂×D₁₇

yes

characteristic

105

#336

—

106

#336

—

107

C₂×C₁₇

#1540

yes

—

108

C₂×C₁₇

#1554

yes

—

109

C₂×C₁₇

#1988

yes

characteristic

110

C₂×C₁₇

#1991

yes

characteristic

111

C₂×C₁₇

#1998

yes

characteristic

112

C₂²×C₈

—

113

C₂²×C₈

—

114

#125

—

115

#125

—

116

C₂×C₄⋊C₄

—

117

C₂×C₄⋊C₄

—

118

C₂×C₄⋊C₄

—

119

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

120

C₂×C₄²

C₂×Dic₁₇

yes

—

121

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

122

C₂×C₄²

C₂²⋊D₁₇

yes

—

123

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

124

C₂×C₄²

C₂²×D₁₇

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

C₄⋊C₈

—

128

C₄×C₈

—

129

#95

—

130

C₁₇

#9899

yes

characteristic, complemented, Sylow

131

C₂²×C₄

—

132

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

133

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

134

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

135

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

136

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

#63228

yes

—

151

C₄²

#63229

yes

—

152

C₄²

Q₈×D₁₇

yes

—

153

C₄²

#63256

yes

—

154

C₂³

C₂×C₂²⋊Dic₁₇

yes

characteristic

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

#10587503

yes

—

165

C₂×C₄

C₂×#63228

yes

characteristic

166

C₂×C₄

C₂×#63229

yes

characteristic

167

C₂×C₄

#10587628

yes

—

168

C₂×C₄

#10587630

yes

characteristic

169

C₂×C₄

Q₈×Dic₁₇

yes

—

170

C₂×C₄

#10587632

yes

—

171

C₂×C₄

#10587633

yes

characteristic

172

C₂×C₄

#10587636

yes

—

173

C₈

—

174

C₈

—

136

—

175

C₂²

#11762716

yes

characteristic

176

C₂²

#11762793

yes

—

177

C₂²

C₂×#10587503

yes

characteristic

178

C₂²

#11762805

yes

—

179

C₂²

#11762818

yes

characteristic

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#11763086

yes

—

185

C₄

#11763091

yes

—

186

C₄

#11763166

yes

—

187

C₄

#11763208

yes

—

188

C₂

#14527433

yes

—

189

C₂

#14527494

yes

—

190

C₂

#14529737

yes

characteristic

191

C₂

#14529750

yes

characteristic

192

C₂

#14529778

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2859	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2859	D₁₇
hypocenter	C₁₇	#9899
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₇

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#2859

C₂

socle, prime radical

C₂³×C₁₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #2859

N	H ≅ G/N	type
130. C₁₇	#9899	semidirect

H ≅ G/N

type

130. C₁₇

#9899

semidirect

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

subgroup	index
#14528901	2
C₄×C₄⋊#3622	2
#14526667	2
C₁₇×#2859	2
#9899	17

subgroup

index

#14528901

C₄×C₄⋊#3622

#14526667

C₁₇×#2859

#9899

Subgroups — #16934454

Conjugacy classes of subgroups

Complements

Maximal subgroups