Group #16927109 — subgroups

Subgroups — #16927109

Order 4352.

456 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 143 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	#14528637	C₂	1	2	yes	maximal
2	C₂×#11762551	C₂	1	2	yes	maximal
3	#14526665	C₂	1	2	yes	maximal
4	C₄×C₁₇×#125	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	—	2	2	no	—
6	C₂×#10587474	C₂²	1	4	yes	—
7	#11762564	—	2	2	no	—
8	#11762551	—	2	4	no	—
9	C₂×C₁₇×#125	C₂²	1	1	yes	characteristic
10	C₁₇×#389	C₄	1	2	yes	—
11	C₁₇×#387	—	2	2	no	—
12	C₁₇×#382	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₇	—	2	2	no	—
14	C₄×C₄⋊C₄×C₁₇	C₄	1	2	yes	—
15	C₄³×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#125	—	2	2	no	—
17	C₁₇×#125	D₄	1	2	yes	—
18	C₁₇×#125	Q₈	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
20	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₁₇	—	2	2	no	—
24	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₇	D₄	1	2	yes	—
27	C₂×C₄²×C₁₇	D₄	1	2	yes	—
28	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
29	C₂²×#3622	—	2	4	no	—
30	C₁₇×#95	—	2	2	no	—
31	C₁₇×#95	—	2	2	no	—
32	C₁₇×#95	—	2	2	no	—
33	C₄⋊#3622	—	2	8	no	—
34	#10587474	—	2	4	no	—
35	#10587474	—	2	4	no	—
36	C₂²×C₄×C₁₇	—	2	2	no	—
37	C₂²×C₄×C₁₇	—	2	2	no	—
38	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₂×D₄	1	2	yes	—
43	C₄⋊C₄×C₁₇	—	2	2	no	—
44	C₄⋊C₄×C₁₇	—	2	4	no	—
45	C₄⋊C₄×C₁₇	—	2	4	no	—
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄⋊C₄×C₁₇	—	4	4	no	—
48	C₄⋊C₄×C₁₇	C₄⋊C₄	1	4	yes	—
49	C₄²×C₁₇	—	2	2	no	—
50	C₄²×C₁₇	—	2	4	no	—
51	C₄²×C₁₇	—	4	1	no	—
52	C₄²×C₁₇	—	4	2	no	—
53	C₄²×C₁₇	—	4	4	no	—
54	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₇	#41	1	2	yes	—
57	C₂×#3622	—	2	8	no	—
58	C₂×#3622	—	4	4	no	—
59	C₂×#3622	—	4	4	no	—
60	#8584	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	—	2	1	no	—
63	C₂×C₄×C₁₇	—	2	1	no	—
64	C₂×C₄×C₁₇	—	2	2	no	—
65	C₂×C₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	—	2	2	no	—
69	C₂×C₄×C₁₇	—	2	4	no	—
70	C₂×C₄×C₁₇	—	2	1	no	—
71	C₂×C₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	4	2	no	—
73	C₂×C₄×C₁₇	—	4	2	no	—
74	C₂×C₄×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	—	4	2	no	—
76	C₂×C₄×C₁₇	—	4	2	no	—
77	C₂×C₄×C₁₇	#101	1	2	yes	—
78	C₂×C₄×C₁₇	#104	1	4	yes	—
79	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₇	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#121	1	2	yes	—
82	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
83	#3622	—	4	8	no	—
84	C₄×#125	D₁₇	1	1	yes	characteristic
85	#1767	—	17	2	no	—
86	C₂×#330	—	17	2	no	—
87	#1295	—	17	2	no	—
88	C₂²×C₁₇	#380	1	1	yes	characteristic
89	C₂²×C₁₇	C₂×#101	1	1	yes	characteristic
90	C₂²×C₁₇	#413	1	1	yes	characteristic
91	C₂²×C₁₇	C₂×#104	1	2	yes	—
92	C₂²×C₁₇	#421	1	2	yes	—
93	C₄×C₁₇	—	2	1	no	—
94	C₄×C₁₇	—	2	1	no	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	—	2	2	no	—
97	C₄×C₁₇	—	2	4	no	—
98	C₄×C₁₇	—	2	2	no	—
99	C₄×C₁₇	—	4	4	no	—
100	C₄×C₁₇	—	4	1	no	—
101	C₄×C₁₇	—	8	2	no	—
102	C₂×#125	C₂×D₁₇	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	34	2	no	—
104	C₂×#97	—	17	4	no	—
105	#389	Dic₁₇	1	2	yes	—
106	#387	—	2	2	no	—
107	#382	C₂×D₁₇	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	2	2	no	—
109	C₄×C₄⋊C₄	Dic₁₇	1	2	yes	—
110	C₄³	C₂×D₁₇	1	1	yes	characteristic
111	#336	—	34	2	no	—
112	#330	—	34	4	no	—
113	C₂×C₁₇	#1526	1	1	yes	characteristic
114	C₂×C₁₇	#1532	1	2	yes	—
115	C₂×C₁₇	#1534	1	1	yes	characteristic
116	C₂×C₁₇	#1801	1	1	yes	characteristic
117	C₂×C₁₇	#1806	1	2	yes	—
118	C₂²×C₈	—	34	4	no	—
119	#125	—	2	2	no	—
120	#125	Dic₃₄	1	2	yes	—
121	#125	D₆₈	1	2	yes	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
125	C₂×C₄⋊C₄	C₂×Dic₁₇	1	2	yes	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₁₇	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
131	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
132	C₄⋊C₈	—	34	8	no	—
133	#97	—	34	4	no	—
134	#97	—	34	4	no	—
135	#95	—	2	2	no	—
136	#95	—	2	2	no	—
137	#95	—	2	2	no	—
138	C₁₇	#8584	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Dic₁₇	1	2	yes	—
144	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₂²⋊D₁₇	1	2	yes	—
146	C₂×C₈	—	34	8	no	—
147	C₂×C₈	—	68	4	no	—
148	C₂×C₈	—	68	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	C₄⋊Dic₁₇	1	4	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	4	1	no	—
158	C₄²	—	4	2	no	—
159	C₄²	—	4	4	no	—
160	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
161	C₄²	C₂×D₆₈	1	1	yes	characteristic
162	C₄²	#63252	1	2	yes	—
163	C₂³	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	#10587507	1	2	yes	—
180	C₂×C₄	#10587508	1	4	yes	—
181	C₂×C₄	#10587610	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
183	C₂×C₄	#10587612	1	1	yes	characteristic
184	C₂×C₄	#10587616	1	2	yes	—
185	C₈	—	68	8	no	—
186	C₂²	#11762669	1	1	yes	characteristic
187	C₂²	C₂×#10587507	1	1	yes	characteristic
188	C₂²	#11762793	1	1	yes	characteristic
189	C₂²	C₂×#10587508	1	2	yes	—
190	C₂²	#11762826	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	2	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	4	no	—
198	C₄	—	4	1	no	—
199	C₄	—	8	2	no	—
200	C₂	#14527379	1	1	yes	characteristic
201	C₂	#14527404	1	2	yes	—
202	C₂	#14527411	1	1	yes	characteristic
203	C₂	#14528780	1	1	yes	characteristic
204	C₂	#14528804	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14528637

C₂

yes

maximal

C₂×#11762551

C₂

yes

maximal

#14526665

C₂

yes

maximal

C₄×C₁₇×#125

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

—

C₂×#10587474

C₂²

yes

—

#11762564

—

#11762551

—

C₂×C₁₇×#125

C₂²

yes

characteristic

C₁₇×#389

C₄

yes

—

C₁₇×#387

—

C₁₇×#382

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₁₇

—

C₄×C₄⋊C₄×C₁₇

C₄

yes

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#125

—

C₁₇×#125

D₄

yes

—

C₁₇×#125

Q₈

yes

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₁₇×#95

—

C₁₇×#95

—

C₁₇×#95

—

C₄⋊#3622

—

#10587474

—

#10587474

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

characteristic

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

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#101

yes

—

C₂×C₄×C₁₇

#104

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

#3622

—

C₄×#125

D₁₇

yes

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#421

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

100

C₄×C₁₇

—

101

C₄×C₁₇

—

102

C₂×#125

C₂×D₁₇

yes

characteristic

103

C₂×C₄⋊C₈

—

104

C₂×#97

—

105

#389

Dic₁₇

yes

—

106

#387

—

107

#382

C₂×D₁₇

yes

characteristic

108

C₄×C₄⋊C₄

—

109

C₄×C₄⋊C₄

Dic₁₇

yes

—

110

C₄³

C₂×D₁₇

yes

characteristic

111

#336

—

112

#330

—

113

C₂×C₁₇

#1526

yes

characteristic

114

C₂×C₁₇

#1532

yes

—

115

C₂×C₁₇

#1534

yes

characteristic

116

C₂×C₁₇

#1801

yes

characteristic

117

C₂×C₁₇

#1806

yes

—

118

C₂²×C₈

—

119

#125

—

120

#125

Dic₃₄

yes

—

121

#125

D₆₈

yes

—

122

C₂×C₄⋊C₄

—

123

C₂×C₄⋊C₄

—

124

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

125

C₂×C₄⋊C₄

C₂×Dic₁₇

yes

—

126

C₂×C₄²

—

127

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

128

C₂×C₄²

C₂×Dic₁₇

yes

characteristic

129

C₂×C₄²

C₂²⋊D₁₇

yes

—

130

C₂×C₄²

C₂²⋊D₁₇

yes

—

131

C₂×C₄²

C₂²×D₁₇

yes

characteristic

132

C₄⋊C₈

—

133

#97

—

134

#97

—

135

#95

—

136

#95

—

137

#95

—

138

C₁₇

#8584

yes

characteristic, complemented, Sylow

139

C₂²×C₄

—

140

C₂²×C₄

—

141

C₂²×C₄

C₂²⋊Dic₁₇

yes

characteristic

142

C₂²×C₄

C₂²⋊Dic₁₇

yes

characteristic

143

C₂²×C₄

C₂²⋊Dic₁₇

yes

—

144

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

145

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

—

146

C₂×C₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

—

151

C₄⋊C₄

—

152

C₄⋊C₄

—

153

C₄⋊C₄

—

154

C₄⋊C₄

C₄⋊Dic₁₇

yes

—

155

C₄²

—

156

C₄²

—

157

C₄²

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×Dic₃₄

yes

characteristic

161

C₄²

C₂×D₆₈

yes

characteristic

162

C₄²

#63252

yes

—

163

C₂³

C₂×C₂²⋊Dic₁₇

yes

characteristic

164

C₂×C₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

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

—

179

C₂×C₄

#10587507

yes

—

180

C₂×C₄

#10587508

yes

—

181

C₂×C₄

#10587610

yes

—

182

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

183

C₂×C₄

#10587612

yes

characteristic

184

C₂×C₄

#10587616

yes

—

185

C₈

—

186

C₂²

#11762669

yes

characteristic

187

C₂²

C₂×#10587507

yes

characteristic

188

C₂²

#11762793

yes

characteristic

189

C₂²

C₂×#10587508

yes

—

190

C₂²

#11762826

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

—

198

C₄

—

199

C₄

—

200

C₂

#14527379

yes

characteristic

201

C₂

#14527404

yes

—

202

C₂

#14527411

yes

characteristic

203

C₂

#14528780

yes

characteristic

204

C₂

#14528804

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄×C₁₇×#125	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄×#125	D₁₇
hypocenter	C₁₇	#8584
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₁₇

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₄×C₁₇×#125

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 ◁ C₄×#125

N	H ≅ G/N	type
138. C₁₇	#8584	semidirect

H ≅ G/N

type

138. C₁₇

#8584

semidirect

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

subgroup	index
#14528637	2
C₂×#11762551	2
#14526665	2
C₄×C₁₇×#125	2
#8584	17

subgroup

index

C₄×C₁₇×#125

Subgroups — #16927109

Conjugacy classes of subgroups

Complements

Maximal subgroups