Group #16928201 — subgroups

Subgroups — #16928201

Order 4352.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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	#14529721	C₂	1	1	yes	maximal, characteristic
2	C₄×#10587507	C₂	1	1	yes	maximal, characteristic
3	#14527537	C₂	1	2	yes	maximal
4	#14526620	C₂	1	2	yes	maximal
5	C₁₇×#2983	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#3622	C₂²	1	1	yes	characteristic
7	C₂×#10587507	C₄	1	2	yes	—
8	C₄×#63223	C₂²	1	2	yes	—
9	C₂×C₄×#3622	C₂²	1	1	yes	characteristic
10	#11762578	C₄	1	1	yes	characteristic
11	#11762578	C₄	1	1	yes	characteristic
12	#11762564	—	2	2	no	—
13	#11762561	—	2	2	no	—
14	#11762559	—	2	2	no	—
15	C₂×C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
16	C₁₇×#446	—	2	2	no	—
17	C₁₇×#446	—	2	2	no	—
18	C₁₇×#432	—	2	2	no	—
19	C₁₇×#431	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₇	—	2	2	no	—
22	C₄×Q₈×C₁₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₁₇	D₄	1	2	yes	—
26	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
27	C₂×#63223	C₂×C₄	1	2	yes	—
28	C₂×#63223	C₂×C₄	1	2	yes	—
29	C₂²×#3622	—	2	1	no	—
30	C₂²×#3622	—	2	1	no	—
31	C₁₇×C₄⋊C₈	—	2	2	no	—
32	C₁₇×C₄⋊C₈	—	2	2	no	—
33	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
34	C₁₇×C₄⋊C₈	D₄	1	2	yes	—
35	C₁₇×C₂²⋊C₈	—	2	2	no	—
36	C₁₇×#97	—	2	4	no	—
37	C₄×C₈×C₁₇	—	2	2	no	—
38	#10587507	—	2	4	no	—
39	C₄⋊#3622	—	2	2	no	—
40	#10587474	—	2	2	no	—
41	C₄×#3622	—	2	2	no	—
42	C₄×#3622	—	2	2	no	—
43	C₂×Q₈×C₁₇	—	2	2	no	—
44	C₂×Q₈×C₁₇	—	2	2	no	—
45	C₂×Q₈×C₁₇	—	2	1	no	—
46	C₂×Q₈×C₁₇	—	2	1	no	—
47	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	—	2	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₄⋊C₄×C₁₇	—	2	2	no	—
55	C₄⋊C₄×C₁₇	—	2	4	no	—
56	C₄²×C₁₇	—	2	2	no	—
57	C₄²×C₁₇	—	2	2	no	—
58	C₄²×C₁₇	C₂×D₄	1	2	yes	—
59	C₄²×C₁₇	#41	1	2	yes	—
60	#63223	—	2	2	no	—
61	#63223	—	2	2	no	—
62	#63223	—	2	2	no	—
63	#63223	M₁₆	1	4	yes	—
64	C₂×#3622	—	2	2	no	—
65	C₂×#3622	—	2	2	no	—
66	C₂×#3622	—	2	2	no	—
67	C₂×#3622	—	4	1	no	—
68	C₂×#3622	—	4	1	no	—
69	C₂×#3622	—	4	2	no	—
70	C₂×#3622	—	4	1	no	—
71	#8838	—	17	1	no	maximal, Sylow
72	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	Q₈×C₁₇	—	4	2	no	—
74	Q₈×C₁₇	—	4	2	no	—
75	C₂×C₄×C₁₇	—	2	1	no	—
76	C₂×C₄×C₁₇	—	2	1	no	—
77	C₂×C₄×C₁₇	—	2	2	no	—
78	C₂×C₄×C₁₇	—	2	2	no	—
79	C₂×C₄×C₁₇	—	2	2	no	—
80	C₂×C₄×C₁₇	—	2	2	no	—
81	C₂×C₄×C₁₇	—	4	1	no	—
82	C₂×C₄×C₁₇	—	4	2	no	—
83	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₁₇	#120	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂²⋊Q₈	1	2	yes	—
86	C₂×C₄×C₁₇	#123	1	1	yes	characteristic
87	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
89	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
90	C₈×C₁₇	—	4	2	no	—
91	C₈×C₁₇	—	4	2	no	—
92	#3622	—	2	4	no	—
93	#3622	—	4	2	no	—
94	#3622	—	4	2	no	—
95	#3622	—	8	1	no	—
96	#2983	D₁₇	1	1	yes	characteristic
97	#1984	—	17	1	no	—
98	C₄×#101	—	17	1	no	—
99	#1565	—	17	2	no	—
100	#1278	—	17	2	no	—
101	C₂²×C₁₇	—	2	1	no	—
102	C₂²×C₁₇	—	2	1	no	—
103	C₂²×C₁₇	#385	1	1	yes	characteristic
104	C₂²×C₁₇	#423	1	1	yes	characteristic
105	C₂²×C₁₇	#424	1	1	yes	characteristic
106	C₄×C₁₇	—	2	1	no	—
107	C₄×C₁₇	—	2	2	no	—
108	C₄×C₁₇	—	4	2	no	—
109	C₄×C₁₇	—	4	2	no	—
110	C₄×C₁₇	#435	1	2	yes	—
111	C₄×C₁₇	#436	1	2	yes	—
112	C₄×C₁₇	#454	1	1	yes	characteristic
113	C₄×C₁₇	#456	1	1	yes	characteristic
114	C₂×C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
115	#446	—	2	2	no	—
116	#446	—	2	2	no	—
117	#432	—	2	2	no	—
118	#431	C₂×D₁₇	1	2	yes	—
119	C₂×C₄⋊C₈	—	17	1	no	—
120	C₂×#101	—	17	2	no	—
121	C₄×M₁₆	—	17	2	no	—
122	C₂×C₄×C₈	—	17	1	no	—
123	#343	—	17	1	no	—
124	#343	—	17	1	no	—
125	#336	—	34	2	no	—
126	#335	—	34	2	no	—
127	#334	—	34	2	no	—
128	C₂×C₁₇	—	2	1	no	—
129	C₂×C₁₇	—	2	1	no	—
130	C₂×C₁₇	#1851	1	1	yes	characteristic
131	C₂×C₁₇	#1857	1	1	yes	characteristic
132	C₂×C₁₇	#1858	1	1	yes	characteristic
133	C₂²×Q₈	C₄×D₁₇	1	1	yes	characteristic
134	C₂×M₁₆	—	17	2	no	—
135	C₂×M₁₆	—	17	2	no	—
136	C₂²×C₈	—	34	1	no	—
137	C₂²×C₈	—	34	1	no	—
138	C₄×Q₈	—	2	2	no	—
139	C₄×Q₈	—	2	2	no	—
140	C₂×C₄⋊C₄	Dic₃₄	1	2	yes	—
141	C₂×C₄⋊C₄	C₄×D₁₇	1	1	yes	characteristic
142	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
143	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	34	2	no	—
147	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
148	C₄⋊C₈	C₂²⋊D₁₇	1	2	yes	—
149	#101	—	34	4	no	—
150	C₂²⋊C₈	—	2	2	no	—
151	#97	—	2	4	no	—
152	#97	—	34	2	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	34	2	no	—
155	C₄×C₈	—	34	2	no	—
156	C₁₇	#8838	1	1	yes	characteristic, complemented, Sylow
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	1	no	—
159	C₂×Q₈	—	2	1	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂²×C₄	#63225	1	2	yes	—
162	C₂²×C₄	#63225	1	2	yes	—
163	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
166	M₁₆	—	17	4	no	—
167	M₁₆	—	34	2	no	—
168	M₁₆	—	34	2	no	—
169	M₁₆	—	34	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	34	2	no	—
173	C₂×C₈	—	34	2	no	—
174	C₂×C₈	—	34	2	no	—
175	C₂×C₈	—	68	1	no	—
176	C₂×C₈	—	68	1	no	—
177	C₂×C₈	—	68	2	no	—
178	C₂×C₈	—	68	1	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	#63252	1	2	yes	—
184	C₄²	C₂×C₂²⋊D₁₇	1	2	yes	—
185	C₂³	C₂×#63225	1	1	yes	characteristic
186	Q₈	—	4	2	no	—
187	Q₈	—	4	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	M₁₆×D₁₇	1	2	yes	—
197	C₂×C₄	#10587593	1	1	yes	characteristic
198	C₂×C₄	#10587593	1	1	yes	characteristic
199	C₂×C₄	#10587610	1	2	yes	—
200	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
201	C₂×C₄	#10587615	1	1	yes	characteristic
202	C₂×C₄	#10587639	1	1	yes	characteristic
203	C₈	—	4	2	no	—
204	C₈	—	4	2	no	—
205	C₈	—	34	4	no	—
206	C₈	—	68	2	no	—
207	C₈	—	68	2	no	—
208	C₈	—	136	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	1	no	—
211	C₂²	#11762682	1	1	yes	characteristic
212	C₂²	#11762832	1	1	yes	characteristic
213	C₂²	#11762837	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	4	2	no	—
217	C₄	—	4	2	no	—
218	C₄	#11762878	1	2	yes	—
219	C₄	#11762885	1	2	yes	—
220	C₄	#11762982	1	1	yes	characteristic
221	C₄	#11762992	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	#14528967	1	1	yes	characteristic
225	C₂	#14528999	1	1	yes	characteristic
226	C₂	#14529004	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14529721

C₂

yes

maximal, characteristic

C₄×#10587507

C₂

yes

maximal, characteristic

#14527537

C₂

yes

maximal

#14526620

C₂

yes

maximal

C₁₇×#2983

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂²

yes

characteristic

C₂×#10587507

C₄

yes

—

C₄×#63223

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

characteristic

#11762578

C₄

yes

characteristic

#11762578

C₄

yes

—

—

—

C₂×C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×#446

—

C₁₇×#446

—

C₁₇×#432

—

C₁₇×#431

C₂²

yes

—

C₂²×Q₈×C₁₇

C₂×C₄

yes

characteristic

C₄×Q₈×C₁₇

—

C₄×Q₈×C₁₇

—

C₂×C₄⋊C₄×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂×#63223

C₂×C₄

yes

—

C₂×#63223

C₂×C₄

yes

—

C₂²×#3622

—

C₂²×#3622

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

D₄

yes

—

C₁₇×C₄⋊C₈

D₄

yes

—

C₁₇×C₂²⋊C₈

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

—

—

—

—

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

M₁₆

yes

—

—

—

—

—

—

—

—

—

C₂³×C₁₇

C₂×C₄⋊C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×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

C₄×D₄

yes

—

C₂×C₄×C₁₇

#120

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₇

#123

yes

characteristic

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

—

D₁₇

yes

—

—

—

100

#1278

—

101

C₂²×C₁₇

—

102

C₂²×C₁₇

—

103

C₂²×C₁₇

#385

yes

characteristic

104

C₂²×C₁₇

#423

yes

characteristic

105

C₂²×C₁₇

#424

yes

characteristic

106

C₄×C₁₇

—

107

C₄×C₁₇

—

108

C₄×C₁₇

—

109

C₄×C₁₇

—

110

C₄×C₁₇

#435

yes

—

111

C₄×C₁₇

#436

yes

—

112

C₄×C₁₇

#454

yes

characteristic

113

C₄×C₁₇

#456

yes

characteristic

114

C₂×C₄×Q₈

C₂×D₁₇

yes

characteristic

115

#446

—

116

#446

—

117

#432

—

118

#431

C₂×D₁₇

yes

—

119

C₂×C₄⋊C₈

—

120

C₂×#101

—

121

C₄×M₁₆

—

122

C₂×C₄×C₈

—

123

#343

—

124

#343

—

125

#336

—

126

#335

—

127

#334

—

128

C₂×C₁₇

—

129

C₂×C₁₇

—

130

C₂×C₁₇

#1851

yes

characteristic

131

C₂×C₁₇

#1857

yes

characteristic

132

C₂×C₁₇

#1858

yes

characteristic

133

C₂²×Q₈

C₄×D₁₇

yes

characteristic

134

C₂×M₁₆

—

135

C₂×M₁₆

—

136

C₂²×C₈

—

137

C₂²×C₈

—

138

C₄×Q₈

—

139

C₄×Q₈

—

140

C₂×C₄⋊C₄

Dic₃₄

yes

—

141

C₂×C₄⋊C₄

C₄×D₁₇

yes

characteristic

142

C₂×C₄²

C₂²⋊D₁₇

yes

—

143

C₂×C₄²

C₂²×D₁₇

yes

characteristic

144

C₄⋊C₈

—

145

C₄⋊C₈

—

146

C₄⋊C₈

—

147

C₄⋊C₈

C₂²⋊D₁₇

yes

—

148

C₄⋊C₈

C₂²⋊D₁₇

yes

—

149

#101

—

150

C₂²⋊C₈

—

151

#97

—

152

#97

—

153

C₄×C₈

—

154

C₄×C₈

—

155

C₄×C₈

—

156

C₁₇

#8838

yes

characteristic, complemented, Sylow

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂²×C₄

#63225

yes

—

162

C₂²×C₄

#63225

yes

—

163

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

164

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

165

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

166

M₁₆

—

167

M₁₆

—

168

M₁₆

—

169

M₁₆

—

170

C₂×C₈

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄²

—

182

C₄²

—

183

C₄²

#63252

yes

—

184

C₄²

C₂×C₂²⋊D₁₇

yes

—

185

C₂³

C₂×#63225

yes

characteristic

186

Q₈

—

187

Q₈

—

188

C₂×C₄

—

189

C₂×C₄

—

190

C₂×C₄

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

M₁₆×D₁₇

yes

—

197

C₂×C₄

#10587593

yes

characteristic

198

C₂×C₄

#10587593

yes

characteristic

199

C₂×C₄

#10587610

yes

—

200

C₂×C₄

C₄×C₂²⋊D₁₇

yes

—

201

C₂×C₄

#10587615

yes

characteristic

202

C₂×C₄

#10587639

yes

characteristic

203

C₈

—

204

C₈

—

205

C₈

—

206

C₈

—

207

C₈

—

208

C₈

—

136

—

209

C₂²

—

210

C₂²

—

211

C₂²

#11762682

yes

characteristic

212

C₂²

#11762832

yes

characteristic

213

C₂²

#11762837

yes

characteristic

214

C₄

—

215

C₄

—

216

C₄

—

217

C₄

—

218

C₄

#11762878

yes

—

219

C₄

#11762885

yes

—

220

C₄

#11762982

yes

characteristic

221

C₄

#11762992

yes

characteristic

222

C₂

—

223

C₂

—

224

C₂

#14528967

yes

characteristic

225

C₂

#14528999

yes

characteristic

226

C₂

#14529004

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#10587639
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#2983	C₂
socle	C₂²×C₁₇	#385
prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#2983	D₁₇
hypocenter	C₁₇	#8838
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2983

subgroup

G/H

center

C₂×C₄

#10587639

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#2983

C₂

socle

C₂²×C₁₇

#385

prime radical

C₂³×C₁₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2983

N	H ≅ G/N	type
156. C₁₇	#8838	semidirect

H ≅ G/N

type

156. C₁₇

#8838

semidirect

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

subgroup	index
#14529721	2
C₄×#10587507	2
#14527537	2
#14526620	2
C₁₇×#2983	2
#8838	17

subgroup

index

C₁₇×#2983

Subgroups — #16928201

Conjugacy classes of subgroups

Complements

Maximal subgroups