Group #16952557 — subgroups

Subgroups — #16952557

Order 4352.

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

normal subgroups only

This group has 590 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	C₂×#11762610	C₂	1	4	yes	maximal, complemented
2	C₂×#11762585	C₂	1	1	yes	maximal, characteristic, complemented
3	#14528948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#493	C₂	1	1	yes	maximal, characteristic
5	C₂²×#63214	C₂²	1	2	yes	—
6	C₂×#10587489	C₂²	1	2	yes	—
7	C₂×#10587488	—	2	1	no	—
8	#11762675	—	2	1	no	—
9	#11762610	—	2	8	no	—
10	#11762585	—	2	2	no	—
11	C₂²×D₈×C₁₇	—	2	1	no	—
12	C₂²×D₈×C₁₇	C₂²	1	2	yes	—
13	C₂×C₁₇×#127	—	2	1	no	—
14	C₁₇×#493	—	2	2	no	—
15	C₁₇×#493	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₁₇	—	2	2	no	—
18	C₂²×D₄×C₁₇	—	4	2	no	—
19	C₂×D₈×C₁₇	—	2	4	no	—
20	C₂×D₈×C₁₇	—	2	4	no	—
21	C₂×D₈×C₁₇	—	2	2	no	—
22	C₂×D₈×C₁₇	—	2	4	no	—
23	C₂×D₈×C₁₇	—	4	2	no	—
24	C₂×D₈×C₁₇	—	4	2	no	—
25	C₂×D₈×C₁₇	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
28	C₁₇×#127	—	2	4	no	—
29	C₁₇×#127	—	4	2	no	—
30	C₂×C₄²×C₁₇	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₁₇	—	2	2	no	—
32	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₇	D₄	1	2	yes	—
34	#10587502	—	4	2	no	—
35	#10587489	—	2	4	no	—
36	#10587488	—	4	2	no	—
37	C₂×#63214	—	2	2	no	—
38	C₂×#63214	—	2	2	no	—
39	C₂×#63214	D₄	1	4	yes	complemented
40	C₂⁴×C₁₇	—	4	2	no	—
41	C₂×D₄×C₁₇	—	2	4	no	—
42	C₂×D₄×C₁₇	—	4	8	no	—
43	C₂×D₄×C₁₇	—	4	2	no	—
44	C₂×D₄×C₁₇	—	4	2	no	—
45	C₂×D₄×C₁₇	—	4	2	no	—
46	C₂×D₄×C₁₇	—	4	2	no	—
47	C₂×D₄×C₁₇	—	4	2	no	—
48	C₂×D₄×C₁₇	—	8	8	no	—
49	C₂²×C₄×C₁₇	D₈	1	2	yes	—
50	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₁₇	—	4	2	no	—
52	D₈×C₁₇	—	2	8	no	—
53	D₈×C₁₇	—	4	8	no	—
54	D₈×C₁₇	—	4	4	no	—
55	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	D₈	1	2	yes	—
57	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
58	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₇	#41	1	2	yes	—
62	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₁₇	SD₁₆	1	2	yes	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	C₄⋊Dic₁₇	—	4	2	no	—
66	#63214	—	2	4	no	—
67	#13206	—	17	1	no	maximal, Sylow
68	C₂³×C₁₇	—	4	4	no	—
69	C₂³×C₁₇	—	8	2	no	—
70	C₂³×C₁₇	—	8	2	no	—
71	C₂³×C₁₇	—	8	8	no	—
72	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
73	D₄×C₁₇	—	4	4	no	—
74	D₄×C₁₇	—	4	4	no	—
75	D₄×C₁₇	—	8	16	no	—
76	D₄×C₁₇	—	8	2	no	—
77	D₄×C₁₇	—	8	2	no	—
78	C₂×C₄×C₁₇	—	2	4	no	—
79	C₂×C₄×C₁₇	#102	1	4	yes	—
80	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
83	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₁₇	—	8	2	no	—
87	C₂×Dic₁₇	—	8	4	no	—
88	C₈×C₁₇	#102	1	4	yes	—
89	C₈×C₁₇	C₄×D₄	1	2	yes	—
90	C₈×C₁₇	#121	1	2	yes	—
91	C₂×#493	D₁₇	1	1	yes	characteristic
92	C₂×#357	—	17	4	no	—
93	C₂×#346	—	17	1	no	—
94	#1846	—	17	1	no	—
95	C₂²×C₁₇	—	8	8	no	—
96	C₂²×C₁₇	—	8	4	no	—
97	C₂²×C₁₇	—	16	4	no	—
98	C₂²×C₁₇	—	16	4	no	—
99	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
100	C₂²×C₁₇	#486	1	1	yes	characteristic
101	C₂²×C₁₇	#493	1	1	yes	characteristic
102	C₂²×C₁₇	#509	1	4	yes	—
103	C₄×C₁₇	—	2	4	no	—
104	C₄×C₁₇	#390	1	1	yes	characteristic
105	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
106	C₄×C₁₇	#486	1	1	yes	characteristic
107	C₄×C₁₇	#492	1	1	yes	characteristic
108	Dic₁₇	—	16	2	no	—
109	C₂²×D₈	—	2	1	no	—
110	C₂²×D₈	C₂×D₁₇	1	2	yes	—
111	C₂×#127	—	2	1	no	—
112	C₂²×C₁₆	—	17	2	no	—
113	#493	—	2	2	no	—
114	#493	Dic₁₇	1	4	yes	complemented
115	C₂×#107	—	17	2	no	—
116	C₂×#106	—	34	1	no	—
117	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
118	#382	—	34	1	no	—
119	#357	—	34	8	no	—
120	#346	—	34	2	no	—
121	C₂×C₁₇	—	16	4	no	—
122	C₂×C₁₇	#1957	1	1	yes	characteristic
123	C₂×C₁₇	#2140	1	2	yes	—
124	C₂×C₁₇	#2244	1	2	yes	—
125	C₂×C₁₇	#2250	1	2	yes	—
126	C₂²×D₄	—	2	2	no	—
127	C₂²×D₄	—	4	2	no	—
128	C₂×D₈	—	2	4	no	—
129	C₂×D₈	—	2	4	no	—
130	C₂×D₈	—	2	4	no	—
131	C₂×D₈	—	2	2	no	—
132	C₂×D₈	—	4	2	no	—
133	C₂×D₈	—	4	2	no	—
134	C₂×D₈	C₂×Dic₁₇	1	4	yes	—
135	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
137	#127	—	2	4	no	—
138	#127	—	4	2	no	—
139	C₂×C₄⋊C₄	—	34	2	no	—
140	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
141	C₂×C₁₆	—	17	4	no	—
142	C₂×C₁₆	—	34	2	no	—
143	C₂×C₁₆	—	34	2	no	—
144	#107	—	34	4	no	—
145	#106	—	68	2	no	—
146	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
147	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
148	#95	—	68	2	no	—
149	C₁₇	#13206	1	1	yes	characteristic, complemented, Sylow
150	C₂⁴	—	4	2	no	—
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	4	8	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	2	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂×D₄	—	8	8	no	—
159	C₂²×C₄	—	68	2	no	—
160	C₂²×C₄	#63228	1	2	yes	—
161	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
162	D₈	—	2	8	no	—
163	D₈	—	4	8	no	—
164	D₈	—	4	4	no	—
165	C₂×C₈	#63228	1	2	yes	—
166	C₂×C₈	#63229	1	2	yes	—
167	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
168	C₂×C₈	D₄×D₁₇	1	2	yes	—
169	C₂×C₈	#63254	1	2	yes	—
170	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
171	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
172	C₄⋊C₄	—	68	2	no	—
173	C₄⋊C₄	—	68	2	no	—
174	C₄²	#63229	1	2	yes	—
175	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
176	C₁₆	—	34	4	no	—
177	C₂³	—	4	4	no	—
178	C₂³	—	8	8	no	—
179	C₂³	—	8	2	no	—
180	C₂³	—	8	2	no	—
181	C₂³	C₂×#63228	1	1	yes	characteristic
182	D₄	—	4	4	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	16	no	—
185	D₄	—	8	2	no	—
186	D₄	—	8	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	136	4	no	—
189	C₂×C₄	—	136	2	no	—
190	C₂×C₄	#10587503	1	4	yes	—
191	C₂×C₄	C₂×#63228	1	1	yes	characteristic
192	C₂×C₄	C₂×#63229	1	1	yes	characteristic
193	C₂×C₄	C₂×#63229	1	1	yes	characteristic
194	C₂×C₄	#10587622	1	1	yes	characteristic
195	C₂×C₄	#10587626	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
197	C₈	#10587503	1	4	yes	—
198	C₈	D₄×Dic₁₇	1	2	yes	—
199	C₈	#10587624	1	2	yes	—
200	C₂²	—	8	4	no	—
201	C₂²	—	8	8	no	—
202	C₂²	—	16	4	no	—
203	C₂²	—	16	4	no	—
204	C₂²	C₂×#10587503	1	1	yes	characteristic
205	C₂²	#11763155	1	1	yes	characteristic
206	C₂²	#11763185	1	1	yes	characteristic
207	C₂²	#11763245	1	4	yes	—
208	C₄	—	2	4	no	—
209	C₄	—	272	2	no	—
210	C₄	#11762712	1	1	yes	characteristic
211	C₄	C₂×#10587503	1	1	yes	characteristic
212	C₄	#11763155	1	1	yes	characteristic
213	C₄	#11763181	1	1	yes	characteristic
214	C₂	—	16	4	no	—
215	C₂	#14529591	1	1	yes	characteristic
216	C₂	#14530500	1	2	yes	—
217	C₂	#14531025	1	2	yes	—
218	C₂	#14531048	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762610

C₂

yes

maximal, complemented

C₂×#11762585

C₂

yes

maximal, characteristic, complemented

#14528948

C₂

yes

maximal, characteristic, complemented

C₂×C₁₇×#493

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

—

—

—

—

C₂²×D₈×C₁₇

—

C₂²×D₈×C₁₇

C₂²

yes

—

C₂×C₁₇×#127

—

C₁₇×#493

—

C₁₇×#493

C₄

yes

complemented

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×D₄×C₁₇

—

C₂²×D₄×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

—

C₂×D₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#127

—

C₁₇×#127

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

—

—

D₄

yes

complemented

C₂⁴×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂×D₄×C₁₇

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

D₈×C₁₇

—

D₈×C₁₇

—

D₈×C₁₇

—

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

D₈

yes

—

C₂×C₈×C₁₇

SD₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

SD₁₆

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

#13206

—

maximal, Sylow

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂×D₈

yes

characteristic

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#102

yes

—

C₂×C₄×C₁₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

#102

yes

—

C₈×C₁₇

C₄×D₄

yes

—

C₈×C₁₇

#121

yes

—

C₂×#493

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#102

yes

characteristic

100

C₂²×C₁₇

#486

yes

characteristic

101

C₂²×C₁₇

#493

yes

characteristic

102

C₂²×C₁₇

#509

yes

—

103

C₄×C₁₇

—

104

C₄×C₁₇

#390

yes

characteristic

105

C₄×C₁₇

C₂×#102

yes

characteristic

106

C₄×C₁₇

#486

yes

characteristic

107

C₄×C₁₇

#492

yes

characteristic

108

Dic₁₇

—

109

C₂²×D₈

—

110

C₂²×D₈

C₂×D₁₇

yes

—

111

C₂×#127

—

112

C₂²×C₁₆

—

113

#493

—

114

#493

Dic₁₇

yes

complemented

115

C₂×#107

—

116

C₂×#106

—

117

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

118

#382

—

119

#357

—

120

#346

—

121

C₂×C₁₇

—

122

C₂×C₁₇

#1957

yes

characteristic

123

C₂×C₁₇

#2140

yes

—

124

C₂×C₁₇

#2244

yes

—

125

C₂×C₁₇

#2250

yes

—

126

C₂²×D₄

—

127

C₂²×D₄

—

128

C₂×D₈

—

129

C₂×D₈

—

130

C₂×D₈

—

131

C₂×D₈

—

132

C₂×D₈

—

133

C₂×D₈

—

134

C₂×D₈

C₂×Dic₁₇

yes

—

135

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

136

C₂²×C₈

C₂²×D₁₇

yes

characteristic

137

#127

—

138

#127

—

139

C₂×C₄⋊C₄

—

140

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

141

C₂×C₁₆

—

142

C₂×C₁₆

—

143

C₂×C₁₆

—

144

#107

—

145

#106

—

146

C₄×C₈

C₂×Dic₁₇

yes

—

147

C₄×C₈

C₂²⋊D₁₇

yes

—

148

#95

—

149

C₁₇

#13206

yes

characteristic, complemented, Sylow

150

C₂⁴

—

151

C₂×D₄

—

152

C₂×D₄

—

153

C₂×D₄

—

154

C₂×D₄

—

155

C₂×D₄

—

156

C₂×D₄

—

157

C₂×D₄

—

158

C₂×D₄

—

159

C₂²×C₄

—

160

C₂²×C₄

#63228

yes

—

161

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

162

D₈

—

163

D₈

—

164

D₈

—

165

C₂×C₈

#63228

yes

—

166

C₂×C₈

#63229

yes

—

167

C₂×C₈

C₂²⋊Dic₁₇

yes

—

168

C₂×C₈

D₄×D₁₇

yes

—

169

C₂×C₈

#63254

yes

—

170

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

171

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

172

C₄⋊C₄

—

173

C₄⋊C₄

—

174

C₄²

#63229

yes

—

175

C₄²

C₂²⋊Dic₁₇

yes

—

176

C₁₆

—

177

C₂³

—

178

C₂³

—

179

C₂³

—

180

C₂³

—

181

C₂³

C₂×#63228

yes

characteristic

182

D₄

—

183

D₄

—

184

D₄

—

185

D₄

—

186

D₄

—

187

C₂×C₄

—

188

C₂×C₄

—

136

—

189

C₂×C₄

—

136

—

190

C₂×C₄

#10587503

yes

—

191

C₂×C₄

C₂×#63228

yes

characteristic

192

C₂×C₄

C₂×#63229

yes

characteristic

193

C₂×C₄

C₂×#63229

yes

characteristic

194

C₂×C₄

#10587622

yes

characteristic

195

C₂×C₄

#10587626

yes

characteristic

196

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

197

C₈

#10587503

yes

—

198

C₈

D₄×Dic₁₇

yes

—

199

C₈

#10587624

yes

—

200

C₂²

—

201

C₂²

—

202

C₂²

—

203

C₂²

—

204

C₂²

C₂×#10587503

yes

characteristic

205

C₂²

#11763155

yes

characteristic

206

C₂²

#11763185

yes

characteristic

207

C₂²

#11763245

yes

—

208

C₄

—

209

C₄

—

272

—

210

C₄

#11762712

yes

characteristic

211

C₄

C₂×#10587503

yes

characteristic

212

C₄

#11763155

yes

characteristic

213

C₄

#11763181

yes

characteristic

214

C₂

—

215

C₂

#14529591

yes

characteristic

216

C₂

#14530500

yes

—

217

C₂

#14531025

yes

—

218

C₂

#14531048

yes

—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting, prime radical	C₂×C₁₇×#493	C₂
socle	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#493	D₁₇
hypocenter	C₁₇	#13206
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

subgroup

G/H

center

C₂³

C₂×#63228

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting, prime radical

C₂×C₁₇×#493

C₂

socle

C₂³×C₁₇

C₂×D₈

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

N	H ≅ G/N	type
1. C₂×#11762610	C₂	semidirect
2. C₂×#11762585	C₂	semidirect
3. #14528948	C₂	semidirect
15. C₁₇×#493	C₄	semidirect
39. C₂×#63214	D₄	semidirect
114. #493	Dic₁₇	semidirect
149. C₁₇	#13206	semidirect

H ≅ G/N

type

1. C₂×#11762610

C₂

semidirect

2. C₂×#11762585

C₂

semidirect

3. #14528948

C₂

semidirect

15. C₁₇×#493

C₄

semidirect

39. C₂×#63214

D₄

semidirect

114. #493

Dic₁₇

semidirect

149. C₁₇

#13206

semidirect

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

subgroup	index
C₂×#11762610	2
C₂×#11762585	2
#14528948	2
C₂×C₁₇×#493	2
#13206	17

subgroup

index

C₂×C₁₇×#493

Subgroups — #16952557

Conjugacy classes of subgroups

Complements

Maximal subgroups