Group #16952561 — subgroups

Subgroups — #16952561

Order 4352.

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

normal subgroups only

This group has 486 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×#11762613	C₂	1	4	yes	maximal
2	C₂×#11762585	C₂	1	1	yes	maximal, characteristic
3	#14528948	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×#494	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	#11762613	—	2	8	no	—
10	#11762585	—	2	2	no	—
11	C₂²×Q₁₆×C₁₇	—	2	1	no	—
12	C₂²×Q₁₆×C₁₇	C₂²	1	2	yes	—
13	C₂×C₁₇×C₄⋊Q₈	—	2	1	no	—
14	C₁₇×#494	—	2	2	no	—
15	C₁₇×#494	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₁₇	—	2	2	no	—
18	C₂×Q₁₆×C₁₇	—	2	4	no	—
19	C₂×Q₁₆×C₁₇	—	2	4	no	—
20	C₂×Q₁₆×C₁₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₇	—	2	4	no	—
22	C₂×Q₁₆×C₁₇	—	4	2	no	—
23	C₂×Q₁₆×C₁₇	—	4	2	no	—
24	C₂×Q₁₆×C₁₇	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₇	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
27	C₁₇×C₄⋊Q₈	—	2	4	no	—
28	C₁₇×C₄⋊Q₈	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₇	—	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	—
40	C₂×Q₈×C₁₇	—	2	4	no	—
41	C₂×Q₈×C₁₇	—	4	2	no	—
42	C₂×Q₈×C₁₇	—	4	2	no	—
43	C₂×Q₈×C₁₇	—	4	2	no	—
44	C₂×Q₈×C₁₇	—	4	2	no	—
45	C₂×Q₈×C₁₇	—	4	2	no	—
46	C₂²×C₄×C₁₇	—	4	2	no	—
47	C₂²×C₄×C₁₇	D₈	1	2	yes	—
48	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₁₇	—	4	2	no	—
50	Q₁₆×C₁₇	—	2	8	no	—
51	Q₁₆×C₁₇	—	4	8	no	—
52	Q₁₆×C₁₇	—	4	4	no	—
53	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₇	D₈	1	2	yes	—
55	C₂×C₈×C₁₇	SD₁₆	1	2	yes	—
56	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₇	—	4	8	no	—
61	C₄²×C₁₇	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₁₇	SD₁₆	1	2	yes	—
63	C₄⋊Dic₁₇	—	4	2	no	—
64	C₄⋊Dic₁₇	—	4	2	no	—
65	#63214	—	2	4	no	—
66	#13207	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₁₇	—	4	4	no	—
69	Q₈×C₁₇	—	4	4	no	—
70	Q₈×C₁₇	—	8	2	no	—
71	Q₈×C₁₇	—	8	2	no	—
72	C₂×C₄×C₁₇	—	2	4	no	—
73	C₂×C₄×C₁₇	—	4	4	no	—
74	C₂×C₄×C₁₇	—	8	2	no	—
75	C₂×C₄×C₁₇	—	8	2	no	—
76	C₂×C₄×C₁₇	#102	1	4	yes	—
77	C₂×C₄×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×Dic₁₇	—	8	2	no	—
84	C₂×Dic₁₇	—	8	4	no	—
85	C₈×C₁₇	#102	1	4	yes	—
86	C₈×C₁₇	C₄×D₄	1	2	yes	—
87	C₈×C₁₇	#121	1	2	yes	—
88	C₂×#494	D₁₇	1	1	yes	characteristic
89	C₂×#358	—	17	4	no	—
90	C₂×#346	—	17	1	no	—
91	#1846	—	17	1	no	—
92	C₂²×C₁₇	C₂×#102	1	1	yes	characteristic
93	C₂²×C₁₇	#486	1	1	yes	characteristic
94	C₂²×C₁₇	#493	1	1	yes	characteristic
95	C₂²×C₁₇	#510	1	4	yes	—
96	C₄×C₁₇	—	2	4	no	—
97	C₄×C₁₇	—	8	4	no	—
98	C₄×C₁₇	#390	1	1	yes	characteristic
99	C₄×C₁₇	C₂×#102	1	1	yes	characteristic
100	C₄×C₁₇	#486	1	1	yes	characteristic
101	C₄×C₁₇	#492	1	1	yes	characteristic
102	Dic₁₇	—	16	2	no	—
103	C₂²×Q₁₆	—	2	1	no	—
104	C₂²×Q₁₆	C₂×D₁₇	1	2	yes	—
105	C₂×C₄⋊Q₈	—	2	1	no	—
106	C₂²×C₁₆	—	17	2	no	—
107	#494	—	2	2	no	—
108	#494	Dic₁₇	1	4	yes	complemented
109	C₂×#107	—	17	2	no	—
110	C₂×#106	—	34	1	no	—
111	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
112	#382	—	34	1	no	—
113	#358	—	34	8	no	—
114	#346	—	34	2	no	—
115	C₂×C₁₇	#1957	1	1	yes	characteristic
116	C₂×C₁₇	#2141	1	2	yes	—
117	C₂×C₁₇	#2245	1	2	yes	—
118	C₂×C₁₇	#2251	1	2	yes	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂×Q₁₆	—	2	4	no	—
121	C₂×Q₁₆	—	2	4	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	2	no	—
124	C₂×Q₁₆	—	4	2	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	C₂×Dic₁₇	1	4	yes	—
127	C₂²×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	4	2	no	—
131	C₂×C₄⋊C₄	—	4	2	no	—
132	C₂×C₄⋊C₄	—	34	2	no	—
133	C₂×C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
134	C₂×C₁₆	—	17	4	no	—
135	C₂×C₁₆	—	34	2	no	—
136	C₂×C₁₆	—	34	2	no	—
137	#107	—	34	4	no	—
138	#106	—	68	2	no	—
139	C₄×C₈	C₂×Dic₁₇	1	2	yes	—
140	C₄×C₈	C₂²⋊D₁₇	1	2	yes	—
141	#95	—	68	2	no	—
142	C₁₇	#13207	1	1	yes	characteristic, complemented, Sylow
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	4	2	no	—
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	68	2	no	—
151	C₂²×C₄	#63228	1	2	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
153	Q₁₆	—	2	8	no	—
154	Q₁₆	—	4	8	no	—
155	Q₁₆	—	4	4	no	—
156	C₂×C₈	#63228	1	2	yes	—
157	C₂×C₈	#63229	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₁₇	1	2	yes	—
159	C₂×C₈	D₄×D₁₇	1	2	yes	—
160	C₂×C₈	#63254	1	2	yes	—
161	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
163	C₄⋊C₄	—	4	8	no	—
164	C₄⋊C₄	—	68	2	no	—
165	C₄⋊C₄	—	68	2	no	—
166	C₄²	#63229	1	2	yes	—
167	C₄²	C₂²⋊Dic₁₇	1	2	yes	—
168	C₁₆	—	34	4	no	—
169	C₂³	C₂×#63228	1	1	yes	characteristic
170	Q₈	—	4	4	no	—
171	Q₈	—	4	4	no	—
172	Q₈	—	8	2	no	—
173	Q₈	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	136	4	no	—
179	C₂×C₄	—	136	2	no	—
180	C₂×C₄	#10587503	1	4	yes	—
181	C₂×C₄	C₂×#63228	1	1	yes	characteristic
182	C₂×C₄	C₂×#63229	1	1	yes	characteristic
183	C₂×C₄	C₂×#63229	1	1	yes	characteristic
184	C₂×C₄	#10587622	1	1	yes	characteristic
185	C₂×C₄	#10587626	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
187	C₈	#10587503	1	4	yes	—
188	C₈	D₄×Dic₁₇	1	2	yes	—
189	C₈	#10587624	1	2	yes	—
190	C₂²	C₂×#10587503	1	1	yes	characteristic
191	C₂²	#11763155	1	1	yes	characteristic
192	C₂²	#11763185	1	1	yes	characteristic
193	C₂²	#11763250	1	4	yes	—
194	C₄	—	2	4	no	—
195	C₄	—	8	4	no	—
196	C₄	—	272	2	no	—
197	C₄	#11762712	1	1	yes	characteristic
198	C₄	C₂×#10587503	1	1	yes	characteristic
199	C₄	#11763155	1	1	yes	characteristic
200	C₄	#11763181	1	1	yes	characteristic
201	C₂	#14529591	1	1	yes	characteristic
202	C₂	#14530504	1	2	yes	—
203	C₂	#14531029	1	2	yes	—
204	C₂	#14531053	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762613

C₂

yes

maximal

C₂×#11762585

C₂

yes

maximal, characteristic

#14528948

C₂

yes

maximal, characteristic

C₂×C₁₇×#494

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

—

C₂×#10587489

C₂²

yes

—

—

—

—

—

C₂²×Q₁₆×C₁₇

—

C₂²×Q₁₆×C₁₇

C₂²

yes

—

C₂×C₁₇×C₄⋊Q₈

—

C₁₇×#494

—

C₁₇×#494

C₄

yes

complemented

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

—

C₂×Q₁₆×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

D₄

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

—

—

—

—

—

D₄

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₈

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

Q₁₆×C₁₇

—

Q₁₆×C₁₇

—

Q₁₆×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

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂²⋊C₄

yes

—

C₄²×C₁₇

SD₁₆

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63214

—

#13207

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×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×#494

D₁₇

yes

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

100

C₄×C₁₇

#486

yes

characteristic

101

C₄×C₁₇

#492

yes

characteristic

102

Dic₁₇

—

103

C₂²×Q₁₆

—

104

C₂²×Q₁₆

C₂×D₁₇

yes

—

105

C₂×C₄⋊Q₈

—

106

C₂²×C₁₆

—

107

#494

—

108

#494

Dic₁₇

yes

complemented

109

C₂×#107

—

110

C₂×#106

—

111

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

112

#382

—

113

#358

—

114

#346

—

115

C₂×C₁₇

#1957

yes

characteristic

116

C₂×C₁₇

#2141

yes

—

117

C₂×C₁₇

#2245

yes

—

118

C₂×C₁₇

#2251

yes

—

119

C₂²×Q₈

—

120

C₂×Q₁₆

—

121

C₂×Q₁₆

—

122

C₂×Q₁₆

—

123

C₂×Q₁₆

—

124

C₂×Q₁₆

—

125

C₂×Q₁₆

—

126

C₂×Q₁₆

C₂×Dic₁₇

yes

—

127

C₂²×C₈

C₂²⋊D₁₇

yes

characteristic

128

C₂²×C₈

C₂²×D₁₇

yes

characteristic

129

C₄⋊Q₈

—

130

C₄⋊Q₈

—

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₄²

C₂²⋊D₁₇

yes

characteristic

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

#107

—

138

#106

—

139

C₄×C₈

C₂×Dic₁₇

yes

—

140

C₄×C₈

C₂²⋊D₁₇

yes

—

141

#95

—

142

C₁₇

#13207

yes

characteristic, complemented, Sylow

143

C₂×Q₈

—

144

C₂×Q₈

—

145

C₂×Q₈

—

146

C₂×Q₈

—

147

C₂×Q₈

—

148

C₂×Q₈

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

#63228

yes

—

152

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

153

Q₁₆

—

154

Q₁₆

—

155

Q₁₆

—

156

C₂×C₈

#63228

yes

—

157

C₂×C₈

#63229

yes

—

158

C₂×C₈

C₂²⋊Dic₁₇

yes

—

159

C₂×C₈

D₄×D₁₇

yes

—

160

C₂×C₈

#63254

yes

—

161

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

162

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄²

#63229

yes

—

167

C₄²

C₂²⋊Dic₁₇

yes

—

168

C₁₆

—

169

C₂³

C₂×#63228

yes

characteristic

170

Q₈

—

171

Q₈

—

172

Q₈

—

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

136

—

179

C₂×C₄

—

136

—

180

C₂×C₄

#10587503

yes

—

181

C₂×C₄

C₂×#63228

yes

characteristic

182

C₂×C₄

C₂×#63229

yes

characteristic

183

C₂×C₄

C₂×#63229

yes

characteristic

184

C₂×C₄

#10587622

yes

characteristic

185

C₂×C₄

#10587626

yes

characteristic

186

C₂×C₄

C₂×C₂²⋊Dic₁₇

yes

characteristic

187

C₈

#10587503

yes

—

188

C₈

D₄×Dic₁₇

yes

—

189

C₈

#10587624

yes

—

190

C₂²

C₂×#10587503

yes

characteristic

191

C₂²

#11763155

yes

characteristic

192

C₂²

#11763185

yes

characteristic

193

C₂²

#11763250

yes

—

194

C₄

—

195

C₄

—

196

C₄

—

272

—

197

C₄

#11762712

yes

characteristic

198

C₄

C₂×#10587503

yes

characteristic

199

C₄

#11763155

yes

characteristic

200

C₄

#11763181

yes

characteristic

201

C₂

#14529591

yes

characteristic

202

C₂

#14530504

yes

—

203

C₂

#14531029

yes

—

204

C₂

#14531053

yes

—

	subgroup	G/H
center	C₂³	C₂×#63228
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#494	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#494	D₁₇
hypocenter	C₁₇	#13207
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

subgroup

G/H

center

C₂³

C₂×#63228

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#494

C₂

socle, prime radical

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×#494

N	H ≅ G/N	type
15. C₁₇×#494	C₄	semidirect
108. #494	Dic₁₇	semidirect
142. C₁₇	#13207	semidirect

H ≅ G/N

type

15. C₁₇×#494

C₄

semidirect

108. #494

Dic₁₇

semidirect

142. C₁₇

#13207

semidirect

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

subgroup	index
C₂×#11762613	2
C₂×#11762585	2
#14528948	2
C₂×C₁₇×#494	2
#13207	17

subgroup

index

C₂×C₁₇×#494

Subgroups — #16952561

Conjugacy classes of subgroups

Complements

Maximal subgroups