Group #16950495 — subgroups

Subgroups — #16950495

Order 4352.

438 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂²×C₁₆×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#11762636	C₂	1	2	yes	maximal, complemented
3	#14528810	C₂	1	2	yes	maximal
4	#14526876	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1380	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×D₁₇	—	2	4	no	—
9	C₂×C₁₆×D₁₇	C₄	1	4	yes	complemented
10	C₂×#10587489	C₂²	1	2	yes	—
11	#11762666	—	2	2	no	—
12	#11762636	—	2	4	no	—
13	#11762571	—	2	2	no	—
14	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
15	C₂×C₁₇×#107	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₇	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
19	C₂×#63227	—	4	2	no	—
20	C₂×C₄⋊Dic₁₇	—	2	2	no	—
21	C₂²×#3622	D₄	1	1	yes	characteristic
22	C₂×C₈×D₁₇	—	2	2	no	—
23	C₂×C₈×D₁₇	—	2	2	no	—
24	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₁₇	Q₈	1	2	yes	—
26	C₂×C₁₆×C₁₇	—	2	2	no	—
27	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
28	C₁₇×#107	—	2	2	no	—
29	C₁₇×#107	—	2	2	no	—
30	#10587502	—	4	2	no	—
31	#10587489	—	2	2	no	—
32	#10587489	—	2	2	no	—
33	C₂×#63214	C₂×C₄	1	2	yes	—
34	C₂×#63214	D₄	1	4	yes	—
35	C₁₆×D₁₇	—	2	8	no	—
36	C₂³×D₁₇	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₇	—	4	2	no	—
38	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₁₇	—	4	2	no	—
40	C₂²×Dic₁₇	D₈	1	1	yes	characteristic
41	C₂×C₄×D₁₇	—	2	2	no	—
42	C₂×C₄×D₁₇	—	2	2	no	—
43	C₂×C₄×D₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×D₁₇	Q₁₆	1	2	yes	—
45	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₁₇	#41	1	2	yes	—
49	C₄⋊C₄×C₁₇	—	4	2	no	—
50	C₄⋊C₄×C₁₇	—	4	2	no	—
51	#63227	—	8	4	no	—
52	C₄⋊Dic₁₇	—	4	2	no	—
53	C₄⋊Dic₁₇	—	4	2	no	—
54	C₂×#3622	—	2	2	no	—
55	C₂×#3622	C₄⋊C₄	1	2	yes	—
56	C₈×D₁₇	—	2	2	no	—
57	C₈×D₁₇	—	2	2	no	—
58	C₈×D₁₇	—	2	4	no	—
59	C₁₆×C₁₇	—	2	2	no	—
60	#63214	C₂²⋊C₄	1	4	yes	complemented
61	#12835	—	17	1	no	maximal, Sylow
62	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
63	C₂²×D₁₇	—	2	2	no	—
64	C₂²×D₁₇	—	2	1	no	—
65	C₂²×D₁₇	—	2	1	no	—
66	C₂²×D₁₇	#107	1	2	yes	—
67	C₂²×D₁₇	SD₃₂	1	4	yes	—
68	C₂×C₄×C₁₇	—	8	2	no	—
69	C₂×C₄×C₁₇	—	8	2	no	—
70	C₂×C₄×C₁₇	—	8	2	no	—
71	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#121	1	2	yes	—
73	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₇	#135	1	2	yes	—
75	C₂×Dic₁₇	—	2	2	no	—
76	C₂×Dic₁₇	—	8	2	no	—
77	C₂×Dic₁₇	—	8	2	no	—
78	C₂×Dic₁₇	—	8	2	no	—
79	C₂×Dic₁₇	#107	1	2	yes	—
80	C₄×D₁₇	—	2	2	no	—
81	C₄×D₁₇	—	2	2	no	—
82	C₄×D₁₇	—	2	4	no	—
83	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₁₇	#117	1	1	yes	characteristic
85	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
86	#3622	—	2	2	no	—
87	C₂³×C₁₆	—	17	1	no	—
88	C₂×#367	—	17	2	no	—
89	#1808	—	17	2	no	—
90	#1380	—	17	1	no	—
91	#1380	D₁₇	1	1	yes	characteristic, complemented
92	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
93	C₂²×C₁₇	#466	1	2	yes	—
94	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
95	C₂²×C₁₇	#508	1	1	yes	characteristic
96	C₂²×C₁₇	#508	1	1	yes	characteristic
97	C₂×D₁₇	—	2	2	no	—
98	C₂×D₁₇	—	2	2	no	—
99	C₂×D₁₇	—	2	4	no	—
100	C₂×D₁₇	—	2	4	no	—
101	C₂×D₁₇	#367	1	4	yes	complemented
102	C₄×C₁₇	—	16	2	no	—
103	C₄×C₁₇	#380	1	1	yes	characteristic
104	C₄×C₁₇	#427	1	1	yes	characteristic
105	C₄×C₁₇	#467	1	2	yes	—
106	Dic₁₇	—	2	2	no	—
107	Dic₁₇	—	16	2	no	—
108	C₂³×C₈	—	17	1	no	—
109	C₂²×C₁₆	—	17	1	no	—
110	C₂²×C₁₆	—	17	4	no	—
111	C₂²×C₁₆	—	34	4	no	—
112	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
113	C₂×#107	—	17	2	no	—
114	C₂×#107	C₂×D₁₇	1	2	yes	—
115	#380	—	34	2	no	—
116	#367	—	34	4	no	—
117	#340	—	34	2	no	—
118	C₂×C₁₇	#1808	1	1	yes	characteristic
119	C₂×C₁₇	C₂×#367	1	1	yes	characteristic
120	C₂×C₁₇	#2158	1	1	yes	characteristic
121	C₂×C₁₇	#2217	1	2	yes	—
122	C₂×C₁₇	#2217	1	2	yes	—
123	D₁₇	—	2	4	no	—
124	C₂³×C₄	—	17	1	no	—
125	C₂²×C₈	—	17	1	no	—
126	C₂²×C₈	—	17	2	no	—
127	C₂²×C₈	—	17	2	no	—
128	C₂²×C₈	—	34	2	no	—
129	C₂²×C₈	—	34	2	no	—
130	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	34	2	no	—
133	C₂×C₂²⋊C₄	—	68	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	17	2	no	—
136	C₂×C₁₆	—	17	4	no	—
137	C₂×C₁₆	—	34	8	no	—
138	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
139	#107	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	—	34	2	no	—
142	#107	—	34	2	no	—
143	#95	—	68	2	no	—
144	C₁₇	#12835	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	—	17	1	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	17	1	no	—
148	C₂²×C₄	—	17	2	no	—
149	C₂²×C₄	—	17	2	no	—
150	C₂²×C₄	—	34	2	no	—
151	C₂²×C₄	—	34	2	no	—
152	C₂²×C₄	—	68	2	no	—
153	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
154	C₂×C₈	—	17	2	no	—
155	C₂×C₈	—	34	2	no	—
156	C₂×C₈	—	34	2	no	—
157	C₂×C₈	—	34	2	no	—
158	C₂×C₈	—	34	4	no	—
159	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
160	C₂×C₈	#63252	1	2	yes	—
161	C₂×C₈	D₄×D₁₇	1	2	yes	—
162	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	68	2	no	—
166	C₄⋊C₄	—	68	2	no	—
167	C₂²⋊C₄	—	136	4	no	—
168	C₁₆	—	2	2	no	—
169	C₁₆	—	17	4	no	—
170	C₂³	—	17	4	no	—
171	C₂³	—	17	2	no	—
172	C₂³	—	34	1	no	—
173	C₂³	—	34	1	no	—
174	C₂³	—	34	2	no	—
175	C₂³	D₈×D₁₇	1	1	yes	characteristic
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	—	8	2	no	—
179	C₂×C₄	—	17	2	no	—
180	C₂×C₄	—	34	2	no	—
181	C₂×C₄	—	34	4	no	—
182	C₂×C₄	—	34	2	no	—
183	C₂×C₄	—	34	2	no	—
184	C₂×C₄	—	136	2	no	—
185	C₂×C₄	—	136	2	no	—
186	C₂×C₄	—	136	2	no	—
187	C₂×C₄	#10587570	1	2	yes	—
188	C₂×C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
189	C₂×C₄	#10587602	1	2	yes	—
190	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
191	C₈	—	34	2	no	—
192	C₈	#10587558	1	1	yes	characteristic
193	C₈	C₂²⋊C₄×D₁₇	1	1	yes	characteristic
194	C₈	#10587582	1	2	yes	—
195	C₂²	—	17	4	no	—
196	C₂²	—	34	2	no	—
197	C₂²	—	34	2	no	—
198	C₂²	—	34	4	no	—
199	C₂²	—	34	4	no	—
200	C₂²	#107×D₁₇	1	1	yes	characteristic
201	C₂²	#11763044	1	2	yes	—
202	C₂²	SD₃₂×D₁₇	1	2	yes	—
203	C₂²	#11763238	1	1	yes	characteristic
204	C₂²	#11763242	1	1	yes	characteristic
205	C₄	—	16	2	no	—
206	C₄	—	272	2	no	—
207	C₄	—	34	2	no	—
208	C₄	#11762668	1	1	yes	characteristic
209	C₄	#11762844	1	1	yes	characteristic
210	C₄	#11763049	1	2	yes	—
211	C₂	—	34	4	no	—
212	C₂	#14528812	1	1	yes	characteristic
213	C₂	D₁₇×#367	1	1	yes	characteristic
214	C₂	#14530565	1	1	yes	characteristic
215	C₂	#14530879	1	2	yes	—
216	C₂	#14530883	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×D₁₇

C₂

yes

maximal, characteristic

C₂×#11762636

C₂

yes

maximal, complemented

#14528810

C₂

yes

maximal

#14526876

C₂

yes

maximal, characteristic, complemented

C₁₇×#1380

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₁₇

C₂²

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

C₂×C₁₆×D₁₇

—

C₂×C₁₆×D₁₇

C₄

yes

complemented

C₂×#10587489

C₂²

yes

—

#11762666

—

#11762636

—

#11762571

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#107

C₂²

yes

—

C₂²×C₄×D₁₇

D₄

yes

characteristic

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×#63227

—

C₂×C₄⋊Dic₁₇

—

C₂²×#3622

D₄

yes

characteristic

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₂×C₈×D₁₇

Q₈

yes

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₁₇×#107

—

C₁₇×#107

—

—

—

—

C₂×C₄

yes

—

C₂×#63214

D₄

yes

—

C₁₆×D₁₇

—

C₂³×D₁₇

D₈

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₈

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₄⋊C₄

yes

—

C₂×C₄×D₁₇

Q₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

#63227

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

C₄⋊C₄

yes

—

C₈×D₁₇

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

C₂²⋊C₄

yes

complemented

#12835

—

maximal, Sylow

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

—

C₂²×D₁₇

#107

yes

—

C₂²×D₁₇

SD₃₂

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#121

yes

—

C₂×C₄×C₁₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₇

#135

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

#107

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

#117

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

#3622

—

C₂³×C₁₆

—

—

—

—

D₁₇

yes

characteristic, complemented

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂²×C₁₇

#466

yes

—

C₂²×C₁₇

C₂×SD₃₂

yes

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

100

C₂×D₁₇

—

101

C₂×D₁₇

#367

yes

complemented

102

C₄×C₁₇

—

103

C₄×C₁₇

#380

yes

characteristic

104

C₄×C₁₇

#427

yes

characteristic

105

C₄×C₁₇

#467

yes

—

106

Dic₁₇

—

107

Dic₁₇

—

108

C₂³×C₈

—

109

C₂²×C₁₆

—

110

C₂²×C₁₆

—

111

C₂²×C₁₆

—

112

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

113

C₂×#107

—

114

C₂×#107

C₂×D₁₇

yes

—

115

#380

—

116

#367

—

117

#340

—

118

C₂×C₁₇

#1808

yes

characteristic

119

C₂×C₁₇

C₂×#367

yes

characteristic

120

C₂×C₁₇

#2158

yes

characteristic

121

C₂×C₁₇

#2217

yes

—

122

C₂×C₁₇

#2217

yes

—

123

D₁₇

—

124

C₂³×C₄

—

125

C₂²×C₈

—

126

C₂²×C₈

—

127

C₂²×C₈

—

128

C₂²×C₈

—

129

C₂²×C₈

—

130

C₂²×C₈

C₂²×D₁₇

yes

characteristic

131

C₂×C₄⋊C₄

—

132

C₂×C₄⋊C₄

—

133

C₂×C₂²⋊C₄

—

134

C₂×C₁₆

—

135

C₂×C₁₆

—

136

C₂×C₁₆

—

137

C₂×C₁₆

—

138

C₂×C₁₆

C₄×D₁₇

yes

—

139

#107

—

140

#107

—

141

#107

—

142

#107

—

143

#95

—

144

C₁₇

#12835

yes

characteristic, complemented, Sylow

145

C₂⁴

—

146

C₂²×C₄

—

147

C₂²×C₄

—

148

C₂²×C₄

—

149

C₂²×C₄

—

150

C₂²×C₄

—

151

C₂²×C₄

—

152

C₂²×C₄

—

153

C₂²×C₄

D₄×D₁₇

yes

characteristic

154

C₂×C₈

—

155

C₂×C₈

—

156

C₂×C₈

—

157

C₂×C₈

—

158

C₂×C₈

—

159

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

160

C₂×C₈

#63252

yes

—

161

C₂×C₈

D₄×D₁₇

yes

—

162

C₂×C₈

Q₈×D₁₇

yes

characteristic

163

C₄⋊C₄

—

164

C₄⋊C₄

—

165

C₄⋊C₄

—

166

C₄⋊C₄

—

167

C₂²⋊C₄

—

136

—

168

C₁₆

—

169

C₁₆

—

170

C₂³

—

171

C₂³

—

172

C₂³

—

173

C₂³

—

174

C₂³

—

175

C₂³

D₈×D₁₇

yes

characteristic

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

—

182

C₂×C₄

—

183

C₂×C₄

—

184

C₂×C₄

—

136

—

185

C₂×C₄

—

136

—

186

C₂×C₄

—

136

—

187

C₂×C₄

#10587570

yes

—

188

C₂×C₄

C₄⋊C₄×D₁₇

yes

characteristic

189

C₂×C₄

#10587602

yes

—

190

C₂×C₄

Q₁₆×D₁₇

yes

characteristic

191

C₈

—

192

C₈

#10587558

yes

characteristic

193

C₈

C₂²⋊C₄×D₁₇

yes

characteristic

194

C₈

#10587582

yes

—

195

C₂²

—

196

C₂²

—

197

C₂²

—

198

C₂²

—

199

C₂²

—

200

C₂²

#107×D₁₇

yes

characteristic

201

C₂²

#11763044

yes

—

202

C₂²

SD₃₂×D₁₇

yes

—

203

C₂²

#11763238

yes

characteristic

204

C₂²

#11763242

yes

characteristic

205

C₄

—

206

C₄

—

272

—

207

C₄

—

208

C₄

#11762668

yes

characteristic

209

C₄

#11762844

yes

characteristic

210

C₄

#11763049

yes

—

211

C₂

—

212

C₂

#14528812

yes

characteristic

213

C₂

D₁₇×#367

yes

characteristic

214

C₂

#14530565

yes

characteristic

215

C₂

#14530879

yes

—

216

C₂

#14530883

yes

—

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1380	C₂
socle	C₂³×C₁₇	C₂×D₈
prime radical	C₂³×D₁₇	D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1380	D₁₇
hypocenter	C₁₇	#12835
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

subgroup

G/H

center

C₂³

D₈×D₁₇

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1380

C₂

socle

C₂³×C₁₇

C₂×D₈

prime radical

C₂³×D₁₇

D₈

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

N	H ≅ G/N	type
2. C₂×#11762636	C₂	semidirect
4. #14526876	C₂	semidirect
5. C₁₇×#1380	C₂	semidirect
9. C₂×C₁₆×D₁₇	C₄	semidirect
60. #63214	C₂²⋊C₄	semidirect
91. #1380	D₁₇	semidirect
101. C₂×D₁₇	#367	semidirect
144. C₁₇	#12835	semidirect

H ≅ G/N

type

2. C₂×#11762636

C₂

semidirect

4. #14526876

C₂

semidirect

5. C₁₇×#1380

C₂

semidirect

9. C₂×C₁₆×D₁₇

C₄

semidirect

60. #63214

C₂²⋊C₄

semidirect

91. #1380

D₁₇

semidirect

101. C₂×D₁₇

#367

semidirect

144. C₁₇

#12835

semidirect

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

subgroup	index
C₂²×C₁₆×D₁₇	2
C₂×#11762636	2
#14528810	2
#14526876	2
C₁₇×#1380	2
#12835	17

subgroup

index

C₂²×C₁₆×D₁₇

C₁₇×#1380

Subgroups — #16950495

Conjugacy classes of subgroups

Complements

Maximal subgroups