Group #16952357 — subgroups

Subgroups — #16952357

Order 4352.

326 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 135 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×#11762637	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#14529515	C₂	1	2	yes	maximal
4	#14526875	C₂	1	2	yes	maximal
5	C₂×C₁₇×#367	C₂	1	1	yes	maximal, characteristic
6	C₂²×#63214	C₂²	1	1	yes	characteristic
7	C₂×#10587489	C₂²	1	2	yes	—
8	C₂×C₈×Dic₁₇	C₂²	1	1	yes	characteristic
9	#11762705	—	2	2	no	—
10	#11762637	C₄	1	4	yes	complemented
11	C₁₆×Dic₁₇	—	2	2	no	—
12	#11762571	—	2	2	no	—
13	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₇×#107	C₂²	1	2	yes	—
15	C₁₇×#367	—	2	2	no	—
16	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
18	C₂×C₄⋊Dic₁₇	—	2	2	no	—
19	C₂×#63225	—	4	2	no	—
20	C₂×C₄×Dic₁₇	D₄	1	1	yes	characteristic
21	C₂²×#3622	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
24	C₂×C₁₆×C₁₇	Q₈	1	2	yes	—
25	C₁₇×#107	—	2	2	no	—
26	C₁₇×#107	—	2	2	no	—
27	#10587502	—	4	2	no	—
28	#10587489	—	2	2	no	—
29	#10587489	C₂×C₄	1	4	yes	—
30	C₈×Dic₁₇	—	2	1	no	—
31	C₈×Dic₁₇	—	2	1	no	—
32	C₂×#63214	—	2	1	no	—
33	C₂×#63214	—	2	1	no	—
34	C₂×#63214	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₇	—	4	2	no	—
36	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₇	—	4	2	no	—
38	C₂²×Dic₁₇	D₈	1	2	yes	—
39	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₁₇	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₁₇	—	4	2	no	—
46	C₄⋊C₄×C₁₇	—	4	2	no	—
47	C₄⋊Dic₁₇	—	2	4	no	—
48	C₄⋊Dic₁₇	—	4	2	no	—
49	#63225	—	8	4	no	—
50	C₄×Dic₁₇	—	2	1	no	—
51	C₄×Dic₁₇	—	2	1	no	—
52	C₂×#3622	—	2	1	no	—
53	C₂×#3622	—	2	1	no	—
54	C₂×#3622	Q₁₆	1	2	yes	—
55	C₁₆×C₁₇	C₄⋊C₄	1	4	yes	complemented
56	#63214	—	2	2	no	—
57	#13166	—	17	1	no	maximal, Sylow
58	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₇	—	8	2	no	—
60	C₂×C₄×C₁₇	—	8	2	no	—
61	C₂×C₄×C₁₇	—	8	2	no	—
62	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
67	C₂×C₄×C₁₇	#135	1	1	yes	characteristic
68	C₂×Dic₁₇	—	2	2	no	—
69	C₂×Dic₁₇	—	2	2	no	—
70	C₂×Dic₁₇	—	4	4	no	—
71	C₂×Dic₁₇	—	8	2	no	—
72	C₂×Dic₁₇	—	8	2	no	—
73	C₂×Dic₁₇	SD₃₂	1	4	yes	—
74	C₈×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₈×C₁₇	C₄×Q₈	1	1	yes	characteristic
76	C₈×C₁₇	C₂²⋊Q₈	1	1	yes	characteristic
77	C₈×C₁₇	#125	1	1	yes	characteristic
78	#3622	—	2	2	no	—
79	C₂×#367	—	17	1	no	—
80	C₂×#367	D₁₇	1	1	yes	characteristic
81	C₂×C₄×C₁₆	—	17	1	no	—
82	#1942	—	17	2	no	—
83	#1380	—	17	2	no	—
84	C₂²×C₁₇	C₄×D₈	1	1	yes	characteristic
85	C₂²×C₁₇	#466	1	1	yes	characteristic
86	C₂²×C₁₇	#500	1	1	yes	characteristic
87	C₂²×C₁₇	C₂×SD₃₂	1	2	yes	—
88	C₂²×C₁₇	#508	1	2	yes	—
89	C₄×C₁₇	—	16	2	no	—
90	C₄×C₁₇	#389	1	1	yes	characteristic
91	C₄×C₁₇	C₄×Q₁₆	1	1	yes	characteristic
92	C₄×C₁₇	#467	1	1	yes	characteristic
93	C₄×C₁₇	#499	1	1	yes	characteristic
94	Dic₁₇	—	2	4	no	—
95	Dic₁₇	—	8	4	no	—
96	C₂²×C₁₆	—	17	1	no	—
97	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
98	C₂×#107	—	17	2	no	—
99	C₂×#107	C₂×D₁₇	1	2	yes	—
100	C₂×C₄×C₈	—	17	1	no	—
101	#389	—	34	2	no	—
102	#367	—	2	2	no	—
103	#367	—	17	4	no	—
104	C₄×C₁₆	—	34	2	no	—
105	#340	—	34	2	no	—
106	C₂×C₁₇	#1942	1	1	yes	characteristic
107	C₂×C₁₇	C₄×SD₃₂	1	2	yes	—
108	C₂×C₁₇	#2217	1	2	yes	—
109	C₂×C₁₇	#2253	1	1	yes	characteristic
110	C₂×C₁₇	#2254	1	1	yes	characteristic
111	C₂²×C₈	—	17	1	no	—
112	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	34	2	no	—
115	C₂×C₄⋊C₄	—	68	2	no	—
116	C₂×C₄²	—	17	1	no	—
117	C₂×C₁₆	—	17	2	no	—
118	C₂×C₁₆	—	34	1	no	—
119	C₂×C₁₆	—	34	1	no	—
120	C₂×C₁₆	Dic₃₄	1	2	yes	—
121	C₂×C₁₆	C₄×D₁₇	1	2	yes	—
122	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	17	4	no	—
126	#107	—	34	2	no	—
127	C₄×C₈	—	34	1	no	—
128	C₄×C₈	—	34	1	no	—
129	#95	—	68	2	no	—
130	C₁₇	#13166	1	1	yes	characteristic, complemented, Sylow
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	17	2	no	—
133	C₂²×C₄	—	68	2	no	—
134	C₂²×C₄	D₄×D₁₇	1	1	yes	characteristic
135	C₂×C₈	—	17	2	no	—
136	C₂×C₈	—	34	1	no	—
137	C₂×C₈	—	34	1	no	—
138	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
140	C₂×C₈	#63254	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₁₇	1	1	yes	characteristic
142	C₂×C₈	#63256	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	34	4	no	—
147	C₄⋊C₄	—	68	2	no	—
148	C₄⋊C₄	—	136	4	no	—
149	C₄²	—	34	1	no	—
150	C₄²	—	34	1	no	—
151	C₁₆	—	34	2	no	—
152	C₁₆	#63225	1	4	yes	complemented
153	C₂³	D₈×D₁₇	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	17	4	no	—
158	C₂×C₄	—	34	2	no	—
159	C₂×C₄	—	34	2	no	—
160	C₂×C₄	—	68	4	no	—
161	C₂×C₄	—	136	2	no	—
162	C₂×C₄	—	136	2	no	—
163	C₂×C₄	#10587574	1	1	yes	characteristic
164	C₂×C₄	#10587579	1	1	yes	characteristic
165	C₂×C₄	#10587598	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₁₇	1	1	yes	characteristic
167	C₂×C₄	#10587605	1	1	yes	characteristic
168	C₂×C₄	#10587624	1	1	yes	characteristic
169	C₈	—	34	2	no	—
170	C₈	#10587573	1	1	yes	characteristic
171	C₈	#10587576	1	1	yes	characteristic
172	C₈	C₂×#63225	1	1	yes	characteristic
173	C₈	#10587632	1	1	yes	characteristic
174	C₂²	#11762891	1	1	yes	characteristic
175	C₂²	#11763042	1	1	yes	characteristic
176	C₂²	#11763213	1	1	yes	characteristic
177	C₂²	SD₃₂×D₁₇	1	2	yes	—
178	C₂²	#11763240	1	2	yes	—
179	C₄	—	16	2	no	—
180	C₄	—	34	4	no	—
181	C₄	—	136	4	no	—
182	C₄	#11762705	1	1	yes	characteristic
183	C₄	#11762903	1	1	yes	characteristic
184	C₄	#11763047	1	1	yes	characteristic
185	C₄	#11763210	1	1	yes	characteristic
186	C₂	#14529515	1	1	yes	characteristic
187	C₂	#14530635	1	2	yes	—
188	C₂	#14530877	1	2	yes	—
189	C₂	#14531062	1	1	yes	characteristic
190	C₂	#14531065	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762637

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₁₇

C₂

yes

maximal, characteristic

#14529515

C₂

yes

maximal

#14526875

C₂

yes

maximal

C₂×C₁₇×#367

C₂

yes

maximal, characteristic

C₂²×#63214

C₂²

yes

characteristic

C₂×#10587489

C₂²

yes

—

C₂×C₈×Dic₁₇

C₂²

yes

characteristic

#11762705

—

#11762637

C₄

yes

complemented

C₁₆×Dic₁₇

—

#11762571

—

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×C₁₇×#107

C₂²

yes

—

C₁₇×#367

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄⋊Dic₁₇

—

C₂×#63225

—

C₂×C₄×Dic₁₇

D₄

yes

characteristic

C₂²×#3622

D₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

Q₈

yes

—

C₁₇×#107

—

C₁₇×#107

—

#10587502

—

#10587489

—

#10587489

C₂×C₄

yes

—

C₈×Dic₁₇

—

C₈×Dic₁₇

—

C₂×#63214

—

C₂×#63214

—

C₂×#63214

Q₈

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₈

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

C₂×C₈×C₁₇

yes

C₂×C₈×C₁₇

yes

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

Q₁₆

yes

—

C₁₆×C₁₇

C₄⋊C₄

yes

—

—

C₂³×C₁₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄×D₄

yes

characteristic

C₂×C₄×C₁₇

#121

yes

characteristic

C₂×C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₁₆

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

SD₃₂

yes

—

C₈×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₄×Q₈

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

C₈×C₁₇

yes

—

—

D₁₇

yes

characteristic

C₂×C₄×C₁₆

—

#1942

—

#1380

—

C₂²×C₁₇

C₄×D₈

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

C₂×SD₃₂

yes

—

C₂²×C₁₇

#508

yes

—

C₄×C₁₇

—

C₄×C₁₇

#389

yes

characteristic

C₄×C₁₇

C₄×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

Dic₁₇

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

C₂×#107

—

C₂×#107

C₂×D₁₇

yes

—

100

C₂×C₄×C₈

—

101

#389

—

102

#367

—

103

#367

—

104

C₄×C₁₆

—

105

#340

—

106

C₂×C₁₇

#1942

yes

characteristic

107

C₂×C₁₇

C₄×SD₃₂

yes

—

108

C₂×C₁₇

#2217

yes

—

109

C₂×C₁₇

#2253

yes

characteristic

110

C₂×C₁₇

#2254

yes

characteristic

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₁₇

yes

characteristic

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄²

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

Dic₃₄

yes

—

121

C₂×C₁₆

C₄×D₁₇

yes

—

122

C₂×C₁₆

C₂²⋊D₁₇

yes

—

123

#107

—

124

#107

—

125

#107

—

126

#107

—

127

C₄×C₈

—

128

C₄×C₈

—

129

#95

—

130

C₁₇

#13166

yes

characteristic, complemented, Sylow

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

D₄×D₁₇

yes

characteristic

135

C₂×C₈

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

C₂×Dic₃₄

yes

characteristic

139

C₂×C₈

C₂×C₄×D₁₇

yes

characteristic

140

C₂×C₈

#63254

yes

characteristic

141

C₂×C₈

Q₈×D₁₇

yes

characteristic

142

C₂×C₈

#63256

yes

characteristic

143

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

136

—

149

C₄²

—

150

C₄²

—

151

C₁₆

—

152

C₁₆

#63225

yes

complemented

153

C₂³

D₈×D₁₇

yes

characteristic

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

136

—

162

C₂×C₄

—

136

—

163

C₂×C₄

#10587574

yes

characteristic

164

C₂×C₄

#10587579

yes

characteristic

165

C₂×C₄

#10587598

yes

characteristic

166

C₂×C₄

Q₁₆×D₁₇

yes

characteristic

167

C₂×C₄

#10587605

yes

characteristic

168

C₂×C₄

#10587624

yes

characteristic

169

C₈

—

170

C₈

#10587573

yes

characteristic

171

C₈

#10587576

yes

characteristic

172

C₈

C₂×#63225

yes

characteristic

173

C₈

#10587632

yes

characteristic

174

C₂²

#11762891

yes

characteristic

175

C₂²

#11763042

yes

characteristic

176

C₂²

#11763213

yes

characteristic

177

C₂²

SD₃₂×D₁₇

yes

—

178

C₂²

#11763240

yes

—

179

C₄

—

180

C₄

—

181

C₄

—

136

—

182

C₄

#11762705

yes

characteristic

183

C₄

#11762903

yes

characteristic

184

C₄

#11763047

yes

characteristic

185

C₄

#11763210

yes

characteristic

186

C₂

#14529515

yes

characteristic

187

C₂

#14530635

yes

—

188

C₂

#14530877

yes

—

189

C₂

#14531062

yes

characteristic

190

C₂

#14531065

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₇
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#367	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₈
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#367	D₁₇
hypocenter	C₁₇	#13166
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂³

D₈×D₁₇

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#367

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 ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
10. #11762637	C₄	semidirect
55. C₁₆×C₁₇	C₄⋊C₄	semidirect
130. C₁₇	#13166	semidirect
152. C₁₆	#63225	semidirect

H ≅ G/N

type

10. #11762637

C₄

semidirect

55. C₁₆×C₁₇

C₄⋊C₄

semidirect

130. C₁₇

#13166

semidirect

152. C₁₆

#63225

semidirect

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

subgroup	index
C₂×#11762637	2
C₂×C₁₆×Dic₁₇	2
#14529515	2
#14526875	2
C₂×C₁₇×#367	2
#13166	17

subgroup

index

C₂×C₁₆×Dic₁₇

C₂×C₁₇×#367

Subgroups — #16952357

Conjugacy classes of subgroups

Complements

Maximal subgroups