Group #16950804 — subgroups

Subgroups — #16950804

Order 4352.

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

normal subgroups only

This group has 370 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	#14530549	C₂	1	2	yes	maximal, complemented
2	#14530398	C₂	1	1	yes	maximal, characteristic
3	#14528584	C₂	1	1	yes	maximal, characteristic
4	#14526874	C₂	1	2	yes	maximal, complemented
5	C₁₇×#3407	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587483	C₂²	1	1	yes	characteristic
7	C₂²×#63214	C₂²	1	1	yes	characteristic
8	#11762793	C₄	1	1	yes	characteristic
9	#11762783	C₄	1	1	yes	characteristic
10	#11762764	C₂²	1	2	yes	—
11	#11762629	—	2	4	no	—
12	C₂²⋊#63214	—	2	2	no	—
13	#11762580	C₄	1	2	yes	—
14	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
15	C₁₇×#504	—	2	4	no	—
16	C₂×C₁₇×M₃₂	—	2	2	no	—
17	C₂×C₁₇×M₃₂	C₂²	1	2	yes	—
18	C₂×#41×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₁₇×#131	—	2	2	no	—
20	C₁₇×#131	—	2	2	no	—
21	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
23	C₂²×C₈×C₁₇	D₄	1	2	yes	—
24	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
25	#10587612	C₂×C₄	1	2	yes	—
26	C₂×#63223	C₂×C₄	1	2	yes	—
27	C₁₇×M₃₂	—	2	4	no	—
28	C₁₇×M₃₂	—	2	2	no	—
29	C₁₇×M₃₂	D₄	1	4	yes	—
30	C₂×C₁₆×C₁₇	—	2	4	no	—
31	C₂×C₁₆×C₁₇	—	2	2	no	—
32	#10587507	C₈	1	2	yes	—
33	#10587505	—	2	2	no	—
34	#10587504	C₈	1	2	yes	—
35	#10587486	—	2	2	no	—
36	C₈×Dic₁₇	—	2	2	no	—
37	#10587483	—	2	2	no	—
38	C₂×#63214	—	2	1	no	—
39	C₂×#63214	—	2	1	no	—
40	C₂×#63214	—	2	2	no	—
41	#41×C₁₇	—	2	2	no	—
42	#41×C₁₇	—	2	2	no	—
43	C₂×Q₈×C₁₇	C₂×C₈	1	1	yes	characteristic
44	C₂×D₄×C₁₇	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₁₇	M₁₆	1	2	yes	—
46	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
48	M₁₆×C₁₇	—	2	2	no	—
49	M₁₆×C₁₇	—	2	4	no	—
50	C₂×C₈×C₁₇	—	2	2	no	—
51	C₂×C₈×C₁₇	—	2	2	no	—
52	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₁₇	#41	1	2	yes	—
57	C₂²⋊Dic₁₇	—	2	2	no	—
58	C₄⋊Dic₁₇	—	2	2	no	—
59	C₄×Dic₁₇	—	2	2	no	—
60	#63223	—	2	2	no	—
61	#63223	C₂×C₈	1	4	yes	—
62	C₂×#3622	—	2	2	no	—
63	C₁₆×C₁₇	—	2	4	no	—
64	#63214	—	4	1	no	—
65	#63214	—	4	1	no	—
66	#12916	—	17	1	no	maximal, Sylow
67	C₂³×C₁₇	C₄⋊C₈	1	2	yes	—
68	C₂³×C₁₇	C₂×M₁₆	1	1	yes	characteristic
69	Q₈×C₁₇	—	2	2	no	—
70	D₄×C₁₇	—	2	4	no	—
71	D₄×C₁₇	—	2	2	no	—
72	C₂×C₄×C₁₇	—	2	2	no	—
73	C₂×C₄×C₁₇	—	2	2	no	—
74	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₇	#131	1	2	yes	—
79	C₂×Dic₁₇	—	4	2	no	—
80	C₈×C₁₇	—	2	2	no	—
81	C₈×C₁₇	#120	1	1	yes	characteristic
82	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
83	C₈×C₁₇	#123	1	1	yes	characteristic
84	#3622	—	2	4	no	—
85	#3407	D₁₇	1	1	yes	characteristic
86	#2153	—	17	2	no	—
87	#2113	—	17	1	no	—
88	#1757	—	17	1	no	—
89	#1379	—	17	2	no	—
90	C₂²×C₁₇	—	2	2	no	—
91	C₂²×C₁₇	—	4	1	no	—
92	C₂²×C₁₇	—	4	2	no	—
93	C₂²×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₂²×C₁₇	#435	1	2	yes	—
95	C₄×C₁₇	—	2	2	no	—
96	C₄×C₁₇	#385	1	1	yes	characteristic
97	C₄×C₁₇	#424	1	1	yes	characteristic
98	C₄×C₁₇	C₈×D₄	1	2	yes	—
99	Dic₁₇	—	8	2	no	—
100	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
101	#504	—	2	4	no	—
102	C₂×M₃₂	—	2	2	no	—
103	C₂×M₃₂	—	17	1	no	—
104	C₂×M₃₂	C₂×D₁₇	1	2	yes	—
105	C₂²×C₁₆	—	17	1	no	—
106	#413	—	17	1	no	—
107	#410	—	17	1	no	—
108	#405	—	17	2	no	—
109	#364	—	34	4	no	—
110	C₂²⋊C₁₆	—	34	2	no	—
111	#344	—	17	2	no	—
112	C₂×C₁₇	—	2	2	no	—
113	C₂×C₁₇	—	2	1	no	—
114	C₂×C₁₇	—	4	2	no	—
115	C₂×C₁₇	#1850	1	1	yes	characteristic
116	C₂×#41	C₄×D₁₇	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	C₂×M₁₆	—	17	2	no	—
120	C₂×M₁₆	Dic₃₄	1	2	yes	—
121	C₂×M₁₆	C₄×D₁₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₁₇	1	2	yes	—
123	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
124	#117	—	17	2	no	—
125	M₃₂	—	2	4	no	—
126	M₃₂	—	2	2	no	—
127	M₃₂	—	34	2	no	—
128	M₃₂	C₂²⋊D₁₇	1	4	yes	—
129	C₂×C₁₆	—	2	4	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	34	2	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	1	no	—
134	#101	—	17	2	no	—
135	#100	—	17	2	no	—
136	#99	—	34	2	no	—
137	#97	—	34	2	no	—
138	C₄×C₈	—	34	2	no	—
139	C₁₇	#12916	1	1	yes	characteristic, complemented, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₈×D₁₇	1	1	yes	characteristic
143	C₂×D₄	C₈×D₁₇	1	1	yes	characteristic
144	C₂×D₄	#63218	1	2	yes	—
145	C₂²×C₄	#63225	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	17	4	no	—
150	M₁₆	—	34	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	34	2	no	—
154	C₂×C₈	#63225	1	2	yes	—
155	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
156	C₂×C₈	#63252	1	2	yes	—
157	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
159	C₄⋊C₄	—	34	2	no	—
160	C₂²⋊C₄	—	34	2	no	—
161	C₄²	—	34	2	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	68	1	no	—
164	C₁₆	—	68	1	no	—
165	C₂³	#10587485	1	2	yes	—
166	C₂³	C₂×#63218	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	D₄	—	2	4	no	—
169	D₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	68	2	no	—
173	C₂×C₄	#10587485	1	2	yes	—
174	C₂×C₄	C₂×C₈×D₁₇	1	1	yes	characteristic
175	C₂×C₄	#10587587	1	2	yes	—
176	C₂×C₄	C₂×#63225	1	1	yes	characteristic
177	C₂×C₄	C₄×C₂²⋊D₁₇	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	34	4	no	—
180	C₈	#10587610	1	2	yes	—
181	C₈	#10587615	1	1	yes	characteristic
182	C₈	#10587639	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×#10587485	1	1	yes	characteristic
187	C₂²	#11762880	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	136	2	no	—
190	C₄	#11762682	1	1	yes	characteristic
191	C₄	#11762835	1	1	yes	characteristic
192	C₄	C₈×C₂²⋊D₁₇	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#14528962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530549

C₂

yes

maximal, complemented

#14530398

C₂

yes

maximal, characteristic

#14528584

C₂

yes

maximal, characteristic

#14526874

C₂

yes

maximal, complemented

C₁₇×#3407

C₂

yes

maximal, characteristic

C₂×#10587483

C₂²

yes

characteristic

C₂²×#63214

C₂²

yes

characteristic

#11762793

C₄

yes

characteristic

#11762783

C₄

yes

characteristic

#11762764

C₂²

yes

—

#11762629

—

C₂²⋊#63214

—

#11762580

C₄

yes

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₁₇×#504

—

C₂×C₁₇×M₃₂

—

C₂×C₁₇×M₃₂

C₂²

yes

—

C₂×#41×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

#10587612

C₂×C₄

yes

—

C₂×#63223

C₂×C₄

yes

—

C₁₇×M₃₂

—

C₁₇×M₃₂

—

C₁₇×M₃₂

D₄

yes

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

#10587507

C₈

yes

—

#10587505

—

#10587504

C₈

yes

—

#10587486

—

C₈×Dic₁₇

—

—

—

—

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₈

yes

characteristic

C₂×D₄×C₁₇

C₂×C₈

yes

characteristic

C₂×D₄×C₁₇

M₁₆

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

#63223

—

#63223

C₂×C₈

yes

—

C₂×#3622

—

C₁₆×C₁₇

—

—

—

—

C₂³×C₁₇

C₄⋊C₈

yes

—

C₂³×C₁₇

C₂×M₁₆

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₇

#131

yes

—

C₂×Dic₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#120

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

yes

—

D₁₇

yes

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#435

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₈×D₄

yes

—

Dic₁₇

—

100

C₂×#131

C₂×D₁₇

yes

characteristic

101

#504

—

102

C₂×M₃₂

—

103

C₂×M₃₂

—

104

C₂×M₃₂

C₂×D₁₇

yes

—

105

C₂²×C₁₆

—

106

#413

—

107

#410

—

108

#405

—

109

#364

—

110

C₂²⋊C₁₆

—

111

#344

—

112

C₂×C₁₇

—

113

C₂×C₁₇

—

114

C₂×C₁₇

—

115

C₂×C₁₇

#1850

yes

characteristic

116

C₂×#41

C₄×D₁₇

yes

characteristic

117

#131

—

118

#131

—

119

C₂×M₁₆

—

120

C₂×M₁₆

Dic₃₄

yes

—

121

C₂×M₁₆

C₄×D₁₇

yes

characteristic

122

C₂²×C₈

C₂²⋊D₁₇

yes

—

123

C₂²×C₈

C₂²×D₁₇

yes

characteristic

124

#117

—

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

C₂²⋊D₁₇

yes

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#101

—

135

#100

—

136

#99

—

137

#97

—

138

C₄×C₈

—

139

C₁₇

#12916

yes

characteristic, complemented, Sylow

140

#41

—

141

#41

—

142

C₂×Q₈

C₈×D₁₇

yes

characteristic

143

C₂×D₄

C₈×D₁₇

yes

characteristic

144

C₂×D₄

#63218

yes

—

145

C₂²×C₄

#63225

yes

—

146

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

#63225

yes

—

155

C₂×C₈

C₂×Dic₃₄

yes

characteristic

156

C₂×C₈

#63252

yes

—

157

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

158

C₂×C₈

C₂×C₂²⋊D₁₇

yes

characteristic

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

#10587485

yes

—

166

C₂³

C₂×#63218

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

#10587485

yes

—

174

C₂×C₄

C₂×C₈×D₁₇

yes

characteristic

175

C₂×C₄

#10587587

yes

—

176

C₂×C₄

C₂×#63225

yes

characteristic

177

C₂×C₄

C₄×C₂²⋊D₁₇

yes

—

178

C₈

—

179

C₈

—

180

C₈

#10587610

yes

—

181

C₈

#10587615

yes

characteristic

182

C₈

#10587639

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×#10587485

yes

characteristic

187

C₂²

#11762880

yes

—

188

C₄

—

189

C₄

—

136

—

190

C₄

#11762682

yes

characteristic

191

C₄

#11762835

yes

characteristic

192

C₄

C₈×C₂²⋊D₁₇

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#14528962

yes

characteristic

	subgroup	G/H
center	C₈	#10587639
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#3407	C₂
socle	C₂×C₁₇	#1850
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#3407	D₁₇
hypocenter	C₁₇	#12916
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₈

#10587639

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#3407

C₂

socle

C₂×C₁₇

#1850

prime radical

C₂×#41×C₁₇

C₂×C₄

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
1. #14530549	C₂	semidirect
4. #14526874	C₂	semidirect
139. C₁₇	#12916	semidirect

H ≅ G/N

type

1. #14530549

C₂

semidirect

4. #14526874

C₂

semidirect

139. C₁₇

#12916

semidirect

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

subgroup	index
#14530549	2
#14530398	2
#14528584	2
#14526874	2
C₁₇×#3407	2
#12916	17

subgroup

index

C₁₇×#3407

Subgroups — #16950804

Conjugacy classes of subgroups

Complements

Maximal subgroups