Group #16950806 — subgroups

Subgroups — #16950806

Order 4352.

370 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 147 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	#14541887	C₂	1	1	yes	maximal, characteristic
2	#14530550	C₂	1	2	yes	maximal, complemented
3	#14530398	C₂	1	1	yes	maximal, characteristic
4	#14526873	C₂	1	2	yes	maximal, complemented
5	C₁₇×#1757	C₂	1	1	yes	maximal, characteristic
6	#11763226	—	2	4	no	—
7	C₂×#10587483	—	2	2	no	—
8	C₂×#10587483	C₂²	1	1	yes	characteristic
9	C₂×#10587483	C₂²	1	2	yes	—
10	C₂²×#63214	C₂²	1	1	yes	characteristic
11	#11762630	—	2	4	no	—
12	C₂²⋊#63214	—	2	2	no	—
13	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
14	C₁₇×#413	C₄	1	1	yes	characteristic
15	C₁₇×#410	C₄	1	1	yes	characteristic
16	C₁₇×#405	C₂²	1	2	yes	—
17	C₁₇×#344	C₄	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	2	yes	—
22	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₇	D₄	1	2	yes	—
25	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
26	C₁₇×#117	C₂×C₄	1	2	yes	—
27	C₁₇×#101	C₈	1	2	yes	—
28	C₁₇×#100	C₈	1	2	yes	—
29	C₁₇×#99	—	2	2	no	—
30	C₁₇×#97	—	2	2	no	—
31	C₄×C₈×C₁₇	—	2	2	no	—
32	#10587483	—	2	4	no	—
33	#10587483	—	2	2	no	—
34	#10587483	—	2	2	no	—
35	#10587483	D₄	1	4	yes	—
36	C₂×#63214	—	2	4	no	—
37	C₂×#63214	—	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	2	no	—
50	M₁₆×C₁₇	—	2	4	no	—
51	M₁₆×C₁₇	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₁₇	—	2	2	no	—
53	C₂×C₈×C₁₇	—	2	2	no	—
54	C₂×C₈×C₁₇	—	2	2	no	—
55	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₇	#41	1	2	yes	—
60	C₄⋊C₄×C₁₇	—	2	2	no	—
61	C₂²⋊C₄×C₁₇	—	2	2	no	—
62	C₄²×C₁₇	—	2	2	no	—
63	#63214	—	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	—	4	2	no	—
75	C₂×C₄×C₁₇	C₄⋊C₈	1	2	yes	—
76	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₇	#131	1	2	yes	—
80	C₈×C₁₇	—	2	2	no	—
81	C₈×C₁₇	—	2	4	no	—
82	C₈×C₁₇	#120	1	1	yes	characteristic
83	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
84	C₈×C₁₇	#123	1	1	yes	characteristic
85	#3407	—	17	1	no	—
86	#2153	—	17	2	no	—
87	#2113	—	17	1	no	—
88	#1757	D₁₇	1	1	yes	characteristic
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	—	8	2	no	—
97	C₄×C₁₇	#385	1	1	yes	characteristic
98	C₄×C₁₇	#424	1	1	yes	characteristic
99	C₄×C₁₇	C₈×D₄	1	2	yes	—
100	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
101	#504	—	34	4	no	—
102	C₂×M₃₂	—	17	2	no	—
103	C₂×M₃₂	—	17	1	no	—
104	C₂×M₃₂	—	34	2	no	—
105	C₂²×C₁₆	—	17	1	no	—
106	#413	Dic₁₇	1	1	yes	characteristic
107	#410	Dic₁₇	1	1	yes	characteristic
108	#405	C₂×D₁₇	1	2	yes	—
109	#364	—	34	4	no	—
110	C₂²⋊C₁₆	—	34	2	no	—
111	#344	Dic₁₇	1	2	yes	—
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	C₂×Dic₁₇	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	C₂×M₁₆	Dic₃₄	1	2	yes	—
120	C₂×M₁₆	C₂×Dic₁₇	1	2	yes	—
121	C₂×M₁₆	C₂×Dic₁₇	1	1	yes	characteristic
122	C₂²×C₈	D₆₈	1	2	yes	—
123	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
124	#117	C₂×Dic₁₇	1	2	yes	—
125	M₃₂	—	17	4	no	—
126	M₃₂	—	34	4	no	—
127	M₃₂	—	34	2	no	—
128	M₃₂	—	34	2	no	—
129	C₂×C₁₆	—	34	4	no	—
130	C₂×C₁₆	—	34	2	no	—
131	C₂×C₁₆	—	34	1	no	—
132	C₂×C₁₆	—	34	1	no	—
133	C₂×C₁₆	—	34	2	no	—
134	#101	#3622	1	2	yes	—
135	#100	#3622	1	2	yes	—
136	#99	—	2	2	no	—
137	#97	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₁₇	#12916	1	1	yes	characteristic, complemented, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×#3622	1	1	yes	characteristic
143	C₂×D₄	C₂×#3622	1	1	yes	characteristic
144	C₂×D₄	#63223	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
146	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
147	M₁₆	—	2	2	no	—
148	M₁₆	—	2	2	no	—
149	M₁₆	—	2	4	no	—
150	M₁₆	C₂×#3622	1	4	yes	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
155	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
156	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
157	C₂×C₈	D₄×D₁₇	1	2	yes	—
158	C₂×C₈	#63254	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	34	4	no	—
163	C₁₆	—	68	1	no	—
164	C₁₆	—	68	1	no	—
165	C₂³	C₄⋊#3622	1	2	yes	—
166	C₂³	C₂×#63223	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	—	4	2	no	—
173	C₂×C₄	C₄⋊#3622	1	2	yes	—
174	C₂×C₄	C₂²×#3622	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
176	C₂×C₄	D₄×Dic₁₇	1	2	yes	—
177	C₂×C₄	#10587634	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	2	2	no	—
180	C₈	#10587564	1	2	yes	—
181	C₈	#10587568	1	1	yes	characteristic
182	C₈	#10587572	1	1	yes	characteristic
183	C₂²	—	2	2	no	—
184	C₂²	—	4	1	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂×C₄⋊#3622	1	1	yes	characteristic
187	C₂²	#11762881	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	8	2	no	—
190	C₄	#11762684	1	1	yes	characteristic
191	C₄	#11762836	1	1	yes	characteristic
192	C₄	D₄×#3622	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	1	no	—
195	C₂	—	4	2	no	—
196	C₂	#14528964	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14541887

C₂

yes

maximal, characteristic

#14530550

C₂

yes

maximal, complemented

#14530398

C₂

yes

maximal, characteristic

#14526873

C₂

yes

maximal, complemented

C₁₇×#1757

C₂

yes

maximal, characteristic

#11763226

—

C₂×#10587483

—

C₂×#10587483

C₂²

yes

characteristic

C₂×#10587483

C₂²

yes

—

C₂²×#63214

C₂²

yes

characteristic

#11762630

—

C₂²⋊#63214

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₁₇×#413

C₄

yes

characteristic

C₁₇×#410

C₄

yes

characteristic

C₁₇×#405

C₂²

yes

—

C₁₇×#344

C₄

yes

—

C₂×#41×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

—

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

D₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

C₁₇×#117

C₂×C₄

yes

—

C₁₇×#101

C₈

yes

—

C₁₇×#100

C₈

yes

—

C₁₇×#99

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

—

—

—

D₄

yes

—

—

—

—

—

—

#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

—

M₁₆×C₁₇

—

M₁₆×C₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₇

C₂×D₄

yes

—

C₂×C₈×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

#41

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×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

—

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

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#120

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₈×C₁₇

yes

—

—

—

D₁₇

yes

characteristic

#1379

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₇

#435

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

C₈×D₄

yes

—

100

C₂×#131

C₂×D₁₇

yes

characteristic

101

#504

—

102

C₂×M₃₂

—

103

C₂×M₃₂

—

104

C₂×M₃₂

—

105

C₂²×C₁₆

—

106

#413

Dic₁₇

yes

characteristic

107

#410

Dic₁₇

yes

characteristic

108

#405

C₂×D₁₇

yes

—

109

#364

—

110

C₂²⋊C₁₆

—

111

#344

Dic₁₇

yes

—

112

C₂×C₁₇

—

113

C₂×C₁₇

—

114

C₂×C₁₇

—

115

C₂×C₁₇

#1850

yes

characteristic

116

C₂×#41

C₂×Dic₁₇

yes

characteristic

117

#131

—

118

#131

—

119

C₂×M₁₆

Dic₃₄

yes

—

120

C₂×M₁₆

C₂×Dic₁₇

yes

—

121

C₂×M₁₆

C₂×Dic₁₇

yes

characteristic

122

C₂²×C₈

D₆₈

yes

—

123

C₂²×C₈

C₂²×D₁₇

yes

characteristic

124

#117

C₂×Dic₁₇

yes

—

125

M₃₂

—

126

M₃₂

—

127

M₃₂

—

128

M₃₂

—

129

C₂×C₁₆

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

—

133

C₂×C₁₆

—

134

#101

#3622

yes

—

135

#100

#3622

yes

—

136

#99

—

137

#97

—

138

C₄×C₈

—

139

C₁₇

#12916

yes

characteristic, complemented, Sylow

140

#41

—

141

#41

—

142

C₂×Q₈

C₂×#3622

yes

characteristic

143

C₂×D₄

C₂×#3622

yes

characteristic

144

C₂×D₄

#63223

yes

—

145

C₂²×C₄

C₄⋊Dic₁₇

yes

—

146

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

147

M₁₆

—

148

M₁₆

—

149

M₁₆

—

150

M₁₆

C₂×#3622

yes

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₄⋊Dic₁₇

yes

—

155

C₂×C₈

C₂×Dic₃₄

yes

characteristic

156

C₂×C₈

C₂×D₆₈

yes

characteristic

157

C₂×C₈

D₄×D₁₇

yes

—

158

C₂×C₈

#63254

yes

—

159

C₄⋊C₄

—

160

C₂²⋊C₄

—

161

C₄²

—

162

C₁₆

—

163

C₁₆

—

164

C₁₆

—

165

C₂³

C₄⋊#3622

yes

—

166

C₂³

C₂×#63223

yes

characteristic

167

Q₈

—

168

D₄

—

169

D₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

C₄⋊#3622

yes

—

174

C₂×C₄

C₂²×#3622

yes

characteristic

175

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

176

C₂×C₄

D₄×Dic₁₇

yes

—

177

C₂×C₄

#10587634

yes

—

178

C₈

—

179

C₈

—

180

C₈

#10587564

yes

—

181

C₈

#10587568

yes

characteristic

182

C₈

#10587572

yes

characteristic

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₂×C₄⋊#3622

yes

characteristic

187

C₂²

#11762881

yes

—

188

C₄

—

189

C₄

—

190

C₄

#11762684

yes

characteristic

191

C₄

#11762836

yes

characteristic

192

C₄

D₄×#3622

yes

—

193

C₂

—

194

C₂

—

195

C₂

—

196

C₂

#14528964

yes

characteristic

	subgroup	G/H
center	C₈	#10587568
commutator	C₂×C₄×C₁₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1757	C₂
socle	C₂×C₁₇	#1850
prime radical	C₂×#41×C₁₇	C₂×C₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1757	D₁₇
hypocenter	C₁₇	#12916
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1757

subgroup

G/H

center

C₈

#10587568

commutator

C₂×C₄×C₁₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₁₇×#1757

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 ◁ #1757

N	H ≅ G/N	type
2. #14530550	C₂	semidirect
4. #14526873	C₂	semidirect
139. C₁₇	#12916	semidirect

H ≅ G/N

type

2. #14530550

C₂

semidirect

4. #14526873

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
#14541887	2
#14530550	2
#14530398	2
#14526873	2
C₁₇×#1757	2
#12916	17

subgroup

index

C₁₇×#1757

Subgroups — #16950806

Conjugacy classes of subgroups

Complements

Maximal subgroups