Group #16950513 — subgroups

Subgroups — #16950513

Order 4352.

362 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 143 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	#14530568	C₂	1	2	yes	maximal, complemented
2	#14528830	C₂	1	2	yes	maximal
3	#14526886	C₂	1	2	yes	maximal, complemented
4	C₂×C₁₇×#504	C₂	1	1	yes	maximal, characteristic
5	#11762847	C₂²	1	4	yes	—
6	#11762783	—	2	2	no	—
7	#11762637	—	2	2	no	—
8	#11762635	—	2	2	no	—
9	#11762566	—	2	2	no	—
10	C₂×C₁₇×#131	C₂²	1	1	yes	characteristic
11	C₁₇×#504	—	2	2	no	—
12	C₁₇×#504	—	2	2	no	—
13	C₂×C₁₇×M₃₂	C₄	1	2	yes	complemented
14	C₂×C₁₇×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₁₇	C₄	1	2	yes	complemented
16	C₂²×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#41×C₁₇	D₄	1	1	yes	characteristic
18	C₁₇×#131	—	2	2	no	—
19	C₁₇×#131	—	2	2	no	—
20	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₇	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
24	#10587612	—	2	4	no	—
25	C₁₇×M₃₂	—	2	4	no	—
26	C₁₇×M₃₂	—	2	2	no	—
27	C₂×C₁₆×C₁₇	—	2	2	no	—
28	C₂×C₁₆×C₁₇	—	2	2	no	—
29	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
33	C₂×C₁₆×C₁₇	D₄	1	2	yes	—
34	#10587505	—	4	4	no	—
35	#10587489	—	2	4	no	—
36	#10587488	—	2	4	no	—
37	#41×C₁₇	—	2	2	no	—
38	#41×C₁₇	—	2	2	no	—
39	C₂×Q₈×C₁₇	Q₁₆	1	1	yes	characteristic
40	C₂×D₄×C₁₇	D₈	1	2	yes	—
41	C₂×D₄×C₁₇	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₇	—	2	2	no	—
45	M₁₆×C₁₇	—	2	4	no	—
46	C₂×C₈×C₁₇	—	2	2	no	—
47	C₂×C₈×C₁₇	—	2	2	no	—
48	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₇	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₇	#41	1	2	yes	—
53	C₂²⋊Dic₁₇	—	4	4	no	—
54	C₄⋊Dic₁₇	—	4	4	no	—
55	C₄×Dic₁₇	—	4	4	no	—
56	C₁₆×C₁₇	—	2	2	no	—
57	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₆×C₁₇	C₂²⋊C₄	1	2	yes	—
60	#12839	—	17	1	no	maximal, Sylow
61	C₂³×C₁₇	#107	1	2	yes	—
62	C₂³×C₁₇	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₁₇	—	2	2	no	—
64	D₄×C₁₇	—	2	4	no	—
65	D₄×C₁₇	—	2	2	no	—
66	C₂×C₄×C₁₇	—	2	2	no	—
67	C₂×C₄×C₁₇	—	2	2	no	—
68	C₂×C₄×C₁₇	#107	1	2	yes	—
69	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₇	#121	1	2	yes	—
71	C₂×C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₇	#135	1	2	yes	—
73	C₂×Dic₁₇	—	8	4	no	—
74	C₈×C₁₇	—	2	2	no	—
75	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₈×C₁₇	#117	1	1	yes	characteristic
77	C₈×C₁₇	C₂²⋊Q₈	1	2	yes	—
78	C₂×#504	D₁₇	1	1	yes	characteristic
79	#2158	—	17	2	no	—
80	#1812	—	17	2	no	—
81	#1384	—	17	2	no	—
82	C₂²×C₁₇	—	2	2	no	—
83	C₂²×C₁₇	—	4	1	no	—
84	C₂²×C₁₇	—	4	2	no	—
85	C₂²×C₁₇	C₂×#107	1	1	yes	characteristic
86	C₂²×C₁₇	#466	1	2	yes	—
87	C₄×C₁₇	—	2	2	no	—
88	C₄×C₁₇	#380	1	1	yes	characteristic
89	C₄×C₁₇	#427	1	1	yes	characteristic
90	C₄×C₁₇	#467	1	2	yes	—
91	Dic₁₇	—	16	4	no	—
92	C₂×#131	C₂×D₁₇	1	1	yes	characteristic
93	#504	—	2	2	no	—
94	#504	—	2	2	no	—
95	C₂×M₃₂	Dic₁₇	1	2	yes	complemented
96	C₂×M₃₂	C₂×D₁₇	1	1	yes	characteristic
97	C₂²×C₁₆	Dic₁₇	1	2	yes	complemented
98	C₂²×C₁₆	C₂×D₁₇	1	1	yes	characteristic
99	#427	—	17	4	no	—
100	#410	—	34	2	no	—
101	#367	—	34	2	no	—
102	#366	—	34	2	no	—
103	H(3,ℤ/4)	—	34	2	no	—
104	C₂×C₁₇	—	2	2	no	—
105	C₂×C₁₇	—	2	1	no	—
106	C₂×C₁₇	—	4	2	no	—
107	C₂×C₁₇	#1808	1	1	yes	characteristic
108	C₂×#41	D₆₈	1	1	yes	characteristic
109	#131	—	2	2	no	—
110	#131	—	2	2	no	—
111	C₂×M₁₆	Dic₃₄	1	2	yes	—
112	C₂×M₁₆	D₆₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
114	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
115	#117	—	34	4	no	—
116	M₃₂	—	2	2	no	—
117	M₃₂	—	2	4	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	C₂×Dic₁₇	1	2	yes	—
121	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
122	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
123	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₁₇	1	2	yes	—
125	#107	—	34	4	no	—
126	#106	—	34	4	no	—
127	#99	—	68	4	no	—
128	C₁₇	#12839	1	1	yes	characteristic, complemented, Sylow
129	#41	—	2	2	no	—
130	#41	—	2	2	no	—
131	C₂×Q₈	Dic₆₈	1	1	yes	characteristic
132	C₂×D₄	D₁₃₆	1	2	yes	—
133	C₂×D₄	Dic₆₈	1	1	yes	characteristic
134	C₂²×C₄	C₄⋊Dic₁₇	1	2	yes	—
135	C₂²×C₄	C₂×D₆₈	1	1	yes	characteristic
136	M₁₆	—	2	2	no	—
137	M₁₆	—	2	4	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₄⋊Dic₁₇	1	2	yes	—
141	C₂×C₈	C₂×Dic₃₄	1	1	yes	characteristic
142	C₂×C₈	#63252	1	2	yes	—
143	C₂×C₈	C₂²×Dic₁₇	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₁₇	1	2	yes	—
145	C₄⋊C₄	—	68	4	no	—
146	C₂²⋊C₄	—	68	4	no	—
147	C₄²	—	68	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊Dic₁₇	1	1	yes	characteristic
150	C₁₆	C₂²⋊Dic₁₇	1	1	yes	characteristic
151	C₁₆	C₂²⋊Dic₁₇	1	2	yes	—
152	C₂³	#10587489	1	2	yes	—
153	C₂³	C₂×D₁₃₆	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	136	4	no	—
160	C₂×C₄	#10587489	1	2	yes	—
161	C₂×C₄	#10587590	1	2	yes	—
162	C₂×C₄	C₂×Dic₆₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
164	C₂×C₄	#10587616	1	2	yes	—
165	C₈	—	2	2	no	—
166	C₈	#10587610	1	2	yes	—
167	C₈	#10587612	1	1	yes	characteristic
168	C₈	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
169	C₂²	—	2	2	no	—
170	C₂²	—	4	1	no	—
171	C₂²	—	4	2	no	—
172	C₂²	C₂×#10587489	1	1	yes	characteristic
173	C₂²	#11763045	1	2	yes	—
174	C₄	—	2	2	no	—
175	C₄	—	272	4	no	—
176	C₄	#11762669	1	1	yes	characteristic
177	C₄	#11762847	1	1	yes	characteristic
178	C₄	#11763050	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	2	1	no	—
181	C₂	—	4	2	no	—
182	C₂	#14528813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14530568

C₂

yes

maximal, complemented

#14528830

C₂

yes

maximal

#14526886

C₂

yes

maximal, complemented

C₂×C₁₇×#504

C₂

yes

maximal, characteristic

#11762847

C₂²

yes

—

—

—

—

—

C₂×C₁₇×#131

C₂²

yes

characteristic

C₁₇×#504

—

C₁₇×#504

—

C₂×C₁₇×M₃₂

C₄

yes

complemented

C₂×C₁₇×M₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₁₇

C₄

yes

complemented

C₂²×C₁₆×C₁₇

C₂²

yes

characteristic

C₂×#41×C₁₇

D₄

yes

characteristic

C₁₇×#131

—

C₁₇×#131

—

C₂×M₁₆×C₁₇

D₄

yes

characteristic

C₂×M₁₆×C₁₇

Q₈

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂³

yes

characteristic

#10587612

—

C₁₇×M₃₂

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₇

D₄

yes

—

C₂×C₁₆×C₁₇

D₄

yes

—

#10587505

—

#10587489

—

#10587488

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

Q₁₆

yes

characteristic

C₂×D₄×C₁₇

D₈

yes

—

C₂×D₄×C₁₇

Q₁₆

yes

characteristic

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

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

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

—

C₁₆×C₁₇

—

C₁₆×C₁₇

C₂²⋊C₄

yes

characteristic

C₁₆×C₁₇

C₂²⋊C₄

yes

characteristic

C₁₆×C₁₇

C₂²⋊C₄

yes

—

#12839

—

maximal, Sylow

C₂³×C₁₇

#107

yes

—

C₂³×C₁₇

C₂×D₈

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#107

yes

—

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

—

C₈×C₁₇

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

#117

yes

characteristic

C₈×C₁₇

C₂²⋊Q₈

yes

—

C₂×#504

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×#107

yes

characteristic

C₂²×C₁₇

#466

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×#131

C₂×D₁₇

yes

characteristic

#504

—

#504

—

C₂×M₃₂

Dic₁₇

yes

complemented

C₂×M₃₂

C₂×D₁₇

yes

characteristic

C₂²×C₁₆

Dic₁₇

yes

complemented

C₂²×C₁₆

C₂×D₁₇

yes

characteristic

#427

—

100

#410

—

101

#367

—

102

#366

—

103

H(3,ℤ/4)

—

104

C₂×C₁₇

—

105

C₂×C₁₇

—

106

C₂×C₁₇

—

107

C₂×C₁₇

#1808

yes

characteristic

108

C₂×#41

D₆₈

yes

characteristic

109

#131

—

110

#131

—

111

C₂×M₁₆

Dic₃₄

yes

—

112

C₂×M₁₆

D₆₈

yes

characteristic

113

C₂²×C₈

C₂×Dic₁₇

yes

—

114

C₂²×C₈

C₂²×D₁₇

yes

characteristic

115

#117

—

116

M₃₂

—

117

M₃₂

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

C₂×C₁₆

C₂×Dic₁₇

yes

—

121

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

122

C₂×C₁₆

C₂×Dic₁₇

yes

characteristic

123

C₂×C₁₆

C₂²⋊D₁₇

yes

—

124

C₂×C₁₆

C₂²⋊D₁₇

yes

—

125

#107

—

126

#106

—

127

#99

—

128

C₁₇

#12839

yes

characteristic, complemented, Sylow

129

#41

—

130

#41

—

131

C₂×Q₈

Dic₆₈

yes

characteristic

132

C₂×D₄

D₁₃₆

yes

—

133

C₂×D₄

Dic₆₈

yes

characteristic

134

C₂²×C₄

C₄⋊Dic₁₇

yes

—

135

C₂²×C₄

C₂×D₆₈

yes

characteristic

136

M₁₆

—

137

M₁₆

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

C₄⋊Dic₁₇

yes

—

141

C₂×C₈

C₂×Dic₃₄

yes

characteristic

142

C₂×C₈

#63252

yes

—

143

C₂×C₈

C₂²×Dic₁₇

yes

characteristic

144

C₂×C₈

C₂×C₂²⋊D₁₇

yes

—

145

C₄⋊C₄

—

146

C₂²⋊C₄

—

147

C₄²

—

148

C₁₆

—

149

C₁₆

C₂²⋊Dic₁₇

yes

characteristic

150

C₁₆

C₂²⋊Dic₁₇

yes

characteristic

151

C₁₆

C₂²⋊Dic₁₇

yes

—

152

C₂³

#10587489

yes

—

153

C₂³

C₂×D₁₃₆

yes

characteristic

154

Q₈

—

155

D₄

—

156

D₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

136

—

160

C₂×C₄

#10587489

yes

—

161

C₂×C₄

#10587590

yes

—

162

C₂×C₄

C₂×Dic₆₈

yes

characteristic

163

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

164

C₂×C₄

#10587616

yes

—

165

C₈

—

166

C₈

#10587610

yes

—

167

C₈

#10587612

yes

characteristic

168

C₈

C₂×C₂²⋊Dic₁₇

yes

characteristic

169

C₂²

—

170

C₂²

—

171

C₂²

—

172

C₂²

C₂×#10587489

yes

characteristic

173

C₂²

#11763045

yes

—

174

C₄

—

175

C₄

—

272

—

176

C₄

#11762669

yes

characteristic

177

C₄

#11762847

yes

characteristic

178

C₄

#11763050

yes

—

179

C₂

—

180

C₂

—

181

C₂

—

182

C₂

#14528813

yes

characteristic

	subgroup	G/H
center	C₄	#11762847
commutator	C₂×C₈×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₂×C₁₇×#504	C₂
socle	C₂×C₁₇	#1808
prime radical	C₂×#41×C₁₇	D₄
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	C₂×#504	D₁₇
hypocenter	C₁₇	#12839
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#11762847

commutator

C₂×C₈×C₁₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₁₇

Fitting

C₂×C₁₇×#504

C₂

socle

C₂×C₁₇

#1808

prime radical

C₂×#41×C₁₇

D₄

cosocle

C₂²×C₈×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #14530568	C₂	semidirect
3. #14526886	C₂	semidirect
13. C₂×C₁₇×M₃₂	C₄	semidirect
15. C₂²×C₁₆×C₁₇	C₄	semidirect
95. C₂×M₃₂	Dic₁₇	semidirect
97. C₂²×C₁₆	Dic₁₇	semidirect
128. C₁₇	#12839	semidirect

H ≅ G/N

type

1. #14530568

C₂

semidirect

3. #14526886

C₂

semidirect

13. C₂×C₁₇×M₃₂

C₄

semidirect

15. C₂²×C₁₆×C₁₇

C₄

semidirect

95. C₂×M₃₂

Dic₁₇

semidirect

97. C₂²×C₁₆

Dic₁₇

semidirect

128. C₁₇

#12839

semidirect

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

subgroup	index
#14530568	2
#14528830	2
#14526886	2
C₂×C₁₇×#504	2
#12839	17

subgroup

index

C₂×C₁₇×#504

Subgroups — #16950513

Conjugacy classes of subgroups

Complements

Maximal subgroups