Group #17962580 — subgroups

Subgroups — #17962580

Order 4352.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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×#11763741	C₂	1	2	yes	maximal, complemented
2	#14543900	C₂	1	4	yes	maximal, complemented
3	C₂×#11763227	C₂	1	1	yes	maximal, characteristic
4	#11763741	—	2	2	no	—
5	#11763741	C₂²	1	4	yes	—
6	C₂×#10587467	—	2	2	no	—
7	C₂×#10587587	C₄	1	1	yes	characteristic
8	#11763227	—	2	2	no	—
9	#11763227	—	2	2	no	—
10	C₂×#10587469	C₂²	1	2	yes	—
11	C₂²×#63214	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₁₇	C₄	1	2	yes	—
13	C₂×C₁₇×M₃₂	C₄	1	1	yes	characteristic
14	C₂×#63252	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#63223	C₈	1	1	yes	characteristic
17	#10587587	—	2	2	no	—
18	#10587587	—	2	2	no	—
19	C₂×#63218	C₈	1	2	yes	—
20	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
21	C₁₇×M₃₂	—	2	2	no	—
22	C₂×C₁₆×C₁₇	C₂×C₄	1	2	yes	—
23	C₂×#63214	C₂×C₄	1	1	yes	characteristic
24	C₂×#63214	D₄	1	2	yes	—
25	C₂×#63214	D₄	1	2	yes	—
26	C₂×#63214	C₂³	1	1	yes	characteristic
27	#10587469	—	2	4	no	—
28	C₁₆×D₁₇	—	2	2	no	—
29	C₁₆×D₁₇	—	2	2	no	—
30	#10587467	—	2	4	no	—
31	C₂²×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₂²⋊D₁₇	C₁₆	1	2	yes	—
33	#63252	—	2	2	no	—
34	#63252	—	2	2	no	—
35	C₂×D₆₈	C₁₆	1	1	yes	characteristic
36	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
37	C₂×Dic₃₄	C₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₇	M₁₆	1	2	yes	—
41	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
42	#63223	—	2	2	no	—
43	C₂×#3622	C₂×C₈	1	2	yes	—
44	#63218	—	2	4	no	—
45	C₈×D₁₇	—	2	2	no	—
46	C₈×D₁₇	—	2	2	no	—
47	C₁₆×C₁₇	—	2	2	no	—
48	#63214	C₂²⋊C₄	1	1	yes	characteristic
49	#63214	C₂²⋊C₄	1	1	yes	characteristic
50	#63214	C₂×D₄	1	2	yes	—
51	#13556	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	C₂×C₁₆	1	1	yes	characteristic
53	C₂²×D₁₇	C₂×C₁₆	1	2	yes	—
54	C₂×C₄×C₁₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
56	C₂×C₄×C₁₇	M₃₂	1	2	yes	—
57	C₂×C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
58	C₂²⋊D₁₇	—	2	4	no	—
59	C₂×Dic₁₇	C₂×C₁₆	1	2	yes	—
60	D₆₈	—	2	2	no	—
61	C₄×D₁₇	—	2	2	no	—
62	C₄×D₁₇	—	2	2	no	—
63	Dic₃₄	—	2	2	no	—
64	C₈×C₁₇	C₂²⋊C₈	1	2	yes	—
65	C₈×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₁₇	C₂×M₁₆	1	1	yes	characteristic
67	#3622	—	2	2	no	—
68	C₂×#504	—	17	1	no	—
69	C₂×M₆₄	—	17	2	no	—
70	#1400	—	17	4	no	—
71	C₂²×C₁₇	—	4	1	no	—
72	C₂²×C₁₇	C₂²⋊C₁₆	1	2	yes	—
73	C₂²×C₁₇	C₂²×C₁₆	1	1	yes	characteristic
74	C₂×D₁₇	—	2	2	no	—
75	C₂×D₁₇	—	4	2	no	—
76	C₄×C₁₇	C₂²⋊C₁₆	1	2	yes	—
77	C₄×C₁₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	C₄×C₁₇	C₂×M₃₂	1	1	yes	characteristic
79	Dic₁₇	—	2	2	no	—
80	C₂×#131	—	17	1	no	—
81	#504	—	34	2	no	—
82	#504	—	34	2	no	—
83	C₂×M₃₂	—	17	2	no	—
84	C₂×M₃₂	#595	1	1	yes	characteristic
85	C₂²×C₁₆	—	17	1	no	—
86	C₂²×C₁₆	—	17	2	no	—
87	M₆₄	—	17	4	no	—
88	M₆₄	—	34	2	no	—
89	C₂×C₃₂	—	34	2	no	—
90	C₂×C₁₇	—	2	1	no	—
91	C₂×C₁₇	—	2	2	no	—
92	C₂×C₁₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
93	D₁₇	—	4	2	no	—
94	C₂×#41	—	17	1	no	—
95	#131	—	34	2	no	—
96	#131	—	34	2	no	—
97	C₂×M₁₆	—	17	1	no	—
98	C₂×M₁₆	—	17	2	no	—
99	C₂²×C₈	—	17	2	no	—
100	C₂²×C₈	C₂×#595	1	1	yes	characteristic
101	M₃₂	—	2	2	no	—
102	M₃₂	—	34	4	no	—
103	C₂×C₁₆	—	17	2	no	—
104	C₂×C₁₆	—	17	2	no	—
105	C₂×C₁₆	—	17	1	no	—
106	C₂×C₁₆	—	17	1	no	—
107	C₂×C₁₆	—	34	2	no	—
108	C₂×C₁₆	—	34	2	no	—
109	C₂×C₁₆	C₂×#595	1	2	yes	—
110	C₃₂	—	34	4	no	—
111	C₁₇	#13556	1	1	yes	characteristic, complemented, Sylow
112	#41	—	34	2	no	—
113	#41	—	34	2	no	—
114	C₂×Q₈	—	17	1	no	—
115	C₂×D₄	—	17	2	no	—
116	C₂×D₄	—	17	1	no	—
117	C₂²×C₄	—	17	2	no	—
118	C₂²×C₄	#63242	1	1	yes	characteristic
119	M₁₆	—	34	2	no	—
120	M₁₆	—	34	4	no	—
121	C₂×C₈	—	17	2	no	—
122	C₂×C₈	—	34	2	no	—
123	C₂×C₈	—	34	2	no	—
124	C₂×C₈	#63242	1	1	yes	characteristic
125	C₂×C₈	#63243	1	2	yes	—
126	C₂×C₈	C₂²⋊#595	1	2	yes	—
127	C₂×C₈	C₂²×#595	1	1	yes	characteristic
128	C₁₆	—	2	2	no	—
129	C₁₆	—	17	2	no	—
130	C₁₆	—	17	1	no	—
131	C₁₆	—	17	1	no	—
132	C₂³	—	17	2	no	—
133	C₂³	#10587529	1	1	yes	characteristic
134	Q₈	—	34	2	no	—
135	D₄	—	34	4	no	—
136	D₄	—	34	2	no	—
137	C₂×C₄	—	17	2	no	—
138	C₂×C₄	—	34	2	no	—
139	C₂×C₄	—	34	2	no	—
140	C₂×C₄	#10587529	1	1	yes	characteristic
141	C₂×C₄	#10587530	1	2	yes	—
142	C₂×C₄	#10587543	1	2	yes	—
143	C₂×C₄	C₂×#63242	1	1	yes	characteristic
144	C₈	—	34	2	no	—
145	C₈	#10587543	1	2	yes	—
146	C₈	C₂×#63243	1	1	yes	characteristic
147	C₈	C₂×C₂²⋊#595	1	1	yes	characteristic
148	C₂²	—	4	1	no	—
149	C₂²	—	34	2	no	—
150	C₂²	—	68	2	no	—
151	C₂²	#11763704	1	2	yes	—
152	C₂²	C₂×#10587529	1	1	yes	characteristic
153	C₄	—	34	2	no	—
154	C₄	#11763704	1	2	yes	—
155	C₄	C₂×#10587543	1	1	yes	characteristic
156	C₄	C₂×#10587530	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	68	2	no	—
160	C₂	C₂×#11763704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11763741

C₂

yes

maximal, complemented

#14543900

C₂

yes

maximal, complemented

C₂×#11763227

C₂

yes

maximal, characteristic

#11763741

—

#11763741

C₂²

yes

—

C₂×#10587467

—

C₂×#10587587

C₄

yes

—

—

C₂²

yes

—

C₂²×#63214

C₂²

yes

characteristic

C₂×C₁₆×D₁₇

C₄

yes

—

C₂×C₁₇×M₃₂

C₄

yes

characteristic

C₂×#63252

C₈

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×#63223

C₈

yes

—

—

C₈

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₁₇×M₃₂

—

C₂×C₁₆×C₁₇

C₂×C₄

yes

—

C₂×#63214

C₂×C₄

yes

characteristic

C₂×#63214

D₄

yes

—

C₂×#63214

D₄

yes

—

C₂×#63214

C₂³

yes

characteristic

#10587469

—

C₁₆×D₁₇

—

C₁₆×D₁₇

—

#10587467

—

C₂²×C₄×C₁₇

C₂×C₈

yes

characteristic

C₂×C₂²⋊D₁₇

C₁₆

yes

—

#63252

—

#63252

—

C₂×D₆₈

C₁₆

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

—

C₂×Dic₃₄

C₁₆

yes

characteristic

C₂×C₈×C₁₇

C₂²⋊C₄

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

characteristic

C₂×C₈×C₁₇

M₁₆

yes

—

C₂×C₈×C₁₇

C₂²×C₄

yes

characteristic

#63223

—

C₂×#3622

C₂×C₈

yes

—

#63218

—

C₈×D₁₇

—

C₈×D₁₇

—

C₁₆×C₁₇

—

#63214

C₂²⋊C₄

yes

characteristic

#63214

C₂²⋊C₄

yes

characteristic

#63214

C₂×D₄

yes

—

#13556

—

maximal, Sylow

C₂³×C₁₇

C₂×C₁₆

yes

characteristic

C₂²×D₁₇

C₂×C₁₆

yes

—

C₂×C₄×C₁₇

C₂²⋊C₈

yes

—

C₂×C₄×C₁₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₁₇

M₃₂

yes

—

C₂×C₄×C₁₇

C₂²×C₈

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₂×C₁₆

yes

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

C₂²⋊C₈

yes

—

C₈×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₁₇

C₂×M₁₆

yes

characteristic

#3622

—

C₂×#504

—

C₂×M₆₄

—

#1400

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂²⋊C₁₆

yes

—

C₂²×C₁₇

C₂²×C₁₆

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

C₂²⋊C₁₆

yes

—

C₄×C₁₇

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₁₇

C₂×M₃₂

yes

characteristic

Dic₁₇

—

C₂×#131

—

#504

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

#595

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₂²⋊C₁₆

yes

characteristic

D₁₇

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

100

C₂²×C₈

C₂×#595

yes

characteristic

101

M₃₂

—

102

M₃₂

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₂×#595

yes

—

110

C₃₂

—

111

C₁₇

#13556

yes

characteristic, complemented, Sylow

112

#41

—

113

#41

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

#63242

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

#63242

yes

characteristic

125

C₂×C₈

#63243

yes

—

126

C₂×C₈

C₂²⋊#595

yes

—

127

C₂×C₈

C₂²×#595

yes

characteristic

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

—

133

C₂³

#10587529

yes

characteristic

134

Q₈

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#10587529

yes

characteristic

141

C₂×C₄

#10587530

yes

—

142

C₂×C₄

#10587543

yes

—

143

C₂×C₄

C₂×#63242

yes

characteristic

144

C₈

—

145

C₈

#10587543

yes

—

146

C₈

C₂×#63243

yes

characteristic

147

C₈

C₂×C₂²⋊#595

yes

characteristic

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#11763704

yes

—

152

C₂²

C₂×#10587529

yes

characteristic

153

C₄

—

154

C₄

#11763704

yes

—

155

C₄

C₂×#10587543

yes

characteristic

156

C₄

C₂×#10587530

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

C₂×#11763704

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊#595
commutator	C₂²×C₁₇	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#595
Fitting	C₂×C₁₇×M₃₂	C₄
socle	C₂×C₁₇	C₂×C₂²⋊C₁₆
prime radical	C₂×#63252	C₈
cosocle	C₂×#63214	C₂³
hypercenter	C₂×M₃₂	#595
hypocenter	C₁₇	#13556
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₂²⋊#595

commutator

C₂²×C₁₇

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#595

Fitting

C₂×C₁₇×M₃₂

C₄

socle

C₂×C₁₇

C₂×C₂²⋊C₁₆

C₈

C₂³

C₂×M₃₂

C₁₇

G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
1. C₂×#11763741	C₂	semidirect
2. #14543900	C₂	semidirect
111. C₁₇	#13556	semidirect

H ≅ G/N

type

1. C₂×#11763741

C₂

semidirect

2. #14543900

C₂

semidirect

111. C₁₇

#13556

semidirect

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

subgroup	index
C₂×#11763741	2
#14543900	2
C₂×#11763227	2
#13556	17

subgroup

index

Subgroups — #17962580

Conjugacy classes of subgroups

Complements

Maximal subgroups