Group #35214756 — subgroups

Subgroups — #35214756

Order 20864.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	#23781713	C₂	1	2	yes	maximal, complemented
2	#23781649	C₂	1	1	yes	maximal, characteristic
3	#23781632	C₂	1	1	yes	maximal, characteristic
4	#23781429	C₂	1	2	yes	maximal, complemented
5	#568×C₁₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579683	C₂²	1	1	yes	characteristic
7	C₂×#14579683	C₂²	1	2	yes	—
8	C₂²×C₁₆₃⋊C₈	C₂²	1	1	yes	characteristic
9	#19104670	—	2	4	no	—
10	#19104654	C₄	1	2	yes	—
11	#19104652	C₄	1	2	yes	—
12	C₂²⋊(C₁₆₃⋊C₈)	—	2	2	no	—
13	C₂×#41×C₁₆₃	C₂²	1	1	yes	characteristic
14	#131×C₁₆₃	—	2	4	no	—
15	C₂×M₁₆×C₁₆₃	—	2	2	no	—
16	C₂×M₁₆×C₁₆₃	C₂²	1	2	yes	—
17	#14579683	—	2	2	no	—
18	#14579683	—	2	2	no	—
19	#14579683	C₂×C₄	1	4	yes	—
20	C₂×C₁₆₃⋊C₈	—	2	2	no	—
21	C₂×C₁₆₃⋊C₈	—	2	1	no	—
22	C₂×C₁₆₃⋊C₈	—	2	1	no	—
23	C₂×C₁₆₃⋊C₈	—	2	2	no	—
24	#41×C₁₆₃	—	2	2	no	—
25	#41×C₁₆₃	—	2	2	no	—
26	C₂×Q₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₁₆₃	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₆₃	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₆₃	D₄	1	2	yes	—
30	C₂²×C₄×C₁₆₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₆₃	—	2	4	no	—
32	M₁₆×C₁₆₃	—	2	2	no	—
33	M₁₆×C₁₆₃	D₄	1	4	yes	—
34	C₂×C₈×C₁₆₃	—	2	4	no	—
35	C₂×C₈×C₁₆₃	—	2	2	no	—
36	C₁₆₃⋊C₈	—	2	4	no	—
37	C₁₆₃⋊C₈	—	4	1	no	—
38	C₁₆₃⋊C₈	—	4	1	no	—
39	C₂³×C₁₆₃	C₄⋊C₄	1	2	yes	—
40	C₂³×C₁₆₃	C₂×Q₈	1	1	yes	characteristic
41	Q₈×C₁₆₃	—	2	2	no	—
42	D₄×C₁₆₃	—	2	4	no	—
43	D₄×C₁₆₃	—	2	2	no	—
44	C₂×C₄×C₁₆₃	—	2	2	no	—
45	C₂×C₄×C₁₆₃	—	2	2	no	—
46	C₂×C₄×C₁₆₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₆₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₆₃	#41	1	2	yes	—
51	C₈×C₁₆₃	—	2	4	no	—
52	C₂²×C₁₆₃	—	2	2	no	—
53	C₂²×C₁₆₃	—	4	1	no	—
54	C₂²×C₁₆₃	—	4	2	no	—
55	C₂²×C₁₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₆₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₆₃	—	2	2	no	—
58	C₄×C₁₆₃	C₄×D₄	1	2	yes	—
59	C₄×C₁₆₃	#120	1	1	yes	characteristic
60	C₄×C₁₆₃	#123	1	1	yes	characteristic
61	C₂×C₁₆₃	—	2	1	no	—
62	C₂×C₁₆₃	—	2	2	no	—
63	C₂×C₁₆₃	—	4	2	no	—
64	C₂×C₁₆₃	#385	1	1	yes	characteristic
65	C₁₆₃	#1858	1	1	yes	characteristic, complemented, Sylow
66	#1858	—	163	1	no	maximal, Sylow
67	#568	D₁₆₃	1	1	yes	characteristic
68	#430	—	163	2	no	—
69	#413	—	163	1	no	—
70	#408	—	163	1	no	—
71	#341	—	163	2	no	—
72	C₂×#41	C₂×D₁₆₃	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	163	1	no	—
76	C₂×M₁₆	—	163	2	no	—
77	C₂×M₁₆	C₂×D₁₆₃	1	2	yes	—
78	C₂²×C₈	—	163	1	no	—
79	#108	—	326	4	no	—
80	#101	—	163	2	no	—
81	#100	—	163	2	no	—
82	C₂²⋊C₈	—	326	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₆₃	1	1	yes	characteristic
86	C₂×D₄	C₄×D₁₆₃	1	1	yes	characteristic
87	C₂×D₄	Dic₃₂₆	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₁₆₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₆₃	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	326	2	no	—
93	M₁₆	—	326	2	no	—
94	M₁₆	—	163	4	no	—
95	M₁₆	C₂²⋊D₁₆₃	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	326	2	no	—
99	C₂×C₈	—	326	1	no	—
100	C₂×C₈	—	326	1	no	—
101	C₂×C₈	—	326	2	no	—
102	C₂³	Dic₁₆₃⋊C₄	1	2	yes	—
103	C₂³	C₂×Dic₃₂₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	Dic₁₆₃⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₆₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₆₃	1	1	yes	characteristic
113	C₂×C₄	#14579699	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	326	4	no	—
116	C₈	—	652	1	no	—
117	C₈	—	652	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×Dic₁₆₃⋊C₄	1	1	yes	characteristic
122	C₂²	#19104714	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₁₆₃	1	2	yes	—
125	C₄	#19104704	1	1	yes	characteristic
126	C₄	#19104718	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#23781538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23781713

C₂

yes

maximal, complemented

#23781649

C₂

yes

maximal, characteristic

#23781632

C₂

yes

maximal, characteristic

#23781429

C₂

yes

maximal, complemented

#568×C₁₆₃

C₂

yes

maximal, characteristic

C₂×#14579683

C₂²

yes

characteristic

C₂×#14579683

C₂²

yes

—

C₂²×C₁₆₃⋊C₈

C₂²

yes

characteristic

#19104670

—

#19104654

C₄

yes

—

#19104652

C₄

yes

—

C₂²⋊(C₁₆₃⋊C₈)

—

C₂×#41×C₁₆₃

C₂²

yes

characteristic

#131×C₁₆₃

—

C₂×M₁₆×C₁₆₃

—

C₂×M₁₆×C₁₆₃

C₂²

yes

—

#14579683

—

#14579683

—

#14579683

C₂×C₄

yes

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

C₂×C₁₆₃⋊C₈

—

#41×C₁₆₃

—

#41×C₁₆₃

—

C₂×Q₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

C₂×C₄

yes

characteristic

C₂×D₄×C₁₆₃

Q₈

yes

—

C₂²×C₄×C₁₆₃

D₄

yes

—

C₂²×C₄×C₁₆₃

C₂³

yes

characteristic

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

—

M₁₆×C₁₆₃

D₄

yes

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₁₆₃⋊C₈

—

C₂³×C₁₆₃

C₄⋊C₄

yes

—

C₂³×C₁₆₃

C₂×Q₈

yes

characteristic

Q₈×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

—

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

—

C₈×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₃

C₂²⋊Q₈

yes

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₄×D₄

yes

—

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

—

C₂×C₁₆₃

#385

yes

characteristic

C₁₆₃

#1858

yes

characteristic, complemented, Sylow

#1858

—

163

maximal, Sylow

#568

D₁₆₃

yes

characteristic

#430

—

163

—

#413

—

163

—

#408

—

163

—

#341

—

163

—

C₂×#41

C₂×D₁₆₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

163

—

C₂×M₁₆

—

163

—

C₂×M₁₆

C₂×D₁₆₃

yes

—

C₂²×C₈

—

163

—

#108

—

326

—

#101

—

163

—

#100

—

163

—

C₂²⋊C₈

—

326

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₆₃

yes

characteristic

C₂×D₄

C₄×D₁₆₃

yes

characteristic

C₂×D₄

Dic₃₂₆

yes

—

C₂²×C₄

C₂²⋊D₁₆₃

yes

—

C₂²×C₄

C₂²×D₁₆₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

326

—

M₁₆

—

326

—

M₁₆

—

163

—

M₁₆

C₂²⋊D₁₆₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

326

—

C₂×C₈

—

326

—

100

C₂×C₈

—

326

—

101

C₂×C₈

—

326

—

102

C₂³

Dic₁₆₃⋊C₄

yes

—

103

C₂³

C₂×Dic₃₂₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₁₆₃⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₁₆₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₆₃

yes

characteristic

113

C₂×C₄

#14579699

yes

—

114

C₈

—

115

C₈

—

326

—

116

C₈

—

652

—

117

C₈

—

652

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×Dic₁₆₃⋊C₄

yes

characteristic

122

C₂²

#19104714

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₁₆₃

yes

—

125

C₄

#19104704

yes

characteristic

126

C₄

#19104718

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#23781538

yes

characteristic

	subgroup	G/H
center	C₄	#19104704
commutator	C₂×C₄×C₁₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₃
Fitting	#568×C₁₆₃	C₂
socle	C₂×C₁₆₃	#385
prime radical	C₂×#41×C₁₆₃	C₂²
cosocle	C₂²×C₄×C₁₆₃	C₂³
hypercenter	#568	D₁₆₃
hypocenter	C₁₆₃	#1858
LCS	G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#19104704

commutator

C₂×C₄×C₁₆₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₃

Fitting

#568×C₁₆₃

C₂

socle

C₂×C₁₆₃

#385

prime radical

C₂×#41×C₁₆₃

C₂²

cosocle

C₂²×C₄×C₁₆₃

C₂³

D₁₆₃

C₁₆₃

G ▷ C₂×C₄×C₁₆₃ ▷ C₂×C₁₆₃ ▷ C₁₆₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #23781713	C₂	semidirect
4. #23781429	C₂	semidirect
65. C₁₆₃	#1858	semidirect

H ≅ G/N

type

1. #23781713

C₂

semidirect

4. #23781429

C₂

semidirect

65. C₁₆₃

#1858

semidirect

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

subgroup	index
#23781713	2
#23781649	2
#23781632	2
#23781429	2
#568×C₁₆₃	2
#1858	163

subgroup

index

#568×C₁₆₃

163

Subgroups — #35214756

Conjugacy classes of subgroups

Complements

Maximal subgroups