Group #35210338 — subgroups

Subgroups — #1580×C₁₆₃ #35210338

Order 20864.

192 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₄×D₈×C₁₆₃	C₂	1	1	yes	maximal, characteristic
2	C₈×D₄×C₁₆₃	C₂	1	2	yes	maximal, complemented
3	#334×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₈×C₁₆₃	C₂	1	2	yes	maximal, complemented
5	#322×C₁₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈×C₁₆₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₆₃	—	2	2	no	—
8	C₄×D₄×C₁₆₃	C₂²	1	2	yes	—
9	#107×C₁₆₃	C₄	1	1	yes	characteristic
10	C₄⋊C₈×C₁₆₃	C₂²	1	2	yes	—
11	#102×C₁₆₃	C₄	1	2	yes	—
12	C₂²⋊C₈×C₁₆₃	—	2	2	no	—
13	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₆₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₆₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₆₃	—	2	2	no	—
18	D₈×C₁₆₃	C₈	1	4	yes	complemented
19	C₂×C₈×C₁₆₃	—	2	2	no	—
20	C₂×C₈×C₁₆₃	—	2	1	no	—
21	C₂×C₈×C₁₆₃	—	4	4	no	—
22	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₆₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₆₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₆₃	C₂×C₄	1	2	yes	—
26	C₂²⋊C₄×C₁₆₃	—	2	2	no	—
27	C₄²×C₁₆₃	C₂³	1	1	yes	characteristic
28	C₂³×C₁₆₃	—	2	2	no	—
29	D₄×C₁₆₃	—	2	2	no	—
30	D₄×C₁₆₃	C₂×C₈	1	4	yes	complemented
31	C₂×C₄×C₁₆₃	—	2	2	no	—
32	C₂×C₄×C₁₆₃	—	4	2	no	—
33	C₂×C₄×C₁₆₃	—	4	2	no	—
34	C₂×C₄×C₁₆₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₆₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₆₃	#41	1	1	yes	characteristic
37	C₈×C₁₆₃	—	2	1	no	—
38	C₈×C₁₆₃	—	2	2	no	—
39	C₈×C₁₆₃	—	4	1	no	—
40	C₈×C₁₆₃	—	4	2	no	—
41	C₈×C₁₆₃	C₂×C₈	1	2	yes	complemented
42	C₂²×C₁₆₃	—	2	4	no	—
43	C₂²×C₁₆₃	—	4	2	no	—
44	C₂²×C₁₆₃	—	4	2	no	—
45	C₂²×C₁₆₃	C₄×D₄	1	1	yes	characteristic
46	C₄×C₁₆₃	—	2	1	no	—
47	C₄×C₁₆₃	—	4	2	no	—
48	C₄×C₁₆₃	C₂²×C₈	1	1	yes	characteristic
49	C₄×C₁₆₃	#131	1	1	yes	characteristic
50	C₄×C₁₆₃	AGL(1,ℤ/8)	1	2	yes	—
51	C₂×C₁₆₃	—	4	4	no	—
52	C₂×C₁₆₃	C₈×D₄	1	1	yes	characteristic
53	C₂×C₁₆₃	#442	1	1	yes	characteristic
54	C₂×C₁₆₃	#444	1	1	yes	characteristic
55	C₁₆₃	#1580	1	1	yes	characteristic, direct product factor, Sylow
56	#1580	C₁₆₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₄×D₈	C₂×C₁₆₃	1	1	yes	characteristic
58	C₈×D₄	C₂×C₁₆₃	1	2	yes	complemented
59	#334	C₂×C₁₆₃	1	1	yes	characteristic, complemented
60	D₄⋊C₈	C₂×C₁₆₃	1	2	yes	complemented
61	#322	C₂×C₁₆₃	1	1	yes	characteristic, complemented
62	C₂×D₈	C₄×C₁₆₃	1	1	yes	characteristic
63	C₂²×C₈	—	2	2	no	—
64	C₄×D₄	C₂²×C₁₆₃	1	2	yes	—
65	#107	C₄×C₁₆₃	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₁₆₃	1	2	yes	—
67	#102	C₄×C₁₆₃	1	2	yes	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₄×C₈	C₂²×C₁₆₃	1	1	yes	characteristic
70	C₄×C₈	C₂²×C₁₆₃	1	1	yes	characteristic
71	C₄×C₈	C₂²×C₁₆₃	1	1	yes	characteristic
72	C₂×D₄	C₂×C₄×C₁₆₃	1	2	yes	—
73	C₂²×C₄	—	2	2	no	—
74	D₈	C₈×C₁₆₃	1	4	yes	complemented
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	C₂×C₄×C₁₆₃	1	1	yes	characteristic
79	C₂×C₈	C₂×C₄×C₁₆₃	1	1	yes	characteristic
80	C₂×C₈	D₄×C₁₆₃	1	2	yes	—
81	C₄⋊C₄	C₂×C₄×C₁₆₃	1	2	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	C₂³×C₁₆₃	1	1	yes	characteristic
84	C₂³	—	2	2	no	—
85	D₄	—	2	2	no	—
86	D₄	C₂×C₈×C₁₆₃	1	4	yes	complemented
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	C₂²×C₄×C₁₆₃	1	1	yes	characteristic
91	C₂×C₄	C₂×D₄×C₁₆₃	1	1	yes	characteristic
92	C₂×C₄	#41×C₁₆₃	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	2	2	no	—
95	C₈	—	4	1	no	—
96	C₈	—	4	2	no	—
97	C₈	C₂×C₈×C₁₆₃	1	2	yes	complemented
98	C₂²	—	2	4	no	—
99	C₂²	—	4	2	no	—
100	C₂²	—	4	2	no	—
101	C₂²	C₄×D₄×C₁₆₃	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	C₂²×C₈×C₁₆₃	1	1	yes	characteristic
105	C₄	#131×C₁₆₃	1	1	yes	characteristic
106	C₄	AGL(1,ℤ/8)×C₁₆₃	1	2	yes	—
107	C₂	—	4	4	no	—
108	C₂	C₈×D₄×C₁₆₃	1	1	yes	characteristic
109	C₂	#442×C₁₆₃	1	1	yes	characteristic
110	C₂	#444×C₁₆₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₈×C₁₆₃

C₂

yes

maximal, characteristic

C₈×D₄×C₁₆₃

C₂

yes

maximal, complemented

#334×C₁₆₃

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₁₆₃

C₂

yes

maximal, complemented

#322×C₁₆₃

C₂

yes

maximal, characteristic, complemented

C₂×D₈×C₁₆₃

C₄

yes

characteristic

C₂²×C₈×C₁₆₃

—

C₄×D₄×C₁₆₃

C₂²

yes

—

#107×C₁₆₃

C₄

yes

characteristic

C₄⋊C₈×C₁₆₃

C₂²

yes

—

#102×C₁₆₃

C₄

yes

—

C₂²⋊C₈×C₁₆₃

—

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₄×C₈×C₁₆₃

C₂²

yes

characteristic

C₂×D₄×C₁₆₃

C₂×C₄

yes

—

C₂²×C₄×C₁₆₃

—

D₈×C₁₆₃

C₈

yes

complemented

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

—

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₃

D₄

yes

—

C₄⋊C₄×C₁₆₃

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₃

—

C₄²×C₁₆₃

C₂³

yes

characteristic

C₂³×C₁₆₃

—

D₄×C₁₆₃

—

D₄×C₁₆₃

C₂×C₈

yes

complemented

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

—

C₂×C₄×C₁₆₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₃

#41

yes

characteristic

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₈×C₁₆₃

—

C₈×C₁₆₃

C₂×C₈

yes

complemented

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

—

C₂²×C₁₆₃

C₄×D₄

yes

characteristic

C₄×C₁₆₃

—

C₄×C₁₆₃

—

C₄×C₁₆₃

C₂²×C₈

yes

C₄×C₁₆₃

yes

C₄×C₁₆₃

yes

—

C₂×C₁₆₃

—

C₂×C₁₆₃

C₈×D₄

yes

C₂×C₁₆₃

yes

C₂×C₁₆₃

yes

C₁₆₃

yes

characteristic, direct product factor, Sylow

#1580

C₁₆₃

yes

maximal, characteristic, direct product factor, Sylow

C₄×D₈

C₂×C₁₆₃

yes

characteristic

C₈×D₄

C₂×C₁₆₃

yes

complemented

#334

C₂×C₁₆₃

yes

characteristic, complemented

D₄⋊C₈

C₂×C₁₆₃

yes

complemented

#322

C₂×C₁₆₃

yes

characteristic, complemented

C₂×D₈

C₄×C₁₆₃

yes

characteristic

C₂²×C₈

—

C₄×D₄

C₂²×C₁₆₃

yes

—

#107

C₄×C₁₆₃

yes

characteristic

C₄⋊C₈

C₂²×C₁₆₃

yes

—

#102

C₄×C₁₆₃

yes

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₁₆₃

yes

characteristic

C₄×C₈

C₂²×C₁₆₃

yes

characteristic

C₄×C₈

C₂²×C₁₆₃

yes

characteristic

C₂×D₄

C₂×C₄×C₁₆₃

yes

—

C₂²×C₄

—

D₈

C₈×C₁₆₃

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₆₃

yes

characteristic

C₂×C₈

C₂×C₄×C₁₆₃

yes

characteristic

C₂×C₈

D₄×C₁₆₃

yes

—

C₄⋊C₄

C₂×C₄×C₁₆₃

yes

—

C₂²⋊C₄

—

C₄²

C₂³×C₁₆₃

yes

characteristic

C₂³

—

D₄

—

D₄

C₂×C₈×C₁₆₃

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₁₆₃

yes

characteristic

C₂×C₄

C₂×D₄×C₁₆₃

yes

characteristic

C₂×C₄

#41×C₁₆₃

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×C₈×C₁₆₃

yes

complemented

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

C₄×D₄×C₁₆₃

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

C₂²×C₈×C₁₆₃

yes

characteristic

105

C₄

#131×C₁₆₃

yes

characteristic

106

C₄

AGL(1,ℤ/8)×C₁₆₃

yes

—

107

C₂

—

108

C₂

C₈×D₄×C₁₆₃

yes

characteristic

109

C₂

#442×C₁₆₃

yes

characteristic

110

C₂

#444×C₁₆₃

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₆₃	C₂×D₄
commutator	C₄	C₂²×C₈×C₁₆₃
Frattini, cosocle	C₄²	C₂³×C₁₆₃
socle	C₂²×C₁₆₃	C₄×D₄
prime radical	C₂×D₈×C₁₆₃	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₆₃ ◁ C₄×C₈×C₁₆₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₆₃

C₂×D₄

commutator

C₄

C₂²×C₈×C₁₆₃

Frattini, cosocle

C₄²

C₂³×C₁₆₃

socle

C₂²×C₁₆₃

C₄×D₄

prime radical

C₂×D₈×C₁₆₃

C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₆₃ ◁ C₄×C₈×C₁₆₃ ◁ G

N	H ≅ G/N	type
2. C₈×D₄×C₁₆₃	C₂	semidirect
3. #334×C₁₆₃	C₂	semidirect
4. D₄⋊C₈×C₁₆₃	C₂	semidirect
5. #322×C₁₆₃	C₂	semidirect
18. D₈×C₁₆₃	C₈	semidirect
30. D₄×C₁₆₃	C₂×C₈	semidirect
41. C₈×C₁₆₃	C₂×C₈	semidirect
55. C₁₆₃	#1580	direct (N × H)
56. #1580	C₁₆₃	direct (N × H)
58. C₈×D₄	C₂×C₁₆₃	semidirect
59. #334	C₂×C₁₆₃	semidirect
60. D₄⋊C₈	C₂×C₁₆₃	semidirect
61. #322	C₂×C₁₆₃	semidirect
74. D₈	C₈×C₁₆₃	semidirect
86. D₄	C₂×C₈×C₁₆₃	semidirect
97. C₈	C₂×C₈×C₁₆₃	semidirect

H ≅ G/N

type

2. C₈×D₄×C₁₆₃

C₂

semidirect

3. #334×C₁₆₃

C₂

semidirect

4. D₄⋊C₈×C₁₆₃

C₂

semidirect

5. #322×C₁₆₃

C₂

semidirect

18. D₈×C₁₆₃

C₈

semidirect

30. D₄×C₁₆₃

C₂×C₈

semidirect

41. C₈×C₁₆₃

C₂×C₈

semidirect

55. C₁₆₃

#1580

direct (N × H)

56. #1580

C₁₆₃

direct (N × H)

58. C₈×D₄

C₂×C₁₆₃

semidirect

59. #334

C₂×C₁₆₃

semidirect

60. D₄⋊C₈

C₂×C₁₆₃

semidirect

61. #322

C₂×C₁₆₃

semidirect

74. D₈

C₈×C₁₆₃

semidirect

86. D₄

C₂×C₈×C₁₆₃

semidirect

97. C₈

C₂×C₈×C₁₆₃

semidirect

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

subgroup	index
C₄×D₈×C₁₆₃	2
C₈×D₄×C₁₆₃	2
#334×C₁₆₃	2
D₄⋊C₈×C₁₆₃	2
#322×C₁₆₃	2
#1580	163

subgroup

index

C₄×D₈×C₁₆₃

C₈×D₄×C₁₆₃

#334×C₁₆₃

D₄⋊C₈×C₁₆₃

#322×C₁₆₃

#1580

163

Subgroups — #1580×C₁₆₃ #35210338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1580×C163 #35210338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1580×C₁₆₃ #35210338