Group #20270652 — subgroups

Subgroups — C₅₃×#1715 #20270652

Order 6784.

220 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#492	C₂	1	2	yes	maximal, complemented
4	C₅₃×#486	C₂	1	2	yes	maximal, complemented
5	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₅₃	—	2	4	no	—
7	C₂×D₈×C₅₃	—	2	2	no	—
8	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	#127×C₅₃	C₂²	1	1	yes	characteristic
10	#127×C₅₃	C₂²	1	1	yes	characteristic
11	#106×C₅₃	—	2	2	no	—
12	#102×C₅₃	—	2	4	no	—
13	C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×D₄×C₅₃	—	2	2	no	—
17	C₂×D₄×C₅₃	—	2	1	no	—
18	C₂×D₄×C₅₃	—	2	1	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	SD₁₆×C₅₃	—	4	8	no	—
21	D₈×C₅₃	—	4	4	no	—
22	C₂×C₈×C₅₃	D₄	1	4	yes	—
23	C₂×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₂³×C₅₃	—	4	1	no	—
28	C₂³×C₅₃	—	4	1	no	—
29	Q₈×C₅₃	—	4	4	no	—
30	D₄×C₅₃	—	4	2	no	—
31	D₄×C₅₃	—	4	2	no	—
32	D₄×C₅₃	—	4	4	no	—
33	D₄×C₅₃	—	4	4	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
37	C₈×C₅₃	—	2	4	no	—
38	C₈×C₅₃	SD₁₆	1	4	yes	complemented
39	C₂²×C₅₃	—	8	1	no	—
40	C₂²×C₅₃	—	8	1	no	—
41	C₂²×C₅₃	—	8	2	no	—
42	C₂²×C₅₃	—	8	2	no	—
43	C₂²×C₅₃	#127	1	1	yes	characteristic
44	C₄×C₅₃	—	8	2	no	—
45	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
46	C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
47	#1715	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅₃	—	16	1	no	—
49	C₂×C₅₃	—	16	1	no	—
50	C₂×C₅₃	#492	1	1	yes	characteristic
51	C₂×C₅₃	#496	1	2	yes	—
52	#498	C₂×C₅₃	1	1	yes	characteristic, complemented
53	#493	C₂×C₅₃	1	1	yes	characteristic
54	#492	C₂×C₅₃	1	2	yes	complemented
55	#486	C₂×C₅₃	1	2	yes	complemented
56	#322	C₂×C₅₃	1	1	yes	characteristic, complemented
57	C₅₃	#1715	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×SD₁₆	—	2	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
61	#127	C₂²×C₅₃	1	1	yes	characteristic
62	#127	C₂²×C₅₃	1	1	yes	characteristic
63	#106	—	2	2	no	—
64	#102	—	2	4	no	—
65	C₄×C₈	C₂²×C₅₃	1	2	yes	—
66	C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	SD₁₆	—	4	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	D₄×C₅₃	1	4	yes	—
75	C₂×C₈	D₄×C₅₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂³×C₅₃	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	SD₁₆×C₅₃	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#127×C₅₃	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
98	C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	C₅₃×#492	1	1	yes	characteristic
102	C₂	C₅₃×#496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#498

C₂

yes

maximal, characteristic, complemented

C₅₃×#493

C₂

yes

maximal, characteristic

C₅₃×#492

C₂

yes

maximal, complemented

C₅₃×#486

C₂

yes

maximal, complemented

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

#127×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#106×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

#1715

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#492

yes

characteristic

C₂×C₅₃

#496

yes

—

#498

C₂×C₅₃

yes

characteristic, complemented

#493

C₂×C₅₃

yes

characteristic

#492

C₂×C₅₃

yes

complemented

#486

C₂×C₅₃

yes

complemented

#322

C₂×C₅₃

yes

characteristic, complemented

C₅₃

#1715

yes

characteristic, direct product factor, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₅₃

yes

—

#127

C₂²×C₅₃

yes

characteristic

#127

C₂²×C₅₃

yes

characteristic

#106

—

#102

—

C₄×C₈

C₂²×C₅₃

yes

—

C₄×C₈

C₂²×C₅₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₅₃

yes

—

C₂×C₈

D₄×C₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂³×C₅₃

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₅₃

yes

characteristic

C₂×C₄

C₂×D₄×C₅₃

yes

—

C₈

—

C₈

SD₁₆×C₅₃

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#127×C₅₃

yes

characteristic

C₄

—

C₄

C₂×SD₁₆×C₅₃

yes

—

C₄

AGL(1,ℤ/8)×C₅₃

yes

—

C₂

—

100

C₂

—

101

C₂

C₅₃×#492

yes

characteristic

102

C₂

C₅₃×#496

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₅₃
prime radical	C₅₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₅₃

prime radical

C₅₃×#493

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#498	C₂	semidirect
3. C₅₃×#492	C₂	semidirect
4. C₅₃×#486	C₂	semidirect
5. C₅₃×#322	C₂	semidirect
38. C₈×C₅₃	SD₁₆	semidirect
47. #1715	C₅₃	direct (N × H)
52. #498	C₂×C₅₃	semidirect
54. #492	C₂×C₅₃	semidirect
55. #486	C₂×C₅₃	semidirect
56. #322	C₂×C₅₃	semidirect
57. C₅₃	#1715	direct (N × H)
90. C₈	SD₁₆×C₅₃	semidirect

H ≅ G/N

type

1. C₅₃×#498

C₂

semidirect

3. C₅₃×#492

C₂

semidirect

4. C₅₃×#486

C₂

semidirect

5. C₅₃×#322

C₂

semidirect

38. C₈×C₅₃

SD₁₆

semidirect

47. #1715

C₅₃

direct (N × H)

52. #498

C₂×C₅₃

semidirect

54. #492

C₂×C₅₃

semidirect

55. #486

C₂×C₅₃

semidirect

56. #322

C₂×C₅₃

semidirect

57. C₅₃

#1715

direct (N × H)

90. C₈

SD₁₆×C₅₃

semidirect

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

subgroup	index
C₅₃×#498	2
C₅₃×#493	2
C₅₃×#492	2
C₅₃×#486	2
C₅₃×#322	2
#1715	53

subgroup

index

C₅₃×#498

C₅₃×#493

C₅₃×#492

C₅₃×#486

C₅₃×#322

#1715

Subgroups — C₅₃×#1715 #20270652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#1715 #20270652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#1715 #20270652