Subgroups — C₂×C₃×D₂₁×C₅₃ #27211331

Order 13356.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₂₁×C₅₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₇×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₂₁×C₅₃	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×D₇×C₅₃	—	3	1	no	maximal
5	C₃²×C₇×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₂₁×C₅₃	C₂×C₃	1	2	yes	direct product factor
7	C₃×D₇×C₅₃	—	3	2	no	—
8	C₂×C₃×C₇×C₅₃	—	2	1	no	—
9	C₂×C₃×C₇×C₅₃	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₇×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃×C₅₃	—	7	1	no	maximal, Hall
12	C₂×D₇×C₅₃	—	3	1	no	Hall
13	C₃×C₇×C₅₃	—	2	1	no	—
14	C₃×C₇×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented
15	C₃×C₇×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented
16	C₂×C₃²×C₅₃	D₇	1	1	yes	characteristic, complemented
17	C₃×D₃×C₅₃	—	7	2	no	—
18	C₂×C₇×C₅₃	C₃×D₃	1	1	yes	characteristic, complemented
19	D₇×C₅₃	—	3	2	no	—
20	C₂²×C₃×C₅₃	—	21	1	no	—
21	C₂×D₃×C₅₃	—	7	1	no	—
22	C₃²×C₅₃	C₂×D₇	1	1	yes	characteristic, complemented, Hall
23	C₇×C₅₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₅₃	—	2	1	no	—
25	C₂×C₃×C₅₃	—	21	2	no	—
26	C₂×C₃×C₅₃	C₃×D₇	1	1	yes	characteristic, complemented
27	C₂×C₃×C₅₃	D₂₁	1	1	yes	characteristic, direct product factor
28	D₃×C₅₃	—	7	2	no	—
29	C₂×C₃×D₂₁	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂²×C₅₃	—	21	1	no	Hall
31	C₃×C₅₃	—	2	1	no	—
32	C₃×C₅₃	C₂×C₃×D₇	1	1	yes	characteristic, complemented
33	C₃×C₅₃	C₂×D₂₁	1	1	yes	characteristic, direct product factor
34	C₂×C₃²×C₇	C₂×C₅₃	1	1	yes	characteristic, complemented
35	C₃×D₂₁	C₂×C₅₃	1	2	yes	direct product factor
36	C₂×C₅₃	—	21	2	no	—
37	C₂×C₅₃	C₃×D₂₁	1	1	yes	characteristic, direct product factor
38	C₂×D₂₁	C₃×C₅₃	1	1	yes	characteristic, direct product factor
39	C₂×C₃×D₇	—	3	1	no	—
40	C₃²×C₇	C₂²×C₅₃	1	1	yes	characteristic, complemented, Hall
41	C₅₃	C₂×C₃×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃×C₇	—	2	1	no	—
43	C₂×C₃×C₇	D₃×C₅₃	1	1	yes	characteristic, complemented
44	C₂×C₃×C₇	C₂×C₃×C₅₃	1	1	yes	characteristic, complemented
45	D₂₁	C₂×C₃×C₅₃	1	2	yes	direct product factor
46	C₃×D₇	—	3	2	no	—
47	C₂×C₃×D₃	—	7	1	no	Hall
48	C₂×D₇	—	3	1	no	Hall
49	C₃×C₇	—	2	1	no	—
50	C₃×C₇	C₂×D₃×C₅₃	1	1	yes	characteristic, complemented
51	C₃×C₇	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented
52	C₂×C₃²	D₇×C₅₃	1	1	yes	characteristic, complemented
53	C₃×D₃	—	7	2	no	—
54	C₂×C₇	C₃×D₃×C₅₃	1	1	yes	characteristic, complemented
55	D₇	—	3	2	no	—
56	C₂²×C₃	—	21	1	no	—
57	C₂×D₃	—	7	1	no	—
58	C₃²	C₂×D₇×C₅₃	1	1	yes	characteristic, complemented, Sylow
59	C₇	C₂×C₃×D₃×C₅₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	21	2	no	—
62	C₂×C₃	C₃×D₇×C₅₃	1	1	yes	characteristic, complemented
63	C₂×C₃	D₂₁×C₅₃	1	1	yes	characteristic, direct product factor
64	D₃	—	7	2	no	—
65	C₂²	—	21	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×C₃×D₇×C₅₃	1	1	yes	characteristic, complemented
68	C₃	C₂×D₂₁×C₅₃	1	1	yes	characteristic, direct product factor
69	C₂	—	21	2	no	—
70	C₂	C₃×D₂₁×C₅₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅₃	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂²×C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₇×C₅₃	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₃×C₅₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×D₂₁×C₅₃	C₂	direct (N × H)
2. C₂×C₃²×C₇×C₅₃	C₂	semidirect
3. C₂×D₂₁×C₅₃	C₃	direct (N × H)
5. C₃²×C₇×C₅₃	C₂²	semidirect
6. D₂₁×C₅₃	C₂×C₃	direct (N × H)
9. C₂×C₃×C₇×C₅₃	D₃	semidirect
10. C₂×C₃×C₇×C₅₃	C₂×C₃	semidirect
14. C₃×C₇×C₅₃	C₂×D₃	semidirect
15. C₃×C₇×C₅₃	C₂²×C₃	semidirect
16. C₂×C₃²×C₅₃	D₇	semidirect
18. C₂×C₇×C₅₃	C₃×D₃	semidirect
22. C₃²×C₅₃	C₂×D₇	semidirect
23. C₇×C₅₃	C₂×C₃×D₃	semidirect
26. C₂×C₃×C₅₃	C₃×D₇	semidirect
27. C₂×C₃×C₅₃	D₂₁	direct (N × H)
29. C₂×C₃×D₂₁	C₅₃	direct (N × H)
32. C₃×C₅₃	C₂×C₃×D₇	semidirect
33. C₃×C₅₃	C₂×D₂₁	direct (N × H)
34. C₂×C₃²×C₇	C₂×C₅₃	semidirect
35. C₃×D₂₁	C₂×C₅₃	direct (N × H)
37. C₂×C₅₃	C₃×D₂₁	direct (N × H)
38. C₂×D₂₁	C₃×C₅₃	direct (N × H)
40. C₃²×C₇	C₂²×C₅₃	semidirect
41. C₅₃	C₂×C₃×D₂₁	direct (N × H)
43. C₂×C₃×C₇	D₃×C₅₃	semidirect
44. C₂×C₃×C₇	C₂×C₃×C₅₃	semidirect
45. D₂₁	C₂×C₃×C₅₃	direct (N × H)
50. C₃×C₇	C₂×D₃×C₅₃	semidirect
51. C₃×C₇	C₂²×C₃×C₅₃	semidirect
52. C₂×C₃²	D₇×C₅₃	semidirect
54. C₂×C₇	C₃×D₃×C₅₃	semidirect
58. C₃²	C₂×D₇×C₅₃	semidirect
59. C₇	C₂×C₃×D₃×C₅₃	semidirect
62. C₂×C₃	C₃×D₇×C₅₃	semidirect
63. C₂×C₃	D₂₁×C₅₃	direct (N × H)
67. C₃	C₂×C₃×D₇×C₅₃	semidirect
68. C₃	C₂×D₂₁×C₅₃	direct (N × H)
70. C₂	C₃×D₂₁×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₂₁×C₅₃	2
C₂×C₃²×C₇×C₅₃	2
C₂×D₂₁×C₅₃	3
C₂×C₃×D₇×C₅₃	3
C₂×C₃×D₃×C₅₃	7
C₂×C₃×D₂₁	53

Subgroups — C2×C3×D21×C53 #27211331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₂₁×C₅₃ #27211331