Subgroups — C₂×C₃×C₅₃×D₁₃⋊C₃ #37613698

Order 24804.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃⋊C₃×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅₃×D₁₃⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×D₁₃×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×D₁₃⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₁₃⋊C₃×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₅₃×D₁₃⋊C₃	C₂×C₃	1	6	yes	direct product factor
7	C₃×D₁₃×C₅₃	C₂×C₃	1	2	yes	complemented
8	C₂×C₁₃⋊C₃×C₅₃	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₁₃×C₅₃	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×D₁₃×C₅₃	C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₁₃×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₁₃⋊C₃×C₅₃	C₂²×C₃	1	3	yes	complemented
13	C₂²×C₃²×C₅₃	—	13	1	no	maximal, Hall
14	C₂×C₁₃×C₅₃	C₂×C₃²	1	1	yes	characteristic, complemented
15	D₁₃×C₅₃	C₂×C₃²	1	2	yes	complemented
16	C₂×C₃²×C₅₃	—	13	2	no	—
17	C₂×C₃²×C₅₃	—	13	1	no	—
18	C₁₃×C₅₃	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
19	C₂²×C₃×C₅₃	—	13	3	no	—
20	C₂²×C₃×C₅₃	—	13	1	no	—
21	C₃²×C₅₃	—	13	1	no	Hall
22	C₂×C₃×D₁₃⋊C₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₃×C₅₃	—	13	6	no	—
24	C₂×C₃×C₅₃	—	13	2	no	—
25	C₂×C₃×C₅₃	—	13	3	no	—
26	C₂×C₃×C₅₃	D₁₃⋊C₃	1	1	yes	characteristic, direct product factor
27	C₂×C₃×C₁₃⋊C₃	C₂×C₅₃	1	1	yes	characteristic, complemented
28	C₃×D₁₃⋊C₃	C₂×C₅₃	1	2	yes	direct product factor
29	C₂²×C₅₃	—	13	1	no	Hall
30	C₃×C₅₃	—	13	3	no	—
31	C₃×C₅₃	C₂×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor
32	C₂×C₃×D₁₃	C₃×C₅₃	1	1	yes	characteristic, complemented
33	C₂×D₁₃⋊C₃	C₃×C₅₃	1	3	yes	direct product factor
34	C₃×C₁₃⋊C₃	C₂²×C₅₃	1	1	yes	characteristic, complemented, Hall
35	C₂×C₅₃	—	13	2	no	—
36	C₂×C₅₃	C₃×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor
37	C₂×C₃×C₁₃	C₂×C₃×C₅₃	1	1	yes	characteristic, complemented
38	C₃×D₁₃	C₂×C₃×C₅₃	1	2	yes	complemented
39	C₂×C₁₃⋊C₃	C₂×C₃×C₅₃	1	3	yes	complemented
40	D₁₃⋊C₃	C₂×C₃×C₅₃	1	6	yes	direct product factor
41	C₅₃	C₂×C₃×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×D₁₃	C₃²×C₅₃	1	1	yes	characteristic, complemented, Hall
43	C₃×C₁₃	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented
44	C₁₃⋊C₃	C₂²×C₃×C₅₃	1	3	yes	complemented
45	C₂²×C₃²	—	13	1	no	Hall
46	C₂×C₁₃	C₂×C₃²×C₅₃	1	1	yes	characteristic, complemented
47	D₁₃	C₂×C₃²×C₅₃	1	2	yes	complemented
48	C₂×C₃²	—	13	2	no	—
49	C₂×C₃²	—	13	1	no	—
50	C₁₃	C₂²×C₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₃	—	13	3	no	—
52	C₂²×C₃	—	13	1	no	—
53	C₃²	—	13	1	no	Sylow
54	C₂×C₃	—	13	6	no	—
55	C₂×C₃	—	13	2	no	—
56	C₂×C₃	—	13	3	no	—
57	C₂×C₃	C₅₃×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor
58	C₂²	—	13	1	no	Sylow
59	C₃	—	13	3	no	—
60	C₃	C₂×C₅₃×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂	—	13	2	no	—
62	C₂	C₃×C₅₃×D₁₃⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅₃	D₁₃⋊C₃
commutator, cosocle, hypocenter	C₁₃	C₂²×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃×C₅₃	C₂×C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂×C₃×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₃⋊C₃×C₅₃	C₂	semidirect
2. C₃×C₅₃×D₁₃⋊C₃	C₂	direct (N × H)
3. C₂×C₃×D₁₃×C₅₃	C₃	semidirect
4. C₂×C₅₃×D₁₃⋊C₃	C₃	direct (N × H)
5. C₃×C₁₃⋊C₃×C₅₃	C₂²	semidirect
6. C₅₃×D₁₃⋊C₃	C₂×C₃	direct (N × H)
7. C₃×D₁₃×C₅₃	C₂×C₃	semidirect
8. C₂×C₁₃⋊C₃×C₅₃	C₂×C₃	semidirect
9. C₂×C₃×C₁₃×C₅₃	C₂×C₃	semidirect
10. C₂×D₁₃×C₅₃	C₃²	semidirect
11. C₃×C₁₃×C₅₃	C₂²×C₃	semidirect
12. C₁₃⋊C₃×C₅₃	C₂²×C₃	semidirect
14. C₂×C₁₃×C₅₃	C₂×C₃²	semidirect
15. D₁₃×C₅₃	C₂×C₃²	semidirect
18. C₁₃×C₅₃	C₂²×C₃²	semidirect
22. C₂×C₃×D₁₃⋊C₃	C₅₃	direct (N × H)
26. C₂×C₃×C₅₃	D₁₃⋊C₃	direct (N × H)
27. C₂×C₃×C₁₃⋊C₃	C₂×C₅₃	semidirect
28. C₃×D₁₃⋊C₃	C₂×C₅₃	direct (N × H)
31. C₃×C₅₃	C₂×D₁₃⋊C₃	direct (N × H)
32. C₂×C₃×D₁₃	C₃×C₅₃	semidirect
33. C₂×D₁₃⋊C₃	C₃×C₅₃	direct (N × H)
34. C₃×C₁₃⋊C₃	C₂²×C₅₃	semidirect
36. C₂×C₅₃	C₃×D₁₃⋊C₃	direct (N × H)
37. C₂×C₃×C₁₃	C₂×C₃×C₅₃	semidirect
38. C₃×D₁₃	C₂×C₃×C₅₃	semidirect
39. C₂×C₁₃⋊C₃	C₂×C₃×C₅₃	semidirect
40. D₁₃⋊C₃	C₂×C₃×C₅₃	direct (N × H)
41. C₅₃	C₂×C₃×D₁₃⋊C₃	direct (N × H)
42. C₂×D₁₃	C₃²×C₅₃	semidirect
43. C₃×C₁₃	C₂²×C₃×C₅₃	semidirect
44. C₁₃⋊C₃	C₂²×C₃×C₅₃	semidirect
46. C₂×C₁₃	C₂×C₃²×C₅₃	semidirect
47. D₁₃	C₂×C₃²×C₅₃	semidirect
50. C₁₃	C₂²×C₃²×C₅₃	semidirect
57. C₂×C₃	C₅₃×D₁₃⋊C₃	direct (N × H)
60. C₃	C₂×C₅₃×D₁₃⋊C₃	direct (N × H)
62. C₂	C₃×C₅₃×D₁₃⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₃⋊C₃×C₅₃	2
C₃×C₅₃×D₁₃⋊C₃	2
C₂×C₃×D₁₃×C₅₃	3
C₂×C₅₃×D₁₃⋊C₃	3
C₂²×C₃²×C₅₃	13
C₂×C₃×D₁₃⋊C₃	53

Subgroups — C2×C3×C53×D13⋊C3 #37613698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅₃×D₁₃⋊C₃ #37613698