Subgroups — C₂²×C₃×C₅₃×C₁₉⋊C₃ #49136331

Order 36252.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅₃×C₁₉⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅₃×C₁₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₁₉×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅₃×C₁₉⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₅₃×C₁₉⋊C₃	C₂×C₃	1	9	yes	direct product factor
6	C₂×C₃×C₁₉×C₅₃	C₂×C₃	1	3	yes	complemented
7	C₂²×C₁₉×C₅₃	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₉×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented
9	C₅₃×C₁₉⋊C₃	C₂²×C₃	1	3	yes	direct product factor
10	C₂×C₁₉×C₅₃	C₂×C₃²	1	3	yes	complemented
11	C₂²×C₃²×C₅₃	—	19	1	no	maximal, Hall
12	C₁₉×C₅₃	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃²×C₅₃	—	19	3	no	—
14	C₂²×C₃×C₁₉⋊C₃	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃×C₅₃	—	19	3	no	—
16	C₂²×C₃×C₅₃	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
17	C₃²×C₅₃	—	19	1	no	Hall
18	C₂×C₃×C₁₉⋊C₃	C₂×C₅₃	1	3	yes	direct product factor
19	C₂×C₃×C₅₃	—	19	9	no	—
20	C₂×C₃×C₅₃	C₂×C₁₉⋊C₃	1	3	yes	direct product factor
21	C₂²×C₃×C₁₉	C₃×C₅₃	1	1	yes	characteristic, complemented
22	C₂²×C₁₉⋊C₃	C₃×C₅₃	1	3	yes	direct product factor
23	C₂²×C₅₃	C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
24	C₃×C₁₉⋊C₃	C₂²×C₅₃	1	1	yes	characteristic, direct product factor, Hall
25	C₃×C₅₃	—	19	3	no	—
26	C₃×C₅₃	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
27	C₂×C₃×C₁₉	C₂×C₃×C₅₃	1	3	yes	complemented
28	C₂×C₁₉⋊C₃	C₂×C₃×C₅₃	1	9	yes	direct product factor
29	C₂×C₅₃	C₂×C₃×C₁₉⋊C₃	1	3	yes	direct product factor
30	C₂²×C₁₉	C₃²×C₅₃	1	1	yes	characteristic, complemented, Hall
31	C₃×C₁₉	C₂²×C₃×C₅₃	1	1	yes	characteristic, complemented
32	C₁₉⋊C₃	C₂²×C₃×C₅₃	1	3	yes	direct product factor
33	C₅₃	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₁₉	C₂×C₃²×C₅₃	1	3	yes	complemented
35	C₂²×C₃²	—	19	1	no	Hall
36	C₁₉	C₂²×C₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃²	—	19	3	no	—
38	C₂²×C₃	—	19	3	no	—
39	C₂²×C₃	C₅₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
40	C₃²	—	19	1	no	Sylow
41	C₂×C₃	—	19	9	no	—
42	C₂×C₃	C₂×C₅₃×C₁₉⋊C₃	1	3	yes	direct product factor
43	C₂²	C₃×C₅₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
44	C₃	—	19	3	no	—
45	C₃	C₂²×C₅₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
46	C₂	C₂×C₃×C₅₃×C₁₉⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅₃	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂²×C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₉×C₅₃	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²×C₃×C₅₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅₃×C₁₉⋊C₃	C₂	direct (N × H)
2. C₂²×C₅₃×C₁₉⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₁₉×C₅₃	C₃	semidirect
4. C₃×C₅₃×C₁₉⋊C₃	C₂²	direct (N × H)
5. C₂×C₅₃×C₁₉⋊C₃	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₉×C₅₃	C₂×C₃	semidirect
7. C₂²×C₁₉×C₅₃	C₃²	semidirect
8. C₃×C₁₉×C₅₃	C₂²×C₃	semidirect
9. C₅₃×C₁₉⋊C₃	C₂²×C₃	direct (N × H)
10. C₂×C₁₉×C₅₃	C₂×C₃²	semidirect
12. C₁₉×C₅₃	C₂²×C₃²	semidirect
14. C₂²×C₃×C₁₉⋊C₃	C₅₃	direct (N × H)
16. C₂²×C₃×C₅₃	C₁₉⋊C₃	direct (N × H)
18. C₂×C₃×C₁₉⋊C₃	C₂×C₅₃	direct (N × H)
20. C₂×C₃×C₅₃	C₂×C₁₉⋊C₃	direct (N × H)
21. C₂²×C₃×C₁₉	C₃×C₅₃	semidirect
22. C₂²×C₁₉⋊C₃	C₃×C₅₃	direct (N × H)
23. C₂²×C₅₃	C₃×C₁₉⋊C₃	direct (N × H)
24. C₃×C₁₉⋊C₃	C₂²×C₅₃	direct (N × H)
26. C₃×C₅₃	C₂²×C₁₉⋊C₃	direct (N × H)
27. C₂×C₃×C₁₉	C₂×C₃×C₅₃	semidirect
28. C₂×C₁₉⋊C₃	C₂×C₃×C₅₃	direct (N × H)
29. C₂×C₅₃	C₂×C₃×C₁₉⋊C₃	direct (N × H)
30. C₂²×C₁₉	C₃²×C₅₃	semidirect
31. C₃×C₁₉	C₂²×C₃×C₅₃	semidirect
32. C₁₉⋊C₃	C₂²×C₃×C₅₃	direct (N × H)
33. C₅₃	C₂²×C₃×C₁₉⋊C₃	direct (N × H)
34. C₂×C₁₉	C₂×C₃²×C₅₃	semidirect
36. C₁₉	C₂²×C₃²×C₅₃	semidirect
39. C₂²×C₃	C₅₃×C₁₉⋊C₃	direct (N × H)
42. C₂×C₃	C₂×C₅₃×C₁₉⋊C₃	direct (N × H)
43. C₂²	C₃×C₅₃×C₁₉⋊C₃	direct (N × H)
45. C₃	C₂²×C₅₃×C₁₉⋊C₃	direct (N × H)
46. C₂	C₂×C₃×C₅₃×C₁₉⋊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×C₅₃×C₁₉⋊C₃	2
C₂²×C₅₃×C₁₉⋊C₃	3
C₂²×C₃×C₁₉×C₅₃	3
C₂²×C₃²×C₅₃	19
C₂²×C₃×C₁₉⋊C₃	53

Subgroups — C22×C3×C53×C19⋊C3 #49136331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅₃×C₁₉⋊C₃ #49136331