Subgroups — C₂²×C₃×C₅²×C₁₉⋊C₃ #30685964

Order 17100.

480 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×C₁₉⋊C₃	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₅²×C₁₉⋊C₃	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₅²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅²×C₁₉⋊C₃	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂²×C₃×C₅×C₁₉⋊C₃	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₁₉⋊C₃	C₂×C₃	1	9	yes	direct product factor
7	C₂×C₃×C₅²×C₁₉	C₂×C₃	1	3	yes	complemented
8	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅×C₁₉⋊C₃	C₂×C₅	1	18	yes	direct product factor
10	C₃×C₅²×C₁₉	C₂²×C₃	1	1	yes	characteristic, complemented
11	C₅²×C₁₉⋊C₃	C₂²×C₃	1	3	yes	direct product factor
12	C₂²×C₃×C₅×C₁₉	C₃×C₅	1	6	yes	complemented
13	C₂²×C₅×C₁₉⋊C₃	C₃×C₅	1	18	yes	direct product factor
14	C₂×C₅²×C₁₉	C₂×C₃²	1	3	yes	complemented
15	C₂²×C₃²×C₅²	—	19	1	no	maximal, Hall
16	C₃×C₅×C₁₉⋊C₃	C₂²×C₅	1	6	yes	direct product factor
17	C₂²×C₃×C₁₉⋊C₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₅×C₁₉	C₂×C₃×C₅	1	18	yes	complemented
19	C₂×C₅×C₁₉⋊C₃	C₂×C₃×C₅	1	54	yes	direct product factor
20	C₅²×C₁₉	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃²×C₅²	—	19	3	no	—
22	C₂²×C₅×C₁₉	C₃²×C₅	1	6	yes	complemented
23	C₂×C₃×C₁₉⋊C₃	C₂×C₅²	1	3	yes	direct product factor
24	C₂²×C₃×C₅²	—	19	3	no	—
25	C₂²×C₃×C₅²	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
26	C₃×C₅×C₁₉	C₂²×C₃×C₅	1	6	yes	complemented
27	C₅×C₁₉⋊C₃	C₂²×C₃×C₅	1	18	yes	direct product factor
28	C₂²×C₃×C₁₉	C₃×C₅²	1	1	yes	characteristic, complemented
29	C₂²×C₁₉⋊C₃	C₃×C₅²	1	3	yes	direct product factor
30	C₃²×C₅²	—	19	1	no	Hall
31	C₂×C₅×C₁₉	C₂×C₃²×C₅	1	18	yes	complemented
32	C₂²×C₃²×C₅	—	19	6	no	—
33	C₃×C₁₉⋊C₃	C₂²×C₅²	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₅²	—	19	9	no	—
35	C₂×C₃×C₅²	C₂×C₁₉⋊C₃	1	3	yes	direct product factor
36	C₂×C₃×C₁₉	C₂×C₃×C₅²	1	3	yes	complemented
37	C₂×C₁₉⋊C₃	C₂×C₃×C₅²	1	9	yes	direct product factor
38	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Hall
39	C₅×C₁₉	C₂²×C₃²×C₅	1	6	yes	complemented
40	C₂×C₃²×C₅	—	19	18	no	—
41	C₂²×C₁₉	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
42	C₃×C₅²	—	19	3	no	—
43	C₃×C₅²	C₂²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
44	C₂²×C₃×C₅	—	19	18	no	—
45	C₂²×C₃×C₅	C₅×C₁₉⋊C₃	1	6	yes	direct product factor
46	C₃×C₁₉	C₂²×C₃×C₅²	1	1	yes	characteristic, complemented
47	C₁₉⋊C₃	C₂²×C₃×C₅²	1	3	yes	direct product factor
48	C₂×C₅²	C₂×C₃×C₁₉⋊C₃	1	3	yes	direct product factor
49	C₃²×C₅	—	19	6	no	—
50	C₂×C₁₉	C₂×C₃²×C₅²	1	3	yes	complemented
51	C₂²×C₃²	—	19	1	no	Hall
52	C₂×C₃×C₅	—	19	54	no	—
53	C₂×C₃×C₅	C₂×C₅×C₁₉⋊C₃	1	18	yes	direct product factor
54	C₅²	C₂²×C₃×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
55	C₂²×C₅	C₃×C₅×C₁₉⋊C₃	1	6	yes	direct product factor
56	C₁₉	C₂²×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃²	—	19	3	no	—
58	C₃×C₅	—	19	18	no	—
59	C₃×C₅	C₂²×C₅×C₁₉⋊C₃	1	6	yes	direct product factor
60	C₂²×C₃	—	19	3	no	—
61	C₂²×C₃	C₅²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
62	C₂×C₅	C₂×C₃×C₅×C₁₉⋊C₃	1	18	yes	direct product factor
63	C₃²	—	19	1	no	Sylow
64	C₂×C₃	—	19	9	no	—
65	C₂×C₃	C₂×C₅²×C₁₉⋊C₃	1	3	yes	direct product factor
66	C₅	C₂²×C₃×C₅×C₁₉⋊C₃	1	6	yes	direct product factor
67	C₂²	C₃×C₅²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
68	C₃	—	19	3	no	—
69	C₃	C₂²×C₅²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
70	C₂	C₂×C₃×C₅²×C₁₉⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₅²	C₁₉⋊C₃
commutator, cosocle, hypocenter	C₁₉	C₂²×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₅²×C₁₉	C₃
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²×C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²×C₁₉⋊C₃	C₂	direct (N × H)
2. C₂²×C₅²×C₁₉⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₅²×C₁₉	C₃	semidirect
4. C₃×C₅²×C₁₉⋊C₃	C₂²	direct (N × H)
5. C₂²×C₃×C₅×C₁₉⋊C₃	C₅	direct (N × H)
6. C₂×C₅²×C₁₉⋊C₃	C₂×C₃	direct (N × H)
7. C₂×C₃×C₅²×C₁₉	C₂×C₃	semidirect
8. C₂²×C₅²×C₁₉	C₃²	semidirect
9. C₂×C₃×C₅×C₁₉⋊C₃	C₂×C₅	direct (N × H)
10. C₃×C₅²×C₁₉	C₂²×C₃	semidirect
11. C₅²×C₁₉⋊C₃	C₂²×C₃	direct (N × H)
12. C₂²×C₃×C₅×C₁₉	C₃×C₅	semidirect
13. C₂²×C₅×C₁₉⋊C₃	C₃×C₅	direct (N × H)
14. C₂×C₅²×C₁₉	C₂×C₃²	semidirect
16. C₃×C₅×C₁₉⋊C₃	C₂²×C₅	direct (N × H)
17. C₂²×C₃×C₁₉⋊C₃	C₅²	direct (N × H)
18. C₂×C₃×C₅×C₁₉	C₂×C₃×C₅	semidirect
19. C₂×C₅×C₁₉⋊C₃	C₂×C₃×C₅	direct (N × H)
20. C₅²×C₁₉	C₂²×C₃²	semidirect
22. C₂²×C₅×C₁₉	C₃²×C₅	semidirect
23. C₂×C₃×C₁₉⋊C₃	C₂×C₅²	direct (N × H)
25. C₂²×C₃×C₅²	C₁₉⋊C₃	direct (N × H)
26. C₃×C₅×C₁₉	C₂²×C₃×C₅	semidirect
27. C₅×C₁₉⋊C₃	C₂²×C₃×C₅	direct (N × H)
28. C₂²×C₃×C₁₉	C₃×C₅²	semidirect
29. C₂²×C₁₉⋊C₃	C₃×C₅²	direct (N × H)
31. C₂×C₅×C₁₉	C₂×C₃²×C₅	semidirect
33. C₃×C₁₉⋊C₃	C₂²×C₅²	direct (N × H)
35. C₂×C₃×C₅²	C₂×C₁₉⋊C₃	direct (N × H)
36. C₂×C₃×C₁₉	C₂×C₃×C₅²	semidirect
37. C₂×C₁₉⋊C₃	C₂×C₃×C₅²	direct (N × H)
38. C₂²×C₅²	C₃×C₁₉⋊C₃	direct (N × H)
39. C₅×C₁₉	C₂²×C₃²×C₅	semidirect
41. C₂²×C₁₉	C₃²×C₅²	semidirect
43. C₃×C₅²	C₂²×C₁₉⋊C₃	direct (N × H)
45. C₂²×C₃×C₅	C₅×C₁₉⋊C₃	direct (N × H)
46. C₃×C₁₉	C₂²×C₃×C₅²	semidirect
47. C₁₉⋊C₃	C₂²×C₃×C₅²	direct (N × H)
48. C₂×C₅²	C₂×C₃×C₁₉⋊C₃	direct (N × H)
50. C₂×C₁₉	C₂×C₃²×C₅²	semidirect
53. C₂×C₃×C₅	C₂×C₅×C₁₉⋊C₃	direct (N × H)
54. C₅²	C₂²×C₃×C₁₉⋊C₃	direct (N × H)
55. C₂²×C₅	C₃×C₅×C₁₉⋊C₃	direct (N × H)
56. C₁₉	C₂²×C₃²×C₅²	semidirect
59. C₃×C₅	C₂²×C₅×C₁₉⋊C₃	direct (N × H)
61. C₂²×C₃	C₅²×C₁₉⋊C₃	direct (N × H)
62. C₂×C₅	C₂×C₃×C₅×C₁₉⋊C₃	direct (N × H)
65. C₂×C₃	C₂×C₅²×C₁₉⋊C₃	direct (N × H)
66. C₅	C₂²×C₃×C₅×C₁₉⋊C₃	direct (N × H)
67. C₂²	C₃×C₅²×C₁₉⋊C₃	direct (N × H)
69. C₃	C₂²×C₅²×C₁₉⋊C₃	direct (N × H)
70. C₂	C₂×C₃×C₅²×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×C₅²×C₁₉⋊C₃	2
C₂²×C₅²×C₁₉⋊C₃	3
C₂²×C₃×C₅²×C₁₉	3
C₂²×C₃×C₅×C₁₉⋊C₃	5
C₂²×C₃²×C₅²	19

Subgroups — C22×C3×C52×C19⋊C3 #30685964

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅²×C₁₉⋊C₃ #30685964