Subgroups — C₅²×A₄×C₁₉⋊C₃ #30685819

Order 17100.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₂²⋊(C₁₉⋊C₃)	C₃	1	2	yes	maximal, complemented
2	C₅²×A₄×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅²×C₁₉⋊C₃	—	4	1	no	maximal, Hall
5	C₅×A₄×C₁₉⋊C₃	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₁₉⋊C₃	—	3	1	no	—
7	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₅²×C₁₉	—	4	1	no	—
9	C₅²×C₁₉⋊C₃	—	4	2	no	—
10	C₅²×C₁₉⋊C₃	A₄	1	1	yes	characteristic, direct product factor
11	C₅×C₂²⋊(C₁₉⋊C₃)	C₃×C₅	1	12	yes	complemented
12	C₅×A₄×C₁₉	C₃×C₅	1	6	yes	complemented
13	C₂²×C₅×C₁₉⋊C₃	C₃×C₅	1	6	yes	complemented
14	C₂×C₅²×C₁₉	—	3	1	no	—
15	C₃×C₅²×A₄	—	19	1	no	maximal, Hall
16	C₃×C₅×C₁₉⋊C₃	—	4	6	no	—
17	A₄×C₁₉⋊C₃	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₅×C₁₉⋊C₃	—	3	6	no	—
19	C₅²×C₁₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₅×C₁₉	C₃²×C₅	1	6	yes	complemented
21	C₂²×C₃×C₅²	—	19	1	no	—
22	C₅²×A₄	—	19	2	no	—
23	C₅²×A₄	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
24	C₃×C₅×C₁₉	—	4	6	no	—
25	C₅×C₁₉⋊C₃	—	4	12	no	—
26	C₅×C₁₉⋊C₃	C₅×A₄	1	6	yes	direct product factor
27	C₂²⋊(C₁₉⋊C₃)	C₃×C₅²	1	2	yes	complemented
28	A₄×C₁₉	C₃×C₅²	1	1	yes	characteristic, complemented
29	C₂²×C₁₉⋊C₃	C₃×C₅²	1	1	yes	characteristic, complemented
30	C₃²×C₅²	—	76	1	no	Hall
31	C₂×C₅×C₁₉	—	3	6	no	—
32	C₃×C₅×A₄	—	19	6	no	—
33	C₃×C₁₉⋊C₃	—	4	1	no	Hall
34	C₂×C₃×C₅²	—	57	1	no	—
35	C₂×C₁₉⋊C₃	—	3	1	no	—
36	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
37	C₅×C₁₉	C₃×C₅×A₄	1	6	yes	complemented
38	C₂²×C₁₉	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
39	C₃×C₅²	—	4	1	no	—
40	C₃×C₅²	—	19	1	no	—
41	C₃×C₅²	—	76	2	no	—
42	C₂²×C₃×C₅	—	19	6	no	—
43	C₅×A₄	—	19	12	no	—
44	C₅×A₄	C₅×C₁₉⋊C₃	1	6	yes	direct product factor
45	C₃×C₁₉	—	4	1	no	—
46	C₁₉⋊C₃	—	4	2	no	—
47	C₁₉⋊C₃	C₅²×A₄	1	1	yes	characteristic, direct product factor
48	C₂×C₅²	—	3	1	no	—
49	C₃²×C₅	—	76	6	no	—
50	C₂×C₁₉	—	3	1	no	—
51	C₃×A₄	—	19	1	no	Hall
52	C₂×C₃×C₅	—	57	6	no	—
53	C₅²	A₄×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₅	C₃×C₅×C₁₉⋊C₃	1	6	yes	complemented
55	C₁₉	C₃×C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
56	C₃×C₅	—	4	6	no	—
57	C₃×C₅	—	19	6	no	—
58	C₃×C₅	—	76	12	no	—
59	C₂²×C₃	—	19	1	no	—
60	A₄	—	19	2	no	—
61	A₄	C₅²×C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
62	C₂×C₅	—	3	6	no	—
63	C₃²	—	76	1	no	Sylow
64	C₂×C₃	—	57	1	no	—
65	C₅	C₅×A₄×C₁₉⋊C₃	1	6	yes	direct product factor
66	C₂²	C₃×C₅²×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
67	C₃	—	4	1	no	—
68	C₃	—	19	1	no	—
69	C₃	—	76	2	no	—
70	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅²	A₄×C₁₉⋊C₃
commutator, cosocle, hypocenter	C₂²×C₁₉	C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₂²×C₅²×C₁₉	C₃²
LCS	G ▷ C₂²×C₁₉
UCS	C₁ ◁ C₅²

Complements

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

N	H ≅ G/N	type
1. C₅²×C₂²⋊(C₁₉⋊C₃)	C₃	semidirect
2. C₅²×A₄×C₁₉	C₃	semidirect
3. C₂²×C₅²×C₁₉⋊C₃	C₃	semidirect
5. C₅×A₄×C₁₉⋊C₃	C₅	direct (N × H)
7. C₂²×C₅²×C₁₉	C₃²	semidirect
10. C₅²×C₁₉⋊C₃	A₄	direct (N × H)
11. C₅×C₂²⋊(C₁₉⋊C₃)	C₃×C₅	semidirect
12. C₅×A₄×C₁₉	C₃×C₅	semidirect
13. C₂²×C₅×C₁₉⋊C₃	C₃×C₅	semidirect
17. A₄×C₁₉⋊C₃	C₅²	direct (N × H)
19. C₅²×C₁₉	C₃×A₄	semidirect
20. C₂²×C₅×C₁₉	C₃²×C₅	semidirect
23. C₅²×A₄	C₁₉⋊C₃	direct (N × H)
26. C₅×C₁₉⋊C₃	C₅×A₄	direct (N × H)
27. C₂²⋊(C₁₉⋊C₃)	C₃×C₅²	semidirect
28. A₄×C₁₉	C₃×C₅²	semidirect
29. C₂²×C₁₉⋊C₃	C₃×C₅²	semidirect
36. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
37. C₅×C₁₉	C₃×C₅×A₄	semidirect
38. C₂²×C₁₉	C₃²×C₅²	semidirect
44. C₅×A₄	C₅×C₁₉⋊C₃	direct (N × H)
47. C₁₉⋊C₃	C₅²×A₄	direct (N × H)
53. C₅²	A₄×C₁₉⋊C₃	direct (N × H)
54. C₂²×C₅	C₃×C₅×C₁₉⋊C₃	semidirect
55. C₁₉	C₃×C₅²×A₄	semidirect
61. A₄	C₅²×C₁₉⋊C₃	direct (N × H)
65. C₅	C₅×A₄×C₁₉⋊C₃	direct (N × H)
66. C₂²	C₃×C₅²×C₁₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₅²×C₂²⋊(C₁₉⋊C₃)	3
C₅²×A₄×C₁₉	3
C₂²×C₅²×C₁₉⋊C₃	3
C₃×C₅²×C₁₉⋊C₃	4
C₅×A₄×C₁₉⋊C₃	5
C₃×C₅²×A₄	19

Subgroups — C52×A4×C19⋊C3 #30685819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×A₄×C₁₉⋊C₃ #30685819