Subgroups — C₃×C₅²×C₂²⋊(C₁₉⋊C₃) #30685960

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	3	yes	maximal, direct product factor
2	C₂²×C₃×C₅²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅²×C₁₉⋊C₃	—	4	1	no	maximal, Hall
4	C₃×C₅×C₂²⋊(C₁₉⋊C₃)	C₅	1	6	yes	maximal, direct product factor
5	C₂×C₃×C₅²×C₁₉	—	3	1	no	—
6	C₂²×C₅²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
7	C₃×C₅²×C₁₉	A₄	1	1	yes	characteristic, complemented
8	C₅²×C₁₉⋊C₃	—	4	3	no	—
9	C₂²×C₃×C₅×C₁₉	C₃×C₅	1	6	yes	complemented
10	C₅×C₂²⋊(C₁₉⋊C₃)	C₃×C₅	1	18	yes	direct product factor
11	C₂×C₅²×C₁₉	—	3	1	no	—
12	C₃×C₅²×A₄	—	19	1	no	maximal, Hall
13	C₃×C₅×C₁₉⋊C₃	—	4	6	no	—
14	C₃×C₂²⋊(C₁₉⋊C₃)	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₂×C₃×C₅×C₁₉	—	3	6	no	—
16	C₅²×C₁₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₅×C₁₉	C₃²×C₅	1	6	yes	complemented
18	C₂²×C₃×C₅²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
19	C₅²×A₄	—	19	3	no	—
20	C₃×C₅×C₁₉	C₅×A₄	1	6	yes	complemented
21	C₅×C₁₉⋊C₃	—	4	18	no	—
22	C₂²×C₃×C₁₉	C₃×C₅²	1	1	yes	characteristic, complemented
23	C₂²⋊(C₁₉⋊C₃)	C₃×C₅²	1	3	yes	direct product factor
24	C₃²×C₅²	—	76	1	no	Hall
25	C₂×C₅×C₁₉	—	3	6	no	—
26	C₃×C₅×A₄	—	19	6	no	—
27	C₃×C₁₉⋊C₃	—	4	1	no	Hall
28	C₂×C₃×C₅²	—	3	1	no	—
29	C₂×C₃×C₁₉	—	3	1	no	—
30	C₂²×C₅²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
31	C₅×C₁₉	C₃×C₅×A₄	1	6	yes	complemented
32	C₂²×C₁₉	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
33	C₃×C₅²	—	76	3	no	—
34	C₃×C₅²	C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, direct product factor
35	C₂²×C₃×C₅	C₅×C₁₉⋊C₃	1	6	yes	complemented
36	C₅×A₄	—	19	18	no	—
37	C₃×C₁₉	C₅²×A₄	1	1	yes	characteristic, complemented
38	C₁₉⋊C₃	—	4	3	no	—
39	C₂×C₅²	—	3	1	no	—
40	C₃²×C₅	—	76	6	no	—
41	C₂×C₁₉	—	3	1	no	—
42	C₃×A₄	—	19	1	no	Hall
43	C₂×C₃×C₅	—	3	6	no	—
44	C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₅	C₃×C₅×C₁₉⋊C₃	1	6	yes	complemented
46	C₁₉	C₃×C₅²×A₄	1	1	yes	characteristic, complemented, Sylow
47	C₃×C₅	—	76	18	no	—
48	C₃×C₅	C₅×C₂²⋊(C₁₉⋊C₃)	1	6	yes	direct product factor
49	C₂²×C₃	C₅²×C₁₉⋊C₃	1	1	yes	characteristic, complemented
50	A₄	—	19	3	no	—
51	C₂×C₅	—	3	6	no	—
52	C₃²	—	76	1	no	Sylow
53	C₂×C₃	—	3	1	no	—
54	C₅	C₃×C₅×C₂²⋊(C₁₉⋊C₃)	1	6	yes	direct product factor
55	C₂²	C₃×C₅²×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₃	—	76	3	no	—
57	C₃	C₅²×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, direct product factor
58	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₅²×C₂²⋊(C₁₉⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₅²×C₁₉	C₃	semidirect
4. C₃×C₅×C₂²⋊(C₁₉⋊C₃)	C₅	direct (N × H)
6. C₂²×C₅²×C₁₉	C₃²	semidirect
7. C₃×C₅²×C₁₉	A₄	semidirect
9. C₂²×C₃×C₅×C₁₉	C₃×C₅	semidirect
10. C₅×C₂²⋊(C₁₉⋊C₃)	C₃×C₅	direct (N × H)
14. C₃×C₂²⋊(C₁₉⋊C₃)	C₅²	direct (N × H)
16. C₅²×C₁₉	C₃×A₄	semidirect
17. C₂²×C₅×C₁₉	C₃²×C₅	semidirect
18. C₂²×C₃×C₅²	C₁₉⋊C₃	semidirect
20. C₃×C₅×C₁₉	C₅×A₄	semidirect
22. C₂²×C₃×C₁₉	C₃×C₅²	semidirect
23. C₂²⋊(C₁₉⋊C₃)	C₃×C₅²	direct (N × H)
30. C₂²×C₅²	C₃×C₁₉⋊C₃	semidirect
31. C₅×C₁₉	C₃×C₅×A₄	semidirect
32. C₂²×C₁₉	C₃²×C₅²	semidirect
34. C₃×C₅²	C₂²⋊(C₁₉⋊C₃)	direct (N × H)
35. C₂²×C₃×C₅	C₅×C₁₉⋊C₃	semidirect
37. C₃×C₁₉	C₅²×A₄	semidirect
44. C₅²	C₃×C₂²⋊(C₁₉⋊C₃)	direct (N × H)
45. C₂²×C₅	C₃×C₅×C₁₉⋊C₃	semidirect
46. C₁₉	C₃×C₅²×A₄	semidirect
48. C₃×C₅	C₅×C₂²⋊(C₁₉⋊C₃)	direct (N × H)
49. C₂²×C₃	C₅²×C₁₉⋊C₃	semidirect
54. C₅	C₃×C₅×C₂²⋊(C₁₉⋊C₃)	direct (N × H)
55. C₂²	C₃×C₅²×C₁₉⋊C₃	semidirect
57. C₃	C₅²×C₂²⋊(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
C₅²×C₂²⋊(C₁₉⋊C₃)	3
C₂²×C₃×C₅²×C₁₉	3
C₃×C₅²×C₁₉⋊C₃	4
C₃×C₅×C₂²⋊(C₁₉⋊C₃)	5
C₃×C₅²×A₄	19

Subgroups — C3×C52×C22⋊(C19⋊C3) #30685960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×C₂²⋊(C₁₉⋊C₃) #30685960