Subgroups — C₃×C₅²×(C₁₉⋊C₃)⋊C₄ #30685646

Order 17100.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×C₁₉⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅²×Dic₁₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅²×(C₁₉⋊C₃)⋊C₄	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅²×C₁₉⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×C₅×(C₁₉⋊C₃)⋊C₄	C₅	1	6	yes	maximal, direct product factor
6	C₂×C₅²×C₁₉⋊C₃	C₂×C₃	1	3	yes	—
7	C₂×C₃×C₅²×C₁₉	C₂×C₃	1	1	yes	characteristic
8	C₅²×Dic₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅×C₁₉⋊C₃	C₂×C₅	1	6	yes	—
10	C₃×C₅²×C₁₉	C₃×C₄	1	1	yes	characteristic, complemented
11	C₅²×C₁₉⋊C₃	C₃×C₄	1	3	yes	complemented
12	C₃×C₅×Dic₁₉	C₃×C₅	1	6	yes	complemented
13	C₅×(C₁₉⋊C₃)⋊C₄	C₃×C₅	1	18	yes	direct product factor
14	C₂×C₅²×C₁₉	C₂×C₃²	1	1	yes	characteristic
15	C₃²×C₄×C₅²	—	19	1	no	maximal, Hall
16	C₃×C₅×C₁₉⋊C₃	C₄×C₅	1	6	yes	complemented
17	C₃×(C₁₉⋊C₃)⋊C₄	C₅²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₅×C₁₉	C₂×C₃×C₅	1	6	yes	—
19	C₂×C₅×C₁₉⋊C₃	C₂×C₃×C₅	1	18	yes	—
20	C₅²×C₁₉	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃²×C₅²	—	19	1	no	—
22	C₅×Dic₁₉	C₃²×C₅	1	6	yes	complemented
23	C₂×C₃×C₁₉⋊C₃	C₂×C₅²	1	1	yes	characteristic
24	C₃×C₄×C₅²	—	19	3	no	—
25	C₃×C₄×C₅²	—	19	1	no	—
26	C₃×C₅×C₁₉	C₃×C₄×C₅	1	6	yes	complemented
27	C₅×C₁₉⋊C₃	C₃×C₄×C₅	1	18	yes	complemented
28	C₃×Dic₁₉	C₃×C₅²	1	1	yes	characteristic, complemented
29	(C₁₉⋊C₃)⋊C₄	C₃×C₅²	1	3	yes	direct product factor
30	C₃²×C₅²	—	19	1	no	Hall
31	C₂×C₅×C₁₉	C₂×C₃²×C₅	1	6	yes	—
32	C₃²×C₄×C₅	—	19	6	no	—
33	C₃×C₁₉⋊C₃	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×C₅²	—	19	3	no	—
35	C₂×C₃×C₅²	D₁₉⋊C₃	1	1	yes	characteristic
36	C₂×C₃×C₁₉	C₂×C₃×C₅²	1	1	yes	characteristic
37	C₂×C₁₉⋊C₃	C₂×C₃×C₅²	1	3	yes	—
38	C₄×C₅²	—	19	1	no	Hall
39	C₅×C₁₉	C₃²×C₄×C₅	1	6	yes	complemented
40	C₂×C₃²×C₅	—	19	6	no	—
41	Dic₁₉	C₃²×C₅²	1	1	yes	characteristic, complemented, Hall
42	C₃×C₅²	—	19	3	no	—
43	C₃×C₅²	(C₁₉⋊C₃)⋊C₄	1	1	yes	characteristic, direct product factor
44	C₃×C₄×C₅	—	19	18	no	—
45	C₃×C₄×C₅	—	19	6	no	—
46	C₃×C₁₉	C₃×C₄×C₅²	1	1	yes	characteristic, complemented
47	C₁₉⋊C₃	C₃×C₄×C₅²	1	3	yes	complemented
48	C₂×C₅²	C₃×D₁₉⋊C₃	1	1	yes	characteristic
49	C₃²×C₅	—	19	6	no	—
50	C₂×C₁₉	C₂×C₃²×C₅²	1	1	yes	characteristic
51	C₃²×C₄	—	19	1	no	Hall
52	C₂×C₃×C₅	—	19	18	no	—
53	C₂×C₃×C₅	C₅×D₁₉⋊C₃	1	6	yes	—
54	C₅²	C₃×(C₁₉⋊C₃)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
55	C₄×C₅	—	19	6	no	—
56	C₁₉	C₃²×C₄×C₅²	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₃²	—	19	1	no	—
58	C₃×C₅	—	19	18	no	—
59	C₃×C₅	C₅×(C₁₉⋊C₃)⋊C₄	1	6	yes	direct product factor
60	C₃×C₄	—	19	3	no	—
61	C₃×C₄	—	19	1	no	—
62	C₂×C₅	C₃×C₅×D₁₉⋊C₃	1	6	yes	—
63	C₃²	—	19	1	no	Sylow
64	C₂×C₃	—	19	3	no	—
65	C₂×C₃	C₅²×D₁₉⋊C₃	1	1	yes	characteristic
66	C₅	C₃×C₅×(C₁₉⋊C₃)⋊C₄	1	6	yes	direct product factor
67	C₄	—	19	1	no	Sylow
68	C₃	—	19	3	no	—
69	C₃	C₅²×(C₁₉⋊C₃)⋊C₄	1	1	yes	characteristic, direct product factor
70	C₂	C₃×C₅²×D₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₅²	D₁₉⋊C₃
commutator, hypocenter	C₁₉	C₃²×C₄×C₅²
Frattini	C₂	C₃×C₅²×D₁₉⋊C₃
Fitting, socle	C₂×C₃×C₅²×C₁₉	C₂×C₃
prime radical	C₂×C₃×C₅²×C₁₉⋊C₃	C₂
cosocle	C₂×C₁₉	C₂×C₃²×C₅²
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂×C₃×C₅²

Complements

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

N	H ≅ G/N	type
2. C₃×C₅²×Dic₁₉	C₃	semidirect
3. C₅²×(C₁₉⋊C₃)⋊C₄	C₃	direct (N × H)
4. C₃×C₅²×C₁₉⋊C₃	C₄	semidirect
5. C₃×C₅×(C₁₉⋊C₃)⋊C₄	C₅	direct (N × H)
8. C₅²×Dic₁₉	C₃²	semidirect
10. C₃×C₅²×C₁₉	C₃×C₄	semidirect
11. C₅²×C₁₉⋊C₃	C₃×C₄	semidirect
12. C₃×C₅×Dic₁₉	C₃×C₅	semidirect
13. C₅×(C₁₉⋊C₃)⋊C₄	C₃×C₅	direct (N × H)
16. C₃×C₅×C₁₉⋊C₃	C₄×C₅	semidirect
17. C₃×(C₁₉⋊C₃)⋊C₄	C₅²	direct (N × H)
20. C₅²×C₁₉	C₃²×C₄	semidirect
22. C₅×Dic₁₉	C₃²×C₅	semidirect
26. C₃×C₅×C₁₉	C₃×C₄×C₅	semidirect
27. C₅×C₁₉⋊C₃	C₃×C₄×C₅	semidirect
28. C₃×Dic₁₉	C₃×C₅²	semidirect
29. (C₁₉⋊C₃)⋊C₄	C₃×C₅²	direct (N × H)
33. C₃×C₁₉⋊C₃	C₄×C₅²	semidirect
39. C₅×C₁₉	C₃²×C₄×C₅	semidirect
41. Dic₁₉	C₃²×C₅²	semidirect
43. C₃×C₅²	(C₁₉⋊C₃)⋊C₄	direct (N × H)
46. C₃×C₁₉	C₃×C₄×C₅²	semidirect
47. C₁₉⋊C₃	C₃×C₄×C₅²	semidirect
54. C₅²	C₃×(C₁₉⋊C₃)⋊C₄	direct (N × H)
56. C₁₉	C₃²×C₄×C₅²	semidirect
59. C₃×C₅	C₅×(C₁₉⋊C₃)⋊C₄	direct (N × H)
66. C₅	C₃×C₅×(C₁₉⋊C₃)⋊C₄	direct (N × H)
69. C₃	C₅²×(C₁₉⋊C₃)⋊C₄	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
C₃×C₅²×Dic₁₉	3
C₅²×(C₁₉⋊C₃)⋊C₄	3
C₃×C₅×(C₁₉⋊C₃)⋊C₄	5
C₃²×C₄×C₅²	19

Subgroups — C3×C52×(C19⋊C3)⋊C4 #30685646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅²×(C₁₉⋊C₃)⋊C₄ #30685646