Subgroups — C₁₉×#11744719 #30685917

Order 17100.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₉×C₃⋊D₁₅	C₂	1	2	yes	maximal, complemented
2	C₃²×C₁₉×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×D₁₅⋊D₅	—	3	4	no	maximal
4	C₃²×C₅²×C₁₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₅×C₃⋊D₃×C₁₉	—	5	2	no	maximal
6	C₅×C₁₉×D₁₅	—	3	8	no	—
7	C₃×C₁₉×C₅⋊D₅	D₃	1	4	yes	complemented
8	D₅²×C₁₉	—	9	1	no	Hall
9	C₁₉×C₃⋊D₁₅	D₅	1	2	yes	complemented
10	C₅×C₃⋊D₃×C₁₉	—	5	2	no	—
11	C₃²×D₅×C₁₉	—	5	2	no	—
12	C₃²×D₅×C₁₉	—	10	2	no	—
13	C₃×C₅²×C₁₉	C₂×D₃	1	4	yes	complemented
14	D₃×D₅×C₁₉	—	15	8	no	—
15	C₁₉×C₅⋊D₅	C₃⋊D₃	1	1	yes	characteristic, complemented
16	C₅×D₅×C₁₉	—	9	2	no	—
17	#11744719	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₃²×C₅×C₁₉	—	2	2	no	—
19	C₃²×C₅×C₁₉	C₂×D₅	1	2	yes	complemented
20	C₂×C₃⋊D₃×C₁₉	—	25	1	no	Hall
21	C₁₉×D₁₅	—	3	8	no	—
22	C₅×D₃×C₁₉	—	15	8	no	—
23	C₃×D₅×C₁₉	—	5	8	no	—
24	C₃×D₅×C₁₉	—	10	8	no	—
25	C₅²×C₁₉	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
26	C₅×C₃⋊D₁₅	C₂×C₁₉	1	2	yes	complemented
27	C₃²×C₅⋊D₅	C₂×C₁₉	1	1	yes	characteristic, complemented
28	C₂×D₅×C₁₉	—	45	2	no	—
29	C₂×C₃²×C₁₉	—	25	1	no	—
30	C₃⋊D₃×C₁₉	—	5	2	no	—
31	D₁₅⋊D₅	—	3	4	no	—
32	C₃×C₅×C₁₉	—	2	8	no	—
33	C₃×C₅×C₁₉	D₃×D₅	1	8	yes	complemented
34	C₂×D₃×C₁₉	—	75	4	no	—
35	C₃²×C₅²	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
36	C₂×C₅×C₁₉	—	45	2	no	—
37	D₅×C₁₉	—	5	2	no	—
38	D₅×C₁₉	—	9	2	no	—
39	D₅×C₁₉	—	10	2	no	—
40	D₅×C₃⋊D₃	—	5	2	no	—
41	C₃²×C₁₉	D₅²	1	1	yes	characteristic, complemented, Hall
42	C₅×D₁₅	—	3	8	no	—
43	C₃×C₅⋊D₅	D₃×C₁₉	1	4	yes	complemented
44	C₂×C₃×C₁₉	—	25	4	no	—
45	D₃×C₁₉	—	15	8	no	—
46	D₅²	—	9	1	no	Hall
47	C₅×C₁₉	—	2	2	no	—
48	C₅×C₁₉	D₅×C₃⋊D₃	1	2	yes	complemented
49	C₃⋊D₁₅	D₅×C₁₉	1	2	yes	complemented
50	C₅×C₃⋊D₃	—	5	2	no	—
51	C₃²×D₅	—	5	2	no	—
52	C₃²×D₅	—	10	2	no	—
53	C₂²×C₁₉	—	225	1	no	Hall
54	C₃×C₅²	C₂×D₃×C₁₉	1	4	yes	complemented
55	D₃×D₅	—	15	8	no	—
56	C₃×C₁₉	D₁₅⋊D₅	1	4	yes	complemented
57	C₅⋊D₅	C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented
58	C₅×D₅	—	9	2	no	—
59	C₃²×C₅	—	2	2	no	—
60	C₃²×C₅	C₂×D₅×C₁₉	1	2	yes	complemented
61	C₂×C₁₉	—	25	1	no	—
62	C₂×C₁₉	—	45	2	no	—
63	C₂×C₃⋊D₃	—	25	1	no	Hall
64	D₁₅	—	3	8	no	—
65	C₃×D₅	—	5	8	no	—
66	C₃×D₅	—	10	8	no	—
67	C₅×D₃	—	15	8	no	—
68	C₅²	C₂×C₃⋊D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
69	C₂×D₅	—	45	2	no	—
70	C₁₉	#11744719	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃²	—	25	1	no	—
72	C₃⋊D₃	—	5	2	no	—
73	C₃×C₅	—	2	8	no	—
74	C₃×C₅	D₃×D₅×C₁₉	1	8	yes	complemented
75	C₂×D₃	—	75	4	no	—
76	C₂×C₅	—	45	2	no	—
77	D₅	—	5	2	no	—
78	D₅	—	9	2	no	—
79	D₅	—	10	2	no	—
80	C₃²	D₅²×C₁₉	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	—	25	4	no	—
82	D₃	—	15	8	no	—
83	C₅	—	2	2	no	—
84	C₅	D₅×C₃⋊D₃×C₁₉	1	2	yes	complemented
85	C₂²	—	225	1	no	Sylow
86	C₃	C₁₉×D₁₅⋊D₅	1	4	yes	complemented
87	C₂	—	25	1	no	—
88	C₂	—	45	2	no	—

	subgroup	G/H
center, hypercenter	C₁₉	#11744719
commutator, cosocle, hypocenter	C₃²×C₅²	C₂²×C₁₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₁₉	C₂²
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₁₉

Complements

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

N	H ≅ G/N	type
1. C₅×C₁₉×C₃⋊D₁₅	C₂	semidirect
2. C₃²×C₁₉×C₅⋊D₅	C₂	semidirect
4. C₃²×C₅²×C₁₉	C₂²	semidirect
7. C₃×C₁₉×C₅⋊D₅	D₃	semidirect
9. C₁₉×C₃⋊D₁₅	D₅	semidirect
13. C₃×C₅²×C₁₉	C₂×D₃	semidirect
15. C₁₉×C₅⋊D₅	C₃⋊D₃	semidirect
17. #11744719	C₁₉	direct (N × H)
19. C₃²×C₅×C₁₉	C₂×D₅	semidirect
25. C₅²×C₁₉	C₂×C₃⋊D₃	semidirect
26. C₅×C₃⋊D₁₅	C₂×C₁₉	semidirect
27. C₃²×C₅⋊D₅	C₂×C₁₉	semidirect
33. C₃×C₅×C₁₉	D₃×D₅	semidirect
35. C₃²×C₅²	C₂²×C₁₉	semidirect
41. C₃²×C₁₉	D₅²	semidirect
43. C₃×C₅⋊D₅	D₃×C₁₉	semidirect
48. C₅×C₁₉	D₅×C₃⋊D₃	semidirect
49. C₃⋊D₁₅	D₅×C₁₉	semidirect
54. C₃×C₅²	C₂×D₃×C₁₉	semidirect
56. C₃×C₁₉	D₁₅⋊D₅	semidirect
57. C₅⋊D₅	C₃⋊D₃×C₁₉	semidirect
60. C₃²×C₅	C₂×D₅×C₁₉	semidirect
68. C₅²	C₂×C₃⋊D₃×C₁₉	semidirect
70. C₁₉	#11744719	direct (N × H)
74. C₃×C₅	D₃×D₅×C₁₉	semidirect
80. C₃²	D₅²×C₁₉	semidirect
84. C₅	D₅×C₃⋊D₃×C₁₉	semidirect
86. C₃	C₁₉×D₁₅⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₁₉×C₃⋊D₁₅	2
C₃²×C₁₉×C₅⋊D₅	2
C₁₉×D₁₅⋊D₅	3
D₅×C₃⋊D₃×C₁₉	5
#11744719	19

Subgroups — C19×#11744719 #30685917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11744719 #30685917