Subgroups — C₂×C₃×C₁₉×D₆₉ #30628655

Order 15732.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₁₉×D₆₉	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₁₉×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×D₆₉	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×C₁₉×D₂₃	—	3	1	no	maximal
5	C₃²×C₁₉×C₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₉×D₆₉	C₂×C₃	1	2	yes	direct product factor
7	C₃×C₁₉×D₂₃	—	3	2	no	—
8	C₂×C₃×C₁₉×C₂₃	—	2	1	no	—
9	C₂×C₃×C₁₉×C₂₃	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₁₉×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₁₉×D₂₃	—	3	1	no	Hall
12	C₃×C₁₉×C₂₃	—	2	1	no	—
13	C₃×C₁₉×C₂₃	C₂×D₃	1	1	yes	characteristic, complemented
14	C₃×C₁₉×C₂₃	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₁₉×C₂₃	C₃×D₃	1	1	yes	characteristic, complemented
16	C₁₉×D₂₃	—	3	2	no	—
17	C₂×C₃×D₆₉	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₃×D₃×C₁₉	—	23	1	no	maximal, Hall
19	C₁₉×C₂₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃²×C₂₃	C₂×C₁₉	1	1	yes	characteristic, complemented
21	C₃×D₆₉	C₂×C₁₉	1	2	yes	direct product factor
22	C₂×C₃²×C₁₉	D₂₃	1	1	yes	characteristic, complemented
23	C₃×D₃×C₁₉	—	23	2	no	—
24	C₂×D₆₉	C₃×C₁₉	1	1	yes	characteristic, direct product factor
25	C₂×C₃×D₂₃	—	3	1	no	—
26	C₂²×C₃×C₁₉	—	69	1	no	—
27	C₂×D₃×C₁₉	—	23	1	no	—
28	C₃²×C₂₃	C₂²×C₁₉	1	1	yes	characteristic, complemented, Hall
29	C₃²×C₁₉	C₂×D₂₃	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₂₃	—	2	1	no	—
31	C₂×C₃×C₂₃	D₃×C₁₉	1	1	yes	characteristic, complemented
32	C₂×C₃×C₂₃	C₂×C₃×C₁₉	1	1	yes	characteristic, complemented
33	D₆₉	C₂×C₃×C₁₉	1	2	yes	direct product factor
34	C₃×D₂₃	—	3	2	no	—
35	C₂×C₃×C₁₉	—	2	1	no	—
36	C₂×C₃×C₁₉	—	69	2	no	—
37	C₂×C₃×C₁₉	C₃×D₂₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₁₉	D₆₉	1	1	yes	characteristic, direct product factor
39	D₃×C₁₉	—	23	2	no	—
40	C₂×D₂₃	—	3	1	no	Hall
41	C₂²×C₁₉	—	69	1	no	Hall
42	C₃×C₂₃	—	2	1	no	—
43	C₃×C₂₃	C₂×D₃×C₁₉	1	1	yes	characteristic, complemented
44	C₃×C₂₃	C₂²×C₃×C₁₉	1	1	yes	characteristic, complemented
45	C₃×C₁₉	—	2	1	no	—
46	C₃×C₁₉	C₂×C₃×D₂₃	1	1	yes	characteristic, complemented
47	C₃×C₁₉	C₂×D₆₉	1	1	yes	characteristic, direct product factor
48	C₂×C₂₃	C₃×D₃×C₁₉	1	1	yes	characteristic, complemented
49	D₂₃	—	3	2	no	—
50	C₂×C₁₉	—	69	2	no	—
51	C₂×C₁₉	C₃×D₆₉	1	1	yes	characteristic, direct product factor
52	C₂×C₃×D₃	—	23	1	no	Hall
53	C₂₃	C₂×C₃×D₃×C₁₉	1	1	yes	characteristic, complemented, Sylow
54	C₁₉	C₂×C₃×D₆₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃²	C₁₉×D₂₃	1	1	yes	characteristic, complemented
56	C₃×D₃	—	23	2	no	—
57	C₂²×C₃	—	69	1	no	—
58	C₂×D₃	—	23	1	no	—
59	C₃²	C₂×C₁₉×D₂₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	69	2	no	—
62	C₂×C₃	C₃×C₁₉×D₂₃	1	1	yes	characteristic, complemented
63	C₂×C₃	C₁₉×D₆₉	1	1	yes	characteristic, direct product factor
64	D₃	—	23	2	no	—
65	C₂²	—	69	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂×C₃×C₁₉×D₂₃	1	1	yes	characteristic, complemented
68	C₃	C₂×C₁₉×D₆₉	1	1	yes	characteristic, direct product factor
69	C₂	—	69	2	no	—
70	C₂	C₃×C₁₉×D₆₉	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₉	D₆₉
commutator, cosocle, hypocenter	C₃×C₂₃	C₂²×C₃×C₁₉
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₉×C₂₃	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₂×C₃×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₃×C₁₉×D₆₉	C₂	direct (N × H)
2. C₂×C₃²×C₁₉×C₂₃	C₂	semidirect
3. C₂×C₁₉×D₆₉	C₃	direct (N × H)
5. C₃²×C₁₉×C₂₃	C₂²	semidirect
6. C₁₉×D₆₉	C₂×C₃	direct (N × H)
9. C₂×C₃×C₁₉×C₂₃	D₃	semidirect
10. C₂×C₃×C₁₉×C₂₃	C₂×C₃	semidirect
13. C₃×C₁₉×C₂₃	C₂×D₃	semidirect
14. C₃×C₁₉×C₂₃	C₂²×C₃	semidirect
15. C₂×C₁₉×C₂₃	C₃×D₃	semidirect
17. C₂×C₃×D₆₉	C₁₉	direct (N × H)
19. C₁₉×C₂₃	C₂×C₃×D₃	semidirect
20. C₂×C₃²×C₂₃	C₂×C₁₉	semidirect
21. C₃×D₆₉	C₂×C₁₉	direct (N × H)
22. C₂×C₃²×C₁₉	D₂₃	semidirect
24. C₂×D₆₉	C₃×C₁₉	direct (N × H)
28. C₃²×C₂₃	C₂²×C₁₉	semidirect
29. C₃²×C₁₉	C₂×D₂₃	semidirect
31. C₂×C₃×C₂₃	D₃×C₁₉	semidirect
32. C₂×C₃×C₂₃	C₂×C₃×C₁₉	semidirect
33. D₆₉	C₂×C₃×C₁₉	direct (N × H)
37. C₂×C₃×C₁₉	C₃×D₂₃	semidirect
38. C₂×C₃×C₁₉	D₆₉	direct (N × H)
43. C₃×C₂₃	C₂×D₃×C₁₉	semidirect
44. C₃×C₂₃	C₂²×C₃×C₁₉	semidirect
46. C₃×C₁₉	C₂×C₃×D₂₃	semidirect
47. C₃×C₁₉	C₂×D₆₉	direct (N × H)
48. C₂×C₂₃	C₃×D₃×C₁₉	semidirect
51. C₂×C₁₉	C₃×D₆₉	direct (N × H)
53. C₂₃	C₂×C₃×D₃×C₁₉	semidirect
54. C₁₉	C₂×C₃×D₆₉	direct (N × H)
55. C₂×C₃²	C₁₉×D₂₃	semidirect
59. C₃²	C₂×C₁₉×D₂₃	semidirect
62. C₂×C₃	C₃×C₁₉×D₂₃	semidirect
63. C₂×C₃	C₁₉×D₆₉	direct (N × H)
67. C₃	C₂×C₃×C₁₉×D₂₃	semidirect
68. C₃	C₂×C₁₉×D₆₉	direct (N × H)
70. C₂	C₃×C₁₉×D₆₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₁₉×D₆₉	2
C₂×C₃²×C₁₉×C₂₃	2
C₂×C₁₉×D₆₉	3
C₂×C₃×C₁₉×D₂₃	3
C₂×C₃×D₆₉	19
C₂×C₃×D₃×C₁₉	23

Subgroups — C2×C3×C19×D69 #30628655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₉×D₆₉ #30628655