Subgroups — C₃×C₅²×D₆₉ #23780388

Order 10350.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×D₆₉	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₅²×D₂₃	—	3	1	no	maximal
4	C₃×C₅×D₆₉	C₅	1	6	yes	maximal, direct product factor
5	C₃×C₅²×C₂₃	—	2	1	no	—
6	C₃×C₅²×C₂₃	D₃	1	1	yes	characteristic, complemented
7	C₃×C₅²×C₂₃	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅²×D₂₃	—	3	1	no	Hall
9	C₃²×C₅×C₂₃	C₂×C₅	1	6	yes	complemented
10	C₅×D₆₉	C₃×C₅	1	6	yes	direct product factor
11	C₃×C₅×D₂₃	—	3	6	no	—
12	C₅²×C₂₃	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₃×C₅²×D₃	—	23	1	no	maximal, Hall
14	C₃×D₆₉	C₅²	1	1	yes	characteristic, direct product factor, Hall
15	C₃×C₅×C₂₃	—	2	6	no	—
16	C₃×C₅×C₂₃	C₅×D₃	1	6	yes	complemented
17	C₃×C₅×C₂₃	C₂×C₃×C₅	1	6	yes	complemented
18	C₅×D₂₃	—	3	6	no	—
19	C₃²×C₅²	D₂₃	1	1	yes	characteristic, complemented, Hall
20	C₃²×C₂₃	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₅²	—	69	1	no	—
22	C₅²×D₃	—	23	1	no	—
23	D₆₉	C₃×C₅²	1	1	yes	characteristic, direct product factor
24	C₃×D₂₃	—	3	1	no	—
25	C₅×C₂₃	C₃×C₅×D₃	1	6	yes	complemented
26	C₃×C₅×D₃	—	23	6	no	—
27	C₃×C₅²	—	2	1	no	—
28	C₃×C₅²	C₃×D₂₃	1	1	yes	characteristic, complemented
29	C₃×C₅²	D₆₉	1	1	yes	characteristic, direct product factor
30	C₃×C₂₃	—	2	1	no	—
31	C₃×C₂₃	C₅²×D₃	1	1	yes	characteristic, complemented
32	C₃×C₂₃	C₂×C₃×C₅²	1	1	yes	characteristic, complemented
33	C₂×C₅²	—	69	1	no	Hall
34	D₂₃	—	3	1	no	Hall
35	C₃²×C₅	C₅×D₂₃	1	6	yes	complemented
36	C₂×C₃×C₅	—	69	6	no	—
37	C₅×D₃	—	23	6	no	—
38	C₅²	C₃×D₆₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂₃	C₃×C₅²×D₃	1	1	yes	characteristic, complemented, Sylow
40	C₃×D₃	—	23	1	no	Hall
41	C₃×C₅	—	2	6	no	—
42	C₃×C₅	C₃×C₅×D₂₃	1	6	yes	complemented
43	C₃×C₅	C₅×D₆₉	1	6	yes	direct product factor
44	C₂×C₅	—	69	6	no	—
45	C₃²	C₅²×D₂₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃	—	69	1	no	—
47	D₃	—	23	1	no	—
48	C₅	C₃×C₅×D₆₉	1	6	yes	direct product factor
49	C₃	—	2	1	no	—
50	C₃	C₃×C₅²×D₂₃	1	1	yes	characteristic, complemented
51	C₃	C₅²×D₆₉	1	1	yes	characteristic, direct product factor
52	C₂	—	69	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅²	D₆₉
commutator, cosocle, hypocenter	C₃×C₂₃	C₂×C₃×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₂₃	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁ ◁ C₃×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₂₃	C₂	semidirect
2. C₅²×D₆₉	C₃	direct (N × H)
4. C₃×C₅×D₆₉	C₅	direct (N × H)
6. C₃×C₅²×C₂₃	D₃	semidirect
7. C₃×C₅²×C₂₃	C₂×C₃	semidirect
9. C₃²×C₅×C₂₃	C₂×C₅	semidirect
10. C₅×D₆₉	C₃×C₅	direct (N × H)
12. C₅²×C₂₃	C₃×D₃	semidirect
14. C₃×D₆₉	C₅²	direct (N × H)
16. C₃×C₅×C₂₃	C₅×D₃	semidirect
17. C₃×C₅×C₂₃	C₂×C₃×C₅	semidirect
19. C₃²×C₅²	D₂₃	semidirect
20. C₃²×C₂₃	C₂×C₅²	semidirect
23. D₆₉	C₃×C₅²	direct (N × H)
25. C₅×C₂₃	C₃×C₅×D₃	semidirect
28. C₃×C₅²	C₃×D₂₃	semidirect
29. C₃×C₅²	D₆₉	direct (N × H)
31. C₃×C₂₃	C₅²×D₃	semidirect
32. C₃×C₂₃	C₂×C₃×C₅²	semidirect
35. C₃²×C₅	C₅×D₂₃	semidirect
38. C₅²	C₃×D₆₉	direct (N × H)
39. C₂₃	C₃×C₅²×D₃	semidirect
42. C₃×C₅	C₃×C₅×D₂₃	semidirect
43. C₃×C₅	C₅×D₆₉	direct (N × H)
45. C₃²	C₅²×D₂₃	semidirect
48. C₅	C₃×C₅×D₆₉	direct (N × H)
50. C₃	C₃×C₅²×D₂₃	semidirect
51. C₃	C₅²×D₆₉	direct (N × H)

Subgroups — C3×C52×D69 #23780388

Conjugacy classes of subgroups

Complements

Subgroups — C₃×C₅²×D₆₉ #23780388