Subgroups — C₅²×C₁₃×D₂₁ #28331663

Order 13650.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₇×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅²×C₁₃×D₇	—	3	1	no	maximal, Hall
3	C₅×C₁₃×D₂₁	C₅	1	6	yes	maximal, direct product factor
4	C₅²×C₇×C₁₃	D₃	1	1	yes	characteristic, complemented, Hall
5	C₅²×D₃×C₁₃	—	7	1	no	maximal, Hall
6	C₃×C₅×C₇×C₁₃	C₂×C₅	1	6	yes	complemented
7	C₅²×D₂₁	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×C₅²×C₁₃	D₇	1	1	yes	characteristic, complemented, Hall
9	C₅×C₁₃×D₇	—	3	6	no	—
10	C₂×C₅²×C₁₃	—	21	1	no	Hall
11	C₁₃×D₂₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₅²×C₇	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
13	C₅×C₇×C₁₃	C₅×D₃	1	6	yes	complemented
14	C₅×D₃×C₁₃	—	7	6	no	—
15	C₅²×D₇	—	3	1	no	Hall
16	C₅²×C₁₃	D₂₁	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₇×C₁₃	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
18	C₅×D₂₁	C₅×C₁₃	1	6	yes	direct product factor
19	C₃×C₅×C₁₃	C₅×D₇	1	6	yes	complemented
20	C₁₃×D₇	—	3	1	no	Hall
21	C₅²×C₇	D₃×C₁₃	1	1	yes	characteristic, complemented, Hall
22	C₅²×D₃	—	7	1	no	Hall
23	C₂×C₅×C₁₃	—	21	6	no	—
24	C₃×C₅×C₇	C₂×C₅×C₁₃	1	6	yes	complemented
25	C₇×C₁₃	C₅²×D₃	1	1	yes	characteristic, complemented, Hall
26	D₃×C₁₃	—	7	1	no	Hall
27	C₃×C₅²	C₁₃×D₇	1	1	yes	characteristic, complemented, Hall
28	C₅×D₇	—	3	6	no	—
29	C₅×C₁₃	C₅×D₂₁	1	6	yes	direct product factor
30	C₂×C₅²	—	21	1	no	Hall
31	D₂₁	C₅²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
32	C₃×C₁₃	C₅²×D₇	1	1	yes	characteristic, complemented, Hall
33	C₅×C₇	C₅×D₃×C₁₃	1	6	yes	complemented
34	C₅×D₃	—	7	6	no	—
35	C₂×C₁₃	—	21	1	no	Hall
36	C₅²	C₁₃×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
37	C₃×C₇	C₂×C₅²×C₁₃	1	1	yes	characteristic, complemented, Hall
38	C₃×C₅	C₅×C₁₃×D₇	1	6	yes	complemented
39	D₇	—	3	1	no	Hall
40	C₁₃	C₅²×D₂₁	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×C₅	—	21	6	no	—
42	C₇	C₅²×D₃×C₁₃	1	1	yes	characteristic, complemented, Sylow
43	D₃	—	7	1	no	Hall
44	C₅	C₅×C₁₃×D₂₁	1	6	yes	direct product factor
45	C₃	C₅²×C₁₃×D₇	1	1	yes	characteristic, complemented, Sylow
46	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²×C₁₃	D₂₁
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₅²×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇×C₁₃	C₂
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₅²×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₇×C₁₃	C₂	semidirect
3. C₅×C₁₃×D₂₁	C₅	direct (N × H)
4. C₅²×C₇×C₁₃	D₃	semidirect
6. C₃×C₅×C₇×C₁₃	C₂×C₅	semidirect
7. C₅²×D₂₁	C₁₃	direct (N × H)
8. C₃×C₅²×C₁₃	D₇	semidirect
11. C₁₃×D₂₁	C₅²	direct (N × H)
12. C₃×C₅²×C₇	C₂×C₁₃	semidirect
13. C₅×C₇×C₁₃	C₅×D₃	semidirect
16. C₅²×C₁₃	D₂₁	direct (N × H)
17. C₃×C₇×C₁₃	C₂×C₅²	semidirect
18. C₅×D₂₁	C₅×C₁₃	direct (N × H)
19. C₃×C₅×C₁₃	C₅×D₇	semidirect
21. C₅²×C₇	D₃×C₁₃	semidirect
24. C₃×C₅×C₇	C₂×C₅×C₁₃	semidirect
25. C₇×C₁₃	C₅²×D₃	semidirect
27. C₃×C₅²	C₁₃×D₇	semidirect
29. C₅×C₁₃	C₅×D₂₁	direct (N × H)
31. D₂₁	C₅²×C₁₃	direct (N × H)
32. C₃×C₁₃	C₅²×D₇	semidirect
33. C₅×C₇	C₅×D₃×C₁₃	semidirect
36. C₅²	C₁₃×D₂₁	direct (N × H)
37. C₃×C₇	C₂×C₅²×C₁₃	semidirect
38. C₃×C₅	C₅×C₁₃×D₇	semidirect
40. C₁₃	C₅²×D₂₁	direct (N × H)
42. C₇	C₅²×D₃×C₁₃	semidirect
44. C₅	C₅×C₁₃×D₂₁	direct (N × H)
45. C₃	C₅²×C₁₃×D₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₇×C₁₃	2
C₅²×C₁₃×D₇	3
C₅×C₁₃×D₂₁	5
C₅²×D₃×C₁₃	7
C₅²×D₂₁	13

Subgroups — C52×C13×D21 #28331663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₁₃×D₂₁ #28331663