Subgroups — C₅²×D₃×C₁₁² #31827712

Order 18150.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²×C₁₁²	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₂×C₅²×C₁₁²	—	3	1	no	maximal, Hall
3	C₅×D₃×C₁₁²	C₅	1	6	yes	maximal, direct product factor
4	C₅²×C₁₁²	D₃	1	1	yes	characteristic, direct product factor, Hall
5	C₃×C₅×C₁₁²	C₂×C₅	1	6	yes	complemented
6	C₅²×D₃×C₁₁	C₁₁	1	12	yes	maximal, direct product factor
7	C₂×C₅×C₁₁²	—	3	6	no	—
8	C₃×C₅²×C₁₁	C₂×C₁₁	1	12	yes	complemented
9	D₃×C₁₁²	C₅²	1	1	yes	characteristic, direct product factor, Hall
10	C₅×C₁₁²	C₅×D₃	1	6	yes	direct product factor
11	C₂×C₅²×C₁₁	—	3	12	no	—
12	C₃×C₁₁²	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
13	C₅×D₃×C₁₁	C₅×C₁₁	1	72	yes	direct product factor
14	C₅²×C₁₁	D₃×C₁₁	1	12	yes	direct product factor
15	C₂×C₁₁²	—	3	1	no	Hall
16	C₃×C₅×C₁₁	C₂×C₅×C₁₁	1	72	yes	complemented
17	C₅²×D₃	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
18	C₁₁²	C₅²×D₃	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₅×C₁₁	—	3	72	no	—
20	C₃×C₅²	C₂×C₁₁²	1	1	yes	characteristic, complemented, Hall
21	D₃×C₁₁	C₅²×C₁₁	1	12	yes	direct product factor
22	C₅×C₁₁	C₅×D₃×C₁₁	1	72	yes	direct product factor
23	C₂×C₅²	—	3	1	no	Hall
24	C₃×C₁₁	C₂×C₅²×C₁₁	1	12	yes	complemented
25	C₅×D₃	C₅×C₁₁²	1	6	yes	direct product factor
26	C₅²	D₃×C₁₁²	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₁₁	—	3	12	no	—
28	C₃×C₅	C₂×C₅×C₁₁²	1	6	yes	complemented
29	C₁₁	C₅²×D₃×C₁₁	1	12	yes	direct product factor
30	C₂×C₅	—	3	6	no	—
31	D₃	C₅²×C₁₁²	1	1	yes	characteristic, direct product factor, Hall
32	C₅	C₅×D₃×C₁₁²	1	6	yes	direct product factor
33	C₃	C₂×C₅²×C₁₁²	1	1	yes	characteristic, complemented, Sylow
34	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₅²×C₁₁²	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₅²×C₁₁²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₁₁²	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₅²×C₁₁²

Complements

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

N	H ≅ G/N	type
1. C₃×C₅²×C₁₁²	C₂	semidirect
3. C₅×D₃×C₁₁²	C₅	direct (N × H)
4. C₅²×C₁₁²	D₃	direct (N × H)
5. C₃×C₅×C₁₁²	C₂×C₅	semidirect
6. C₅²×D₃×C₁₁	C₁₁	direct (N × H)
8. C₃×C₅²×C₁₁	C₂×C₁₁	semidirect
9. D₃×C₁₁²	C₅²	direct (N × H)
10. C₅×C₁₁²	C₅×D₃	direct (N × H)
12. C₃×C₁₁²	C₂×C₅²	semidirect
13. C₅×D₃×C₁₁	C₅×C₁₁	direct (N × H)
14. C₅²×C₁₁	D₃×C₁₁	direct (N × H)
16. C₃×C₅×C₁₁	C₂×C₅×C₁₁	semidirect
17. C₅²×D₃	C₁₁²	direct (N × H)
18. C₁₁²	C₅²×D₃	direct (N × H)
20. C₃×C₅²	C₂×C₁₁²	semidirect
21. D₃×C₁₁	C₅²×C₁₁	direct (N × H)
22. C₅×C₁₁	C₅×D₃×C₁₁	direct (N × H)
24. C₃×C₁₁	C₂×C₅²×C₁₁	semidirect
25. C₅×D₃	C₅×C₁₁²	direct (N × H)
26. C₅²	D₃×C₁₁²	direct (N × H)
28. C₃×C₅	C₂×C₅×C₁₁²	semidirect
29. C₁₁	C₅²×D₃×C₁₁	direct (N × H)
31. D₃	C₅²×C₁₁²	direct (N × H)
32. C₅	C₅×D₃×C₁₁²	direct (N × H)
33. C₃	C₂×C₅²×C₁₁²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅²×C₁₁²	2
C₂×C₅²×C₁₁²	3
C₅×D₃×C₁₁²	5
C₅²×D₃×C₁₁	11

Subgroups — C52×D3×C112 #31827712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×D₃×C₁₁² #31827712