Subgroups — C₃²×C₅²×D₃₁ #28358521

Order 13950.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×D₃₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×D₃₁	C₅	1	6	yes	maximal, direct product factor
4	C₃×C₅²×C₃₁	C₂×C₃	1	4	yes	complemented
5	C₅²×D₃₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₅×C₃₁	C₂×C₅	1	6	yes	complemented
7	C₃×C₅×D₃₁	C₃×C₅	1	24	yes	direct product factor
8	C₅²×C₃₁	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃²×D₃₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₅×C₃₁	C₂×C₃×C₅	1	24	yes	complemented
11	C₂×C₃²×C₅²	—	31	1	no	maximal, Hall
12	C₅×D₃₁	C₃²×C₅	1	6	yes	direct product factor
13	C₃²×C₃₁	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
14	C₃²×C₅²	D₃₁	1	1	yes	characteristic, direct product factor, Hall
15	C₃×D₃₁	C₃×C₅²	1	4	yes	direct product factor
16	C₅×C₃₁	C₂×C₃²×C₅	1	6	yes	complemented
17	C₂×C₃×C₅²	—	31	4	no	—
18	C₃×C₃₁	C₂×C₃×C₅²	1	4	yes	complemented
19	C₂×C₃²×C₅	—	31	6	no	—
20	C₃×C₅²	C₃×D₃₁	1	4	yes	direct product factor
21	D₃₁	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Hall
22	C₂×C₅²	—	31	1	no	Hall
23	C₃²×C₅	C₅×D₃₁	1	6	yes	direct product factor
24	C₃₁	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃×C₅	—	31	24	no	—
26	C₅²	C₃²×D₃₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃²	—	31	1	no	Hall
28	C₃×C₅	C₃×C₅×D₃₁	1	24	yes	direct product factor
29	C₂×C₅	—	31	6	no	—
30	C₃²	C₅²×D₃₁	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	—	31	4	no	—
32	C₅	C₃²×C₅×D₃₁	1	6	yes	direct product factor
33	C₃	C₃×C₅²×D₃₁	1	4	yes	direct product factor
34	C₂	—	31	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅²	D₃₁
commutator, cosocle, hypocenter	C₃₁	C₂×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₃₁	C₂
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₃²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₃₁	C₂	semidirect
2. C₃×C₅²×D₃₁	C₃	direct (N × H)
3. C₃²×C₅×D₃₁	C₅	direct (N × H)
4. C₃×C₅²×C₃₁	C₂×C₃	semidirect
5. C₅²×D₃₁	C₃²	direct (N × H)
6. C₃²×C₅×C₃₁	C₂×C₅	semidirect
7. C₃×C₅×D₃₁	C₃×C₅	direct (N × H)
8. C₅²×C₃₁	C₂×C₃²	semidirect
9. C₃²×D₃₁	C₅²	direct (N × H)
10. C₃×C₅×C₃₁	C₂×C₃×C₅	semidirect
12. C₅×D₃₁	C₃²×C₅	direct (N × H)
13. C₃²×C₃₁	C₂×C₅²	semidirect
14. C₃²×C₅²	D₃₁	direct (N × H)
15. C₃×D₃₁	C₃×C₅²	direct (N × H)
16. C₅×C₃₁	C₂×C₃²×C₅	semidirect
18. C₃×C₃₁	C₂×C₃×C₅²	semidirect
20. C₃×C₅²	C₃×D₃₁	direct (N × H)
21. D₃₁	C₃²×C₅²	direct (N × H)
23. C₃²×C₅	C₅×D₃₁	direct (N × H)
24. C₃₁	C₂×C₃²×C₅²	semidirect
26. C₅²	C₃²×D₃₁	direct (N × H)
28. C₃×C₅	C₃×C₅×D₃₁	direct (N × H)
30. C₃²	C₅²×D₃₁	direct (N × H)
32. C₅	C₃²×C₅×D₃₁	direct (N × H)
33. C₃	C₃×C₅²×D₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₅²×C₃₁	2
C₃×C₅²×D₃₁	3
C₃²×C₅×D₃₁	5
C₂×C₃²×C₅²	31

Subgroups — C32×C52×D31 #28358521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×D₃₁ #28358521