Subgroups — C₃²×C₅²×D₇₁ #44544079

Order 31950.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅²×D₇₁	C₃	1	4	yes	maximal, direct product factor
3	C₃²×C₅×D₇₁	C₅	1	6	yes	maximal, direct product factor
4	C₃×C₅²×C₇₁	C₂×C₃	1	4	yes	complemented
5	C₅²×D₇₁	C₃²	1	1	yes	characteristic, direct product factor, Hall
6	C₃²×C₅×C₇₁	C₂×C₅	1	6	yes	complemented
7	C₃×C₅×D₇₁	C₃×C₅	1	24	yes	direct product factor
8	C₅²×C₇₁	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃²×D₇₁	C₅²	1	1	yes	characteristic, direct product factor, Hall
10	C₃×C₅×C₇₁	C₂×C₃×C₅	1	24	yes	complemented
11	C₅×D₇₁	C₃²×C₅	1	6	yes	direct product factor
12	C₃²×C₇₁	C₂×C₅²	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃²×C₅²	—	71	1	no	maximal, Hall
14	C₃×D₇₁	C₃×C₅²	1	4	yes	direct product factor
15	C₅×C₇₁	C₂×C₃²×C₅	1	6	yes	complemented
16	C₃²×C₅²	D₇₁	1	1	yes	characteristic, direct product factor, Hall
17	C₃×C₇₁	C₂×C₃×C₅²	1	4	yes	complemented
18	C₂×C₃×C₅²	—	71	4	no	—
19	D₇₁	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Hall
20	C₂×C₃²×C₅	—	71	6	no	—
21	C₃×C₅²	C₃×D₇₁	1	4	yes	direct product factor
22	C₇₁	C₂×C₃²×C₅²	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₅²	—	71	1	no	Hall
24	C₃²×C₅	C₅×D₇₁	1	6	yes	direct product factor
25	C₂×C₃×C₅	—	71	24	no	—
26	C₅²	C₃²×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₂×C₃²	—	71	1	no	Hall
28	C₃×C₅	C₃×C₅×D₇₁	1	24	yes	direct product factor
29	C₂×C₅	—	71	6	no	—
30	C₃²	C₅²×D₇₁	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	—	71	4	no	—
32	C₅	C₃²×C₅×D₇₁	1	6	yes	direct product factor
33	C₃	C₃×C₅²×D₇₁	1	4	yes	direct product factor
34	C₂	—	71	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃²×C₅²	D₇₁
commutator, cosocle, hypocenter	C₇₁	C₂×C₃²×C₅²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇₁	C₂
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₃²×C₅²

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇₁	C₂	semidirect
2. C₃×C₅²×D₇₁	C₃	direct (N × H)
3. C₃²×C₅×D₇₁	C₅	direct (N × H)
4. C₃×C₅²×C₇₁	C₂×C₃	semidirect
5. C₅²×D₇₁	C₃²	direct (N × H)
6. C₃²×C₅×C₇₁	C₂×C₅	semidirect
7. C₃×C₅×D₇₁	C₃×C₅	direct (N × H)
8. C₅²×C₇₁	C₂×C₃²	semidirect
9. C₃²×D₇₁	C₅²	direct (N × H)
10. C₃×C₅×C₇₁	C₂×C₃×C₅	semidirect
11. C₅×D₇₁	C₃²×C₅	direct (N × H)
12. C₃²×C₇₁	C₂×C₅²	semidirect
14. C₃×D₇₁	C₃×C₅²	direct (N × H)
15. C₅×C₇₁	C₂×C₃²×C₅	semidirect
16. C₃²×C₅²	D₇₁	direct (N × H)
17. C₃×C₇₁	C₂×C₃×C₅²	semidirect
19. D₇₁	C₃²×C₅²	direct (N × H)
21. C₃×C₅²	C₃×D₇₁	direct (N × H)
22. C₇₁	C₂×C₃²×C₅²	semidirect
24. C₃²×C₅	C₅×D₇₁	direct (N × H)
26. C₅²	C₃²×D₇₁	direct (N × H)
28. C₃×C₅	C₃×C₅×D₇₁	direct (N × H)
30. C₃²	C₅²×D₇₁	direct (N × H)
32. C₅	C₃²×C₅×D₇₁	direct (N × H)
33. C₃	C₃×C₅²×D₇₁	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×C₇₁	2
C₃×C₅²×D₇₁	3
C₃²×C₅×D₇₁	5
C₂×C₃²×C₅²	71

Subgroups — C32×C52×D71 #44544079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅²×D₇₁ #44544079