Subgroups — C₇₁×C₃⋊(C₃⋊(C₅⋊D₅)) #44544094

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	C₇₁×C₃⋊(C₅⋊D₅)	—	3	4	no	maximal
3	C₇₁×C₃⋊D₁₅	—	5	6	no	maximal
4	C₃×C₅²×C₇₁	D₃	1	4	yes	complemented
5	C₅⋊D₅×C₇₁	—	9	1	no	Hall
6	C₃²×C₅×C₇₁	D₅	1	6	yes	complemented
7	D₁₅×C₇₁	—	15	24	no	—
8	C₅²×C₇₁	C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
9	C₃⋊D₃×C₇₁	—	25	1	no	Hall
10	C₃×C₅×C₇₁	D₁₅	1	24	yes	complemented
11	D₅×C₇₁	—	45	6	no	—
12	C₃²×C₇₁	C₅⋊D₅	1	1	yes	characteristic, complemented, Hall
13	C₃⋊(C₃⋊(C₅⋊D₅))	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	D₃×C₇₁	—	75	4	no	—
15	C₅×C₇₁	C₃⋊D₁₅	1	6	yes	complemented
16	C₃²×C₅²	C₂×C₇₁	1	1	yes	characteristic, complemented, Hall
17	C₃×C₇₁	C₃⋊(C₅⋊D₅)	1	4	yes	complemented
18	C₃⋊(C₅⋊D₅)	—	3	4	no	—
19	C₂×C₇₁	—	225	1	no	Hall
20	C₃⋊D₁₅	—	5	6	no	—
21	C₃×C₅²	D₃×C₇₁	1	4	yes	complemented
22	C₇₁	C₃⋊(C₃⋊(C₅⋊D₅))	1	1	yes	characteristic, direct product factor, Sylow
23	C₅⋊D₅	—	9	1	no	Hall
24	C₃²×C₅	D₅×C₇₁	1	6	yes	complemented
25	D₁₅	—	15	24	no	—
26	C₅²	C₃⋊D₃×C₇₁	1	1	yes	characteristic, complemented, Sylow
27	C₃⋊D₃	—	25	1	no	Hall
28	C₃×C₅	D₁₅×C₇₁	1	24	yes	complemented
29	D₅	—	45	6	no	—
30	C₃²	C₅⋊D₅×C₇₁	1	1	yes	characteristic, complemented, Sylow
31	D₃	—	75	4	no	—
32	C₅	C₇₁×C₃⋊D₁₅	1	6	yes	complemented
33	C₃	C₇₁×C₃⋊(C₅⋊D₅)	1	4	yes	complemented
34	C₂	—	225	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇₁	C₃⋊(C₃⋊(C₅⋊D₅))
commutator, cosocle, hypocenter	C₃²×C₅²	C₂×C₇₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇₁	C₂
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₇₁

Complements

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

N	H ≅ G/N	type
1. C₃²×C₅²×C₇₁	C₂	semidirect
4. C₃×C₅²×C₇₁	D₃	semidirect
6. C₃²×C₅×C₇₁	D₅	semidirect
8. C₅²×C₇₁	C₃⋊D₃	semidirect
10. C₃×C₅×C₇₁	D₁₅	semidirect
12. C₃²×C₇₁	C₅⋊D₅	semidirect
13. C₃⋊(C₃⋊(C₅⋊D₅))	C₇₁	direct (N × H)
15. C₅×C₇₁	C₃⋊D₁₅	semidirect
16. C₃²×C₅²	C₂×C₇₁	semidirect
17. C₃×C₇₁	C₃⋊(C₅⋊D₅)	semidirect
21. C₃×C₅²	D₃×C₇₁	semidirect
22. C₇₁	C₃⋊(C₃⋊(C₅⋊D₅))	direct (N × H)
24. C₃²×C₅	D₅×C₇₁	semidirect
26. C₅²	C₃⋊D₃×C₇₁	semidirect
28. C₃×C₅	D₁₅×C₇₁	semidirect
30. C₃²	C₅⋊D₅×C₇₁	semidirect
32. C₅	C₇₁×C₃⋊D₁₅	semidirect
33. C₃	C₇₁×C₃⋊(C₅⋊D₅)	semidirect

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
C₇₁×C₃⋊(C₅⋊D₅)	3
C₇₁×C₃⋊D₁₅	5
C₃⋊(C₃⋊(C₅⋊D₅))	71

Subgroups — C71×C3⋊(C3⋊(C5⋊D5)) #44544094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×C₃⋊(C₃⋊(C₅⋊D₅)) #44544094