Subgroups — C₃²×C₅×C₉×C₇₁ #72205708

Order 28755.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₅×C₇₁	C₃	1	1	yes	maximal, characteristic
2	C₃×C₅×C₉×C₇₁	C₃	1	12	yes	maximal, direct product factor
3	C₃²×C₉×C₇₁	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃²×C₅×C₇₁	C₉	1	9	yes	direct product factor
5	C₃²×C₅×C₇₁	C₃²	1	4	yes	—
6	C₅×C₉×C₇₁	C₃²	1	9	yes	direct product factor
7	C₃³×C₇₁	C₃×C₅	1	1	yes	characteristic
8	C₃×C₉×C₇₁	C₃×C₅	1	12	yes	direct product factor
9	C₃×C₅×C₇₁	C₃×C₉	1	12	yes	direct product factor
10	C₃×C₅×C₇₁	C₃³	1	1	yes	characteristic
11	C₃²×C₇₁	C₅×C₉	1	9	yes	direct product factor
12	C₃²×C₇₁	C₃²×C₅	1	4	yes	—
13	C₉×C₇₁	C₃²×C₅	1	9	yes	direct product factor
14	C₃²×C₅×C₉	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×C₇₁	C₃²×C₉	1	1	yes	characteristic, direct product factor, Hall
16	C₃×C₇₁	C₃×C₅×C₉	1	12	yes	direct product factor
17	C₃×C₇₁	C₃³×C₅	1	1	yes	characteristic
18	C₃³×C₅	C₃×C₇₁	1	1	yes	characteristic
19	C₃×C₅×C₉	C₃×C₇₁	1	12	yes	direct product factor
20	C₃²×C₉	C₅×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
21	C₇₁	C₃²×C₅×C₉	1	1	yes	characteristic, direct product factor, Sylow
22	C₃²×C₅	C₉×C₇₁	1	9	yes	direct product factor
23	C₃²×C₅	C₃²×C₇₁	1	4	yes	—
24	C₅×C₉	C₃²×C₇₁	1	9	yes	direct product factor
25	C₃³	C₃×C₅×C₇₁	1	1	yes	characteristic
26	C₃×C₉	C₃×C₅×C₇₁	1	12	yes	direct product factor
27	C₃×C₅	C₃×C₉×C₇₁	1	12	yes	direct product factor
28	C₃×C₅	C₃³×C₇₁	1	1	yes	characteristic
29	C₃²	C₅×C₉×C₇₁	1	9	yes	direct product factor
30	C₃²	C₃²×C₅×C₇₁	1	4	yes	—
31	C₉	C₃²×C₅×C₇₁	1	9	yes	direct product factor
32	C₅	C₃²×C₉×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	C₃×C₅×C₉×C₇₁	1	12	yes	direct product factor
34	C₃	C₃³×C₅×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃	C₃³×C₅×C₇₁
socle, prime radical	C₃³×C₅×C₇₁	C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₉×C₇₁	C₃	direct (N × H)
3. C₃²×C₉×C₇₁	C₅	direct (N × H)
4. C₃²×C₅×C₇₁	C₉	direct (N × H)
6. C₅×C₉×C₇₁	C₃²	direct (N × H)
8. C₃×C₉×C₇₁	C₃×C₅	direct (N × H)
9. C₃×C₅×C₇₁	C₃×C₉	direct (N × H)
11. C₃²×C₇₁	C₅×C₉	direct (N × H)
13. C₉×C₇₁	C₃²×C₅	direct (N × H)
14. C₃²×C₅×C₉	C₇₁	direct (N × H)
15. C₅×C₇₁	C₃²×C₉	direct (N × H)
16. C₃×C₇₁	C₃×C₅×C₉	direct (N × H)
19. C₃×C₅×C₉	C₃×C₇₁	direct (N × H)
20. C₃²×C₉	C₅×C₇₁	direct (N × H)
21. C₇₁	C₃²×C₅×C₉	direct (N × H)
22. C₃²×C₅	C₉×C₇₁	direct (N × H)
24. C₅×C₉	C₃²×C₇₁	direct (N × H)
26. C₃×C₉	C₃×C₅×C₇₁	direct (N × H)
27. C₃×C₅	C₃×C₉×C₇₁	direct (N × H)
29. C₃²	C₅×C₉×C₇₁	direct (N × H)
31. C₉	C₃²×C₅×C₇₁	direct (N × H)
32. C₅	C₃²×C₉×C₇₁	direct (N × H)
33. C₃	C₃×C₅×C₉×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₅×C₇₁	3
C₃×C₅×C₉×C₇₁	3
C₃²×C₉×C₇₁	5
C₃²×C₅×C₉	71

Subgroups — C32×C5×C9×C71 #72205708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₉×C₇₁ #72205708