Subgroups — C₃²×3₋¹⁺²×C₁₃₁ #44541439

Order 31833.

576 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₃₁	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₁₃₁	C₃	1	36	yes	maximal, direct product factor
3	C₃²×C₉×C₁₃₁	C₃	1	3	yes	maximal, complemented
4	C₃³×C₁₃₁	—	3	9	no	—
5	C₃³×C₁₃₁	C₃²	1	12	yes	—
6	C₃³×C₁₃₁	C₃²	1	1	yes	characteristic
7	3₋¹⁺²×C₁₃₁	C₃²	1	81	yes	direct product factor
8	C₃×C₉×C₁₃₁	C₃²	1	36	yes	complemented
9	C₃²×C₁₃₁	—	3	36	no	—
10	C₃²×C₁₃₁	3₋¹⁺²	1	9	yes	direct product factor
11	C₃²×C₁₃₁	C₃³	1	9	yes	—
12	C₃²×C₁₃₁	C₃³	1	4	yes	—
13	C₉×C₁₃₁	C₃³	1	27	yes	complemented
14	C₃×C₁₃₁	—	3	9	no	—
15	C₃×C₁₃₁	C₃×3₋¹⁺²	1	12	yes	direct product factor
16	C₃×C₁₃₁	C₃⁴	1	1	yes	characteristic
17	C₃²×3₋¹⁺²	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₁₃₁	C₃²×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
19	C₃⁴	C₃×C₁₃₁	1	1	yes	characteristic
20	C₃×3₋¹⁺²	C₃×C₁₃₁	1	36	yes	direct product factor
21	C₃²×C₉	C₃×C₁₃₁	1	3	yes	complemented
22	C₃³	—	3	9	no	—
23	C₃³	C₃²×C₁₃₁	1	12	yes	—
24	C₃³	C₃²×C₁₃₁	1	1	yes	characteristic
25	3₋¹⁺²	C₃²×C₁₃₁	1	81	yes	direct product factor
26	C₃×C₉	C₃²×C₁₃₁	1	36	yes	complemented
27	C₃²	—	3	36	no	—
28	C₃²	3₋¹⁺²×C₁₃₁	1	9	yes	direct product factor
29	C₃²	C₃³×C₁₃₁	1	9	yes	—
30	C₃²	C₃³×C₁₃₁	1	4	yes	—
31	C₉	C₃³×C₁₃₁	1	27	yes	complemented
32	C₃	—	3	9	no	—
33	C₃	C₃×3₋¹⁺²×C₁₃₁	1	12	yes	direct product factor
34	C₃	C₃⁴×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃³×C₁₃₁	C₃²
commutator, Frattini, cosocle	C₃	C₃⁴×C₁₃₁
prime radical	C₃⁴×C₁₃₁	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×3₋¹⁺²×C₁₃₁	C₃	direct (N × H)
3. C₃²×C₉×C₁₃₁	C₃	semidirect
7. 3₋¹⁺²×C₁₃₁	C₃²	direct (N × H)
8. C₃×C₉×C₁₃₁	C₃²	semidirect
10. C₃²×C₁₃₁	3₋¹⁺²	direct (N × H)
13. C₉×C₁₃₁	C₃³	semidirect
15. C₃×C₁₃₁	C₃×3₋¹⁺²	direct (N × H)
17. C₃²×3₋¹⁺²	C₁₃₁	direct (N × H)
18. C₁₃₁	C₃²×3₋¹⁺²	direct (N × H)
20. C₃×3₋¹⁺²	C₃×C₁₃₁	direct (N × H)
21. C₃²×C₉	C₃×C₁₃₁	semidirect
25. 3₋¹⁺²	C₃²×C₁₃₁	direct (N × H)
26. C₃×C₉	C₃²×C₁₃₁	semidirect
28. C₃²	3₋¹⁺²×C₁₃₁	direct (N × H)
31. C₉	C₃³×C₁₃₁	semidirect
33. C₃	C₃×3₋¹⁺²×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₁₃₁	3
C₃×3₋¹⁺²×C₁₃₁	3
C₃²×C₉×C₁₃₁	3
C₃²×3₋¹⁺²	131

Subgroups — C32×3−1+2×C131 #44541439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²×C₁₃₁ #44541439