Subgroups — C₃²×3₋¹⁺²×C₁₀₁ #37589450

Order 24543.

576 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₀₁	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₁₀₁	C₃	1	36	yes	maximal, direct product factor
3	C₃²×C₉×C₁₀₁	C₃	1	3	yes	maximal, complemented
4	C₃³×C₁₀₁	—	3	9	no	—
5	C₃³×C₁₀₁	C₃²	1	12	yes	—
6	C₃³×C₁₀₁	C₃²	1	1	yes	characteristic
7	3₋¹⁺²×C₁₀₁	C₃²	1	81	yes	direct product factor
8	C₃×C₉×C₁₀₁	C₃²	1	36	yes	complemented
9	C₃²×C₁₀₁	—	3	36	no	—
10	C₃²×C₁₀₁	3₋¹⁺²	1	9	yes	direct product factor
11	C₃²×C₁₀₁	C₃³	1	9	yes	—
12	C₃²×C₁₀₁	C₃³	1	4	yes	—
13	C₉×C₁₀₁	C₃³	1	27	yes	complemented
14	C₃×C₁₀₁	—	3	9	no	—
15	C₃×C₁₀₁	C₃×3₋¹⁺²	1	12	yes	direct product factor
16	C₃×C₁₀₁	C₃⁴	1	1	yes	characteristic
17	C₃²×3₋¹⁺²	C₁₀₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₁₀₁	C₃²×3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
19	C₃⁴	C₃×C₁₀₁	1	1	yes	characteristic
20	C₃×3₋¹⁺²	C₃×C₁₀₁	1	36	yes	direct product factor
21	C₃²×C₉	C₃×C₁₀₁	1	3	yes	complemented
22	C₃³	—	3	9	no	—
23	C₃³	C₃²×C₁₀₁	1	12	yes	—
24	C₃³	C₃²×C₁₀₁	1	1	yes	characteristic
25	3₋¹⁺²	C₃²×C₁₀₁	1	81	yes	direct product factor
26	C₃×C₉	C₃²×C₁₀₁	1	36	yes	complemented
27	C₃²	—	3	36	no	—
28	C₃²	3₋¹⁺²×C₁₀₁	1	9	yes	direct product factor
29	C₃²	C₃³×C₁₀₁	1	9	yes	—
30	C₃²	C₃³×C₁₀₁	1	4	yes	—
31	C₉	C₃³×C₁₀₁	1	27	yes	complemented
32	C₃	—	3	9	no	—
33	C₃	C₃×3₋¹⁺²×C₁₀₁	1	12	yes	direct product factor
34	C₃	C₃⁴×C₁₀₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃³×C₁₀₁	C₃²
commutator, Frattini, cosocle	C₃	C₃⁴×C₁₀₁
prime radical	C₃⁴×C₁₀₁	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₁₀₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×3₋¹⁺²×C₁₀₁	C₃	direct (N × H)
3. C₃²×C₉×C₁₀₁	C₃	semidirect
7. 3₋¹⁺²×C₁₀₁	C₃²	direct (N × H)
8. C₃×C₉×C₁₀₁	C₃²	semidirect
10. C₃²×C₁₀₁	3₋¹⁺²	direct (N × H)
13. C₉×C₁₀₁	C₃³	semidirect
15. C₃×C₁₀₁	C₃×3₋¹⁺²	direct (N × H)
17. C₃²×3₋¹⁺²	C₁₀₁	direct (N × H)
18. C₁₀₁	C₃²×3₋¹⁺²	direct (N × H)
20. C₃×3₋¹⁺²	C₃×C₁₀₁	direct (N × H)
21. C₃²×C₉	C₃×C₁₀₁	semidirect
25. 3₋¹⁺²	C₃²×C₁₀₁	direct (N × H)
26. C₃×C₉	C₃²×C₁₀₁	semidirect
28. C₃²	3₋¹⁺²×C₁₀₁	direct (N × H)
31. C₉	C₃³×C₁₀₁	semidirect
33. C₃	C₃×3₋¹⁺²×C₁₀₁	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×C₁₀₁	3
C₃×3₋¹⁺²×C₁₀₁	3
C₃²×C₉×C₁₀₁	3
C₃²×3₋¹⁺²	101

Subgroups — C32×3−1+2×C101 #37589450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×3₋¹⁺²×C₁₀₁ #37589450