Subgroups — C₃×3₋¹⁺⁶ #72258024

Order 6561.

2059386 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺⁶	C₃	1	729	yes	maximal, direct product factor
2	C₃×#10624943	C₃	1	243	yes	maximal, complemented
3	C₃²×3₋¹⁺⁴	C₃	1	120	yes	maximal, complemented
4	C₃²×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10624943	C₃²	1	59049	yes	complemented
6	C₃×3₋¹⁺⁴	C₃²	1	810	yes	complemented
7	C₃×3₋¹⁺⁴	C₃²	1	6480	yes	complemented
8	C₃×3₋¹⁺⁴	C₃²	1	29160	yes	complemented
9	C₃×H(5,𝔽₃)	C₃²	1	81	yes	—
10	C₃×H(5,𝔽₃)	C₃²	1	243	yes	—
11	C₃²×#737	C₃²	1	3240	yes	complemented
12	C₃³×3₋¹⁺²	C₃²	1	360	yes	complemented
13	C₃³×H(3,𝔽₃)	C₃²	1	40	yes	—
14	C₃⁵	C₃³	1	40	yes	—
15	3₋¹⁺⁴	C₃³	1	65610	yes	complemented
16	3₋¹⁺⁴	C₃³	1	524880	yes	complemented
17	H(5,𝔽₃)	C₃³	1	6561	yes	—
18	C₃×#737	C₃³	1	21870	yes	complemented
19	C₃×#737	C₃³	1	262440	yes	complemented
20	C₃²×3₋¹⁺²	C₃³	1	1080	yes	—
21	C₃²×3₋¹⁺²	C₃³	1	8640	yes	complemented
22	C₃²×3₋¹⁺²	C₃³	1	29160	yes	complemented
23	C₃²×H(3,𝔽₃)	C₃³	1	90	yes	—
24	C₃²×H(3,𝔽₃)	C₃³	1	1080	yes	—
25	C₃²×H(3,𝔽₃)	C₃³	1	3240	yes	—
26	C₃³×C₉	C₃³	1	1080	yes	complemented
27	C₃⁴	—	27	120	no	—
28	C₃⁴	C₃⁴	1	360	yes	—
29	C₃⁴	C₃⁴	1	40	yes	—
30	C₃⁴	C₃⁴	1	1080	yes	—
31	#737	C₃⁴	1	590490	yes	complemented
32	C₃×3₋¹⁺²	C₃⁴	1	81	yes	—
33	C₃×3₋¹⁺²	C₃⁴	1	6480	yes	—
34	C₃×3₋¹⁺²	C₃⁴	1	29160	yes	—
35	C₃×3₋¹⁺²	C₃⁴	1	233280	yes	complemented
36	C₃×H(3,𝔽₃)	C₃⁴	1	810	yes	—
37	C₃×H(3,𝔽₃)	C₃⁴	1	2430	yes	—
38	C₃×H(3,𝔽₃)	C₃⁴	1	29160	yes	—
39	C₃²×C₉	C₃⁴	1	3240	yes	—
40	C₃²×C₉	C₃⁴	1	29160	yes	complemented
41	C₃³	—	9	120	no	—
42	C₃³	—	9	1080	no	—
43	C₃³	—	27	1080	no	—
44	C₃³	C₃⁵	1	120	yes	—
45	C₃³	C₃⁵	1	1	yes	characteristic
46	C₃³	C₃⁵	1	3240	yes	—
47	C₃³	C₃⁵	1	360	yes	—
48	3₋¹⁺²	C₃⁵	1	729	yes	—
49	3₋¹⁺²	C₃⁵	1	58320	yes	—
50	H(3,𝔽₃)	C₃⁵	1	7290	yes	—
51	C₃×C₉	C₃⁵	1	29160	yes	—
52	C₃×C₉	C₃⁵	1	243	yes	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	360	no	—
55	C₃²	—	9	360	no	—
56	C₃²	—	9	3240	no	—
57	C₃²	C₃⁶	1	360	yes	—
58	C₃²	C₃⁶	1	3	yes	—
59	C₃²	C₃⁶	1	1	yes	characteristic
60	C₉	C₃⁶	1	729	yes	—
61	C₃	—	3	3	no	—
62	C₃	—	3	360	no	—
63	C₃	3₋¹⁺⁶	1	3	yes	direct product factor
64	C₃	C₃⁷	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃⁶
commutator, Frattini, cosocle	C₃	C₃⁷
prime radical	C₃²×H(5,𝔽₃)	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺⁶	C₃	direct (N × H)
2. C₃×#10624943	C₃	semidirect
3. C₃²×3₋¹⁺⁴	C₃	semidirect
5. #10624943	C₃²	semidirect
6. C₃×3₋¹⁺⁴	C₃²	semidirect
7. C₃×3₋¹⁺⁴	C₃²	semidirect
8. C₃×3₋¹⁺⁴	C₃²	semidirect
11. C₃²×#737	C₃²	semidirect
12. C₃³×3₋¹⁺²	C₃²	semidirect
15. 3₋¹⁺⁴	C₃³	semidirect
16. 3₋¹⁺⁴	C₃³	semidirect
18. C₃×#737	C₃³	semidirect
19. C₃×#737	C₃³	semidirect
21. C₃²×3₋¹⁺²	C₃³	semidirect
22. C₃²×3₋¹⁺²	C₃³	semidirect
26. C₃³×C₉	C₃³	semidirect
31. #737	C₃⁴	semidirect
35. C₃×3₋¹⁺²	C₃⁴	semidirect
40. C₃²×C₉	C₃⁴	semidirect
63. C₃	3₋¹⁺⁶	direct (N × H)

Maximal subgroups

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

subgroup	index
3₋¹⁺⁶	3
C₃×#10624943	3
C₃²×3₋¹⁺⁴	3
C₃²×H(5,𝔽₃)	3

Subgroups — C3×3−1+6 #72258024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×3₋¹⁺⁶ #72258024