Subgroups — 3₋¹⁺⁸ #72290073

Order 19683.

127972082 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72290071	C₃	1	2187	yes	maximal, complemented
2	C₃×H(7,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺⁶	C₃	1	1092	yes	maximal, complemented
4	3₋¹⁺⁶	C₃²	1	530712	yes	complemented
5	3₋¹⁺⁶	C₃²	1	66339	yes	complemented
6	H(7,𝔽₃)	C₃²	1	729	yes	—
7	C₃×#10624943	C₃²	1	265356	yes	complemented
8	C₃²×3₋¹⁺⁴	C₃²	1	32760	yes	complemented
9	C₃²×H(5,𝔽₃)	C₃²	1	364	yes	—
10	#10624943	C₃³	1	16120377	yes	complemented
11	C₃×3₋¹⁺⁴	C₃³	1	884520	yes	complemented
12	C₃×3₋¹⁺⁴	C₃³	1	7076160	yes	complemented
13	C₃×H(5,𝔽₃)	C₃³	1	7371	yes	—
14	C₃×H(5,𝔽₃)	C₃³	1	88452	yes	—
15	C₃²×#737	C₃³	1	884520	yes	complemented
16	C₃³×3₋¹⁺²	C₃³	1	30240	yes	complemented
17	C₃³×H(3,𝔽₃)	C₃³	1	3640	yes	—
18	C₃⁵	C₃⁴	1	1120	yes	—
19	3₋¹⁺⁴	C₃⁴	1	597051	yes	—
20	3₋¹⁺⁴	C₃⁴	1	47764080	yes	complemented
21	H(5,𝔽₃)	C₃⁴	1	597051	yes	—
22	C₃×#737	C₃⁴	1	23882040	yes	complemented
23	C₃²×3₋¹⁺²	C₃⁴	1	294840	yes	—
24	C₃²×3₋¹⁺²	C₃⁴	1	2358720	yes	complemented
25	C₃²×H(3,𝔽₃)	C₃⁴	1	32760	yes	—
26	C₃²×H(3,𝔽₃)	C₃⁴	1	294840	yes	—
27	C₃³×C₉	C₃⁴	1	90720	yes	complemented
28	C₃⁴	—	81	1120	no	—
29	C₃⁴	C₃⁵	1	30240	yes	—
30	C₃⁴	C₃⁵	1	3640	yes	—
31	#737	C₃⁵	1	16120377	yes	—
32	C₃×3₋¹⁺²	C₃⁵	1	88452	yes	—
33	C₃×3₋¹⁺²	C₃⁵	1	7076160	yes	—
34	C₃×H(3,𝔽₃)	C₃⁵	1	7371	yes	—
35	C₃×H(3,𝔽₃)	C₃⁵	1	884520	yes	—
36	C₃²×C₉	C₃⁵	1	884520	yes	—
37	C₃³	—	27	3640	no	—
38	C₃³	—	27	30240	no	—
39	C₃³	C₃⁶	1	32760	yes	—
40	C₃³	C₃⁶	1	364	yes	—
41	3₋¹⁺²	C₃⁶	1	729	yes	—
42	3₋¹⁺²	C₃⁶	1	530712	yes	—
43	H(3,𝔽₃)	C₃⁶	1	66339	yes	—
44	C₃×C₉	C₃⁶	1	265356	yes	—
45	C₃²	—	9	364	no	—
46	C₃²	—	9	32760	no	—
47	C₃²	C₃⁷	1	1092	yes	—
48	C₃²	C₃⁷	1	1	yes	characteristic
49	C₉	C₃⁷	1	2187	yes	—
50	C₃	—	3	1	no	—
51	C₃	—	3	1092	no	—
52	C₃	C₃⁸	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃	C₃⁸
prime radical	C₃×H(7,𝔽₃)	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ G
LECS	G ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72290071	C₃	semidirect
3. C₃×3₋¹⁺⁶	C₃	semidirect
4. 3₋¹⁺⁶	C₃²	semidirect
5. 3₋¹⁺⁶	C₃²	semidirect
7. C₃×#10624943	C₃²	semidirect
8. C₃²×3₋¹⁺⁴	C₃²	semidirect
10. #10624943	C₃³	semidirect
11. C₃×3₋¹⁺⁴	C₃³	semidirect
12. C₃×3₋¹⁺⁴	C₃³	semidirect
15. C₃²×#737	C₃³	semidirect
16. C₃³×3₋¹⁺²	C₃³	semidirect
20. 3₋¹⁺⁴	C₃⁴	semidirect
22. C₃×#737	C₃⁴	semidirect
24. C₃²×3₋¹⁺²	C₃⁴	semidirect
27. C₃³×C₉	C₃⁴	semidirect

Maximal subgroups

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

subgroup	index
#72290071	3
C₃×H(7,𝔽₃)	3
C₃×3₋¹⁺⁶	3

Subgroups — 3−1+8 #72290073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺⁸ #72290073