Subgroups — C₃²×#6893 #14555255

Order 2187.

7892 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
2	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#6893	C₃	1	108	yes	maximal, direct product factor
5	C₃²×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
6	C₃⁵	C₃²	1	3	yes	—
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×3₋¹⁺²	—	3	54	no	—
9	C₃²×3₋¹⁺²	C₃²	1	72	yes	complemented
10	C₃²×H(3,𝔽₃)	—	3	27	no	—
11	C₃²×H(3,𝔽₃)	C₃²	1	36	yes	—
12	C₃³×C₉	—	3	27	no	—
13	C₃³×C₉	C₃²	1	3	yes	—
14	C₃³×C₉	C₃²	1	6	yes	—
15	C₃³×C₉	C₃²	1	36	yes	complemented
16	#6893	C₃²	1	729	yes	direct product factor
17	C₃×C₃²⋊C₉	C₃²	1	324	yes	complemented
18	C₃⁴	—	3	27	no	—
19	C₃⁴	—	3	27	no	—
20	C₃⁴	—	3	81	no	—
21	C₃⁴	H(3,𝔽₃)	1	27	yes	—
22	C₃⁴	C₃³	1	36	yes	—
23	C₃⁴	C₃³	1	12	yes	—
24	C₃⁴	C₃³	1	1	yes	characteristic
25	C₃×3₋¹⁺²	—	3	648	no	—
26	C₃×3₋¹⁺²	C₃³	1	162	yes	complemented
27	C₃×H(3,𝔽₃)	—	3	324	no	—
28	C₃×H(3,𝔽₃)	C₃³	1	81	yes	—
29	C₃²×C₉	—	3	27	no	—
30	C₃²×C₉	—	3	324	no	—
31	C₃²×C₉	—	3	54	no	—
32	C₃²×C₉	C₃³	1	36	yes	—
33	C₃²×C₉	C₃³	1	72	yes	—
34	C₃²×C₉	C₃³	1	81	yes	complemented
35	C₃²⋊C₉	C₃³	1	729	yes	complemented
36	C₃³	—	3	108	no	—
37	C₃³	—	3	243	no	—
38	C₃³	—	3	108	no	—
39	C₃³	—	3	324	no	—
40	C₃³	—	9	243	no	—
41	C₃³	C₃×H(3,𝔽₃)	1	108	yes	—
42	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
43	C₃³	#737	1	27	yes	—
44	C₃³	C₃⁴	1	27	yes	—
45	C₃³	C₃⁴	1	9	yes	—
46	C₃³	C₃⁴	1	4	yes	—
47	3₋¹⁺²	—	3	486	no	—
48	H(3,𝔽₃)	—	3	243	no	—
49	C₃×C₉	—	3	108	no	—
50	C₃×C₉	—	3	243	no	—
51	C₃×C₉	—	3	216	no	—
52	C₃×C₉	C₃⁴	1	27	yes	—
53	C₃×C₉	C₃⁴	1	54	yes	—
54	C₃²	—	3	324	no	—
55	C₃²	—	3	27	no	—
56	C₃²	—	3	27	no	—
57	C₃²	—	3	81	no	—
58	C₃²	—	9	324	no	—
59	C₃²	#6893	1	81	yes	direct product factor
60	C₃²	C₃²×H(3,𝔽₃)	1	27	yes	—
61	C₃²	C₃²×H(3,𝔽₃)	1	12	yes	—
62	C₃²	C₃×#737	1	36	yes	—
63	C₃²	C₃⁵	1	1	yes	characteristic
64	C₉	—	3	27	no	—
65	C₉	—	3	54	no	—
66	C₃	—	3	27	no	—
67	C₃	—	9	27	no	—
68	C₃	C₃×#6893	1	36	yes	direct product factor
69	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃	C₃²×#737	1	3	yes	—

	subgroup	G/H
center, socle	C₃⁴	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁵
prime radical	C₃³×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×3₋¹⁺²	C₃	semidirect
3. C₃⁴×C₉	C₃	semidirect
4. C₃×#6893	C₃	direct (N × H)
5. C₃²×C₃²⋊C₉	C₃	semidirect
9. C₃²×3₋¹⁺²	C₃²	semidirect
15. C₃³×C₉	C₃²	semidirect
16. #6893	C₃²	direct (N × H)
17. C₃×C₃²⋊C₉	C₃²	semidirect
26. C₃×3₋¹⁺²	C₃³	semidirect
34. C₃²×C₉	C₃³	semidirect
35. C₃²⋊C₉	C₃³	semidirect
59. C₃²	#6893	direct (N × H)
68. C₃	C₃×#6893	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃³×H(3,𝔽₃)	3
C₃⁴×C₉	3
C₃×#6893	3
C₃²×C₃²⋊C₉	3

Subgroups — C32×#6893 #14555255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6893 #14555255