Subgroups — #14550605

Order 2187.

1436 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6911	C₃	1	9	yes	maximal, complemented
2	#10624881	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624880	C₃	1	3	yes	maximal, complemented
4	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
5	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
6	C₃²×H(3,𝔽₃)	C₃²	1	9	yes	—
7	#6911	—	3	81	no	—
8	C₃×#732	—	3	27	no	—
9	#6893	—	3	9	no	—
10	#6891	—	3	3	no	—
11	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
12	C₃²⋊H(3,𝔽₃)	—	3	27	no	—
13	C₃⁴	—	3	9	no	—
14	C₃⁴	H(3,𝔽₃)	1	3	yes	—
15	C₃⁴	C₃³	1	1	yes	characteristic
16	C₃×3₋¹⁺²	—	3	3	no	—
17	C₃×3₋¹⁺²	—	3	6	no	—
18	C₃×3₋¹⁺²	—	3	27	no	—
19	C₃×H(3,𝔽₃)	—	3	27	no	—
20	C₃×H(3,𝔽₃)	—	3	54	no	—
21	C₃×H(3,𝔽₃)	—	3	9	no	—
22	C₃×H(3,𝔽₃)	—	3	9	no	—
23	C₃×H(3,𝔽₃)	—	3	9	no	—
24	C₃×H(3,𝔽₃)	—	3	3	no	—
25	C₃×H(3,𝔽₃)	—	3	81	no	—
26	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	27	yes	complemented
27	C₃²×C₉	H(3,𝔽₃)	1	9	yes	complemented
28	#732	—	3	243	no	—
29	C₃²⋊C₉	—	3	27	no	—
30	C₃³	—	3	3	no	—
31	C₃³	—	3	9	no	—
32	C₃³	—	3	27	no	—
33	C₃³	—	9	27	no	—
34	C₃³	—	27	3	no	—
35	C₃³	—	27	1	no	—
36	C₃³	—	27	27	no	—
37	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
38	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
40	3₋¹⁺²	—	9	18	no	—
41	3₋¹⁺²	—	9	9	no	—
42	H(3,𝔽₃)	—	3	81	no	—
43	H(3,𝔽₃)	—	9	9	no	—
44	H(3,𝔽₃)	—	9	27	no	—
45	H(3,𝔽₃)	—	9	18	no	—
46	H(3,𝔽₃)	—	9	9	no	—
47	H(3,𝔽₃)	—	27	81	no	—
48	C₃×C₉	—	3	9	no	—
49	C₃×C₉	—	3	27	no	—
50	C₃²	—	9	1	no	—
51	C₃²	—	9	9	no	—
52	C₃²	—	9	3	no	—
53	C₃²	—	9	27	no	—
54	C₃²	—	9	81	no	—
55	C₃²	—	9	9	no	—
56	C₃²	—	9	27	no	—
57	C₃²	—	9	3	no	—
58	C₃²	—	27	81	no	—
59	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
60	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
61	C₃²	C₃²⋊H(3,𝔽₃)	1	3	yes	—
62	C₉	—	9	9	no	—
63	C₃	—	3	3	no	—
64	C₃	—	3	1	no	—
65	C₃	—	3	9	no	—
66	C₃	—	27	3	no	—
67	C₃	—	27	27	no	—
68	C₃	#10624562	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624562
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×#6911	C₃	semidirect
2. #10624881	C₃	semidirect
3. #10624880	C₃	semidirect
26. C₃×H(3,𝔽₃)	H(3,𝔽₃)	semidirect
27. C₃²×C₉	H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6911	3
#10624881	3
#10624880	3