Subgroups — #14552000

Order 2187.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6880	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6879	C₃	1	2	yes	maximal, complemented
3	C₉×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624540	C₃	1	9	yes	maximal, complemented
5	C₃×#733	—	3	1	no	—
6	C₃×#732	—	3	2	no	—
7	C₃×C₂₇⋊C₃	—	3	1	no	—
8	C₃×C₉²	C₃²	1	1	yes	characteristic
9	#6880	—	3	1	no	—
10	#6880	—	3	1	no	—
11	#6880	C₃²	1	3	yes	complemented
12	#6879	—	3	2	no	—
13	#6879	—	3	2	no	—
14	#6879	C₃²	1	6	yes	complemented
15	#6864	—	3	1	no	—
16	#6864	C₃²	1	3	yes	complemented
17	C₉×C₂₇	—	3	1	no	—
18	C₃×3₋¹⁺²	—	9	1	no	—
19	C₃×H(3,𝔽₃)	—	9	2	no	—
20	C₃²×C₉	—	3	1	no	—
21	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
22	#733	—	3	3	no	—
23	#733	—	9	2	no	—
24	#732	—	3	6	no	—
25	#732	—	9	4	no	—
26	C₂₇⋊C₃	—	3	3	no	—
27	C₂₇⋊C₃	—	3	6	no	—
28	C₃×C₂₇	—	3	3	no	—
29	C₉²	—	3	2	no	—
30	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
31	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
32	C₉²	C₃³	1	1	yes	characteristic
33	C₃³	—	27	2	no	—
34	C₃³	#732	1	1	yes	characteristic
35	3₋¹⁺²	—	9	3	no	—
36	3₋¹⁺²	—	27	2	no	—
37	H(3,𝔽₃)	—	9	6	no	—
38	H(3,𝔽₃)	—	27	2	no	—
39	H(3,𝔽₃)	—	27	2	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	9	3	no	—
46	C₃×C₉	—	27	1	no	—
47	C₃×C₉	#732	1	2	yes	—
48	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
49	C₂₇	—	3	9	no	—
50	C₃²	—	9	1	no	—
51	C₃²	—	27	6	no	—
52	C₃²	—	81	6	no	—
53	C₃²	#6879	1	1	yes	characteristic
54	C₃²	#6879	1	2	yes	—
55	C₃²	C₃×#732	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	3	1	no	—
58	C₉	—	3	2	no	—
59	C₉	—	9	1	no	—
60	C₉	—	9	1	no	—
61	C₉	—	27	3	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	2	no	—
64	C₃	—	3	1	no	—
65	C₃	—	81	6	no	—
66	C₃	C₃×#6879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	C₃×#6879
commutator, Frattini, cosocle	C₉²	C₃³
LCS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×#6880	C₃	semidirect
2. C₃×#6879	C₃	semidirect
3. C₉×C₂₇⋊C₃	C₃	semidirect
4. #10624540	C₃	semidirect
11. #6880	C₃²	semidirect
14. #6879	C₃²	semidirect
16. #6864	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6880	3
C₃×#6879	3
C₉×C₂₇⋊C₃	3
#10624540	3