Subgroups — #14552016

Order 2187.

1421 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	3	yes	maximal
2	C₃²×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₉×C₉⋊C₉	C₃	1	27	yes	maximal, complemented
4	C₉×C₃²⋊C₉	C₃	1	9	yes	maximal
5	C₃³×C₉	C₃²	1	3	yes	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	#6887	C₉	1	81	yes	complemented
8	C₃×C₉⋊C₉	C₉	1	27	yes	complemented
9	C₃×C₉⋊C₉	C₃²	1	9	yes	—
10	C₃×C₉⋊C₉	C₃²	1	18	yes	—
11	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃×C₉²	C₃²	1	27	yes	—
13	C₃×C₉²	C₃²	1	9	yes	—
14	#6855	C₃²	1	54	yes	—
15	C₃⁴	C₃³	1	1	yes	characteristic
16	C₃²×C₉	—	9	9	no	—
17	C₃²×C₉	C₃×C₉	1	27	yes	—
18	C₃²×C₉	C₃×C₉	1	9	yes	—
19	C₃²×C₉	C₃×C₉	1	27	yes	—
20	C₃²×C₉	C₃³	1	9	yes	—
21	C₃²×C₉	C₃³	1	27	yes	—
22	C₃²×C₉	C₃³	1	3	yes	—
23	C₉⋊C₉	C₃×C₉	1	81	yes	—
24	C₉⋊C₉	C₃×C₉	1	162	yes	complemented
25	C₃²⋊C₉	C₃×C₉	1	81	yes	—
26	C₉²	—	9	27	no	—
27	C₉²	C₃×C₉	1	81	yes	complemented
28	C₃³	—	3	9	no	—
29	C₃³	C₃²×C₉	1	9	yes	—
30	C₃³	C₃²×C₉	1	3	yes	—
31	C₃³	C₃⁴	1	1	yes	characteristic
32	C₃×C₉	—	3	9	no	—
33	C₃×C₉	—	3	81	no	—
34	C₃×C₉	—	3	18	no	—
35	C₃×C₉	—	9	27	no	—
36	C₃×C₉	C₉²	1	81	yes	complemented
37	C₃×C₉	C₃²×C₉	1	81	yes	—
38	C₃×C₉	C₃²×C₉	1	27	yes	—
39	C₃×C₉	C₃²×C₉	1	9	yes	—
40	C₃×C₉	C₃²×C₉	1	9	yes	—
41	C₃×C₉	C₃²×C₉	1	18	yes	—
42	C₃²	—	3	9	no	—
43	C₃²	—	3	27	no	—
44	C₃²	C₃×C₉²	1	9	yes	—
45	C₃²	C₃³×C₉	1	1	yes	characteristic
46	C₃²	C₃³×C₉	1	3	yes	—
47	C₃²	3₋¹⁺⁴	1	9	yes	—
48	C₉	—	3	27	no	—
49	C₉	—	3	81	no	—
50	C₉	C₃×C₉²	1	27	yes	—
51	C₃	—	3	9	no	—
52	C₃	C₃²×C₉²	1	1	yes	characteristic
53	C₃	#10624863	1	3	yes	—
54	C₃	#10624864	1	9	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃	C₃²×C₉²
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₉×C₉⋊C₉	C₃	semidirect
7. #6887	C₉	semidirect
8. C₃×C₉⋊C₉	C₉	semidirect
24. C₉⋊C₉	C₃×C₉	semidirect
27. C₉²	C₃×C₉	semidirect
36. C₃×C₉	C₉²	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6887	3
C₃²×C₉⋊C₉	3
C₉×C₉⋊C₉	3
C₉×C₃²⋊C₉	3