Subgroups — #14550788

Order 2187.

249 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624885	C₃	1	1	yes	maximal, characteristic
2	#10624692	C₃	1	3	yes	maximal
3	#10624470	C₃	1	9	yes	maximal, complemented
4	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
5	#6896	—	3	9	no	—
6	#6895	—	3	1	no	—
7	#6895	—	3	1	no	—
8	#6891	—	3	1	no	—
9	#6887	C₃²	1	9	yes	—
10	#6887	C₃²	1	3	yes	—
11	#6868	—	3	3	no	—
12	#6867	—	3	3	no	—
13	#6866	—	3	3	no	—
14	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃×3₋¹⁺²	—	3	3	no	—
16	C₃×3₋¹⁺²	—	3	3	no	—
17	C₃×3₋¹⁺²	—	3	9	no	—
18	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₉	C₃³	1	1	yes	characteristic
21	C₉⋊C₉	—	3	9	no	—
22	C₉⋊C₉	—	3	9	no	—
23	C₉⋊C₉	—	3	3	no	—
24	C₉⋊C₉	—	3	18	no	—
25	C₉⋊C₉	3₋¹⁺²	1	9	yes	—
26	C₃²⋊C₉	—	3	6	no	—
27	C₃²⋊C₉	—	3	3	no	—
28	C₃²⋊C₉	—	3	9	no	—
29	C₉²	—	3	3	no	—
30	C₉²	—	3	9	no	—
31	C₃³	—	3	1	no	—
32	C₃³	—	3	3	no	—
33	C₃³	—	27	1	no	—
34	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
35	3₋¹⁺²	—	9	9	no	—
36	3₋¹⁺²	—	27	3	no	—
37	C₃×C₉	—	3	9	no	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	—	9	9	no	—
43	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
44	C₃×C₉	#737	1	9	yes	—
45	C₃²	—	9	1	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	9	2	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	27	3	no	—
50	C₃²	#6891	1	1	yes	characteristic
51	C₃²	#6893	1	3	yes	—
52	C₉	—	9	9	no	—
53	C₉	—	9	1	no	—
54	C₉	—	9	1	no	—
55	C₉	—	27	9	no	—
56	C₉	#6899	1	9	yes	—
57	C₃	—	3	1	no	—
58	C₃	—	3	3	no	—
59	C₃	—	27	1	no	—
60	C₃	#10624570	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624570
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃⁴	H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624470	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624885	3
#10624692	3
#10624470	3