Subgroups — #14554237

Order 2187.

540 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624872	C₃	1	1	yes	maximal, characteristic
2	#10624729	C₃	1	27	yes	maximal, complemented
3	#10624713	C₃	1	9	yes	maximal, complemented
4	#10624683	C₃	1	3	yes	maximal
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×C₂₇⋊C₃	C₃²	1	9	yes	—
7	#6895	C₉	1	6	yes	—
8	#6894	C₉	1	18	yes	—
9	#6891	C₉	1	3	yes	—
10	C₉×3₋¹⁺²	C₃²	1	9	yes	—
11	#6887	C₃²	1	3	yes	—
12	C₉⋊C₂₇	C₃²	1	54	yes	complemented
13	C₃²⋊C₂₇	C₃²	1	9	yes	—
14	C₃²⋊C₂₇	C₃²	1	18	yes	—
15	#6864	C₃²	1	18	yes	complemented
16	C₉×C₂₇	C₃²	1	9	yes	complemented
17	C₃⁴	C₃×C₉	1	1	yes	characteristic
18	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
19	C₃²×C₉	—	9	3	no	—
20	C₃²×C₉	C₃×C₉	1	6	yes	—
21	C₃²×C₉	C₃×C₉	1	2	yes	—
22	C₃²×C₉	C₃³	1	1	yes	characteristic
23	C₃²×C₉	C₃³	1	3	yes	—
24	C₂₇⋊C₃	—	9	9	no	—
25	C₃×C₂₇	C₃³	1	27	yes	—
26	C₉⋊C₉	C₃×C₉	1	18	yes	—
27	C₉⋊C₉	C₃×C₉	1	54	yes	—
28	C₃²⋊C₉	C₃×C₉	1	9	yes	—
29	C₃²⋊C₉	C₃×C₉	1	18	yes	—
30	C₉²	C₃³	1	9	yes	—
31	C₃³	—	9	1	no	—
32	C₃³	—	9	3	no	—
33	C₃³	C₃²×C₉	1	1	yes	characteristic
34	C₃³	C₃²×C₉	1	3	yes	—
35	3₋¹⁺²	—	9	9	no	—
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	3	3	no	—
38	C₃×C₉	—	3	6	no	—
39	C₃×C₉	—	9	6	no	—
40	C₃×C₉	—	9	2	no	—
41	C₃×C₉	C₃²×C₉	1	27	yes	—
42	C₃×C₉	C₃²×C₉	1	2	yes	—
43	C₃×C₉	C₃²×C₉	1	6	yes	—
44	C₃×C₉	C₃⁴	1	1	yes	characteristic
45	C₂₇	—	3	27	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	3	3	no	—
50	C₃²	—	3	6	no	—
51	C₃²	—	9	9	no	—
52	C₃²	C₃³×C₉	1	1	yes	characteristic
53	C₉	—	3	6	no	—
54	C₉	—	3	2	no	—
55	C₉	—	3	27	no	—
56	C₉	3₋¹⁺⁴	1	3	yes	—
57	C₃	—	9	3	no	—
58	C₃	—	9	1	no	—
59	C₃	#10624863	1	1	yes	characteristic
60	C₃	#10624918	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #10624729	C₃	semidirect
3. #10624713	C₃	semidirect
12. C₉⋊C₂₇	C₃²	semidirect
15. #6864	C₃²	semidirect
16. C₉×C₂₇	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624872	3
#10624729	3
#10624713	3
#10624683	3