Subgroups — #14554132

Order 2187.

1537 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624943	C₃	1	1	yes	maximal, characteristic
2	#10624918	C₃	1	2	yes	maximal, complemented
3	C₃×#6903	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624702	C₃	1	24	yes	maximal, complemented
5	C₃×#6872	C₃	1	12	yes	maximal, complemented
6	3₋¹⁺⁴	—	3	18	no	—
7	3₋¹⁺⁴	C₉	1	24	yes	—
8	3₋¹⁺⁴	C₉	1	2	yes	—
9	H(5,𝔽₃)	C₉	1	1	yes	characteristic
10	C₃×#737	—	3	9	no	—
11	C₃×#737	C₃²	1	12	yes	—
12	C₃×#737	C₃²	1	1	yes	characteristic
13	#6903	—	3	9	no	—
14	#6903	—	3	18	no	—
15	C₃×C₂₇⋊C₃	C₃²	1	24	yes	complemented
16	C₃×C₂₇⋊C₃	C₃²	1	8	yes	complemented
17	C₃²×C₂₇	C₃²	1	4	yes	complemented
18	#6872	C₃²	1	9	yes	complemented
19	#6872	C₃²	1	72	yes	complemented
20	#737	—	3	144	no	—
21	#737	—	3	18	no	—
22	#737	—	3	24	no	—
23	#737	—	3	48	no	—
24	#737	C₃×C₉	1	72	yes	—
25	#737	H(3,𝔽₃)	1	9	yes	—
26	#737	3₋¹⁺²	1	18	yes	—
27	#737	C₃³	1	9	yes	—
28	C₃×3₋¹⁺²	—	3	72	no	—
29	C₃×3₋¹⁺²	—	3	24	no	—
30	C₃×3₋¹⁺²	C₃×C₉	1	24	yes	—
31	C₃×3₋¹⁺²	C₃×C₉	1	8	yes	—
32	C₃×H(3,𝔽₃)	—	3	9	no	—
33	C₃×H(3,𝔽₃)	C₃×C₉	1	12	yes	—
34	C₃×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
35	C₃²×C₉	—	3	12	no	—
36	C₃²×C₉	C₃³	1	4	yes	—
37	C₂₇⋊C₃	—	3	48	no	—
38	C₂₇⋊C₃	—	3	24	no	—
39	C₂₇⋊C₃	C₃³	1	18	yes	—
40	C₃×C₂₇	—	3	12	no	—
41	C₃×C₂₇	—	3	24	no	—
42	C₃×C₂₇	C₃³	1	1	yes	characteristic
43	C₃×C₂₇	C₃³	1	8	yes	—
44	C₃³	—	3	12	no	—
45	C₃³	C₃²×C₉	1	4	yes	—
46	3₋¹⁺²	—	3	48	no	—
47	3₋¹⁺²	—	3	18	no	—
48	3₋¹⁺²	—	3	144	no	—
49	3₋¹⁺²	C₃²⋊C₉	1	72	yes	—
50	3₋¹⁺²	C₃²×C₉	1	18	yes	—
51	H(3,𝔽₃)	—	3	24	no	—
52	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
53	H(3,𝔽₃)	C₃²×C₉	1	9	yes	—
54	C₃×C₉	—	3	24	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	72	no	—
57	C₃×C₉	C₃²×C₉	1	8	yes	—
58	C₃×C₉	C₃×H(3,𝔽₃)	1	12	yes	—
59	C₃×C₉	C₃×3₋¹⁺²	1	24	yes	—
60	C₃×C₉	C₃⁴	1	1	yes	characteristic
61	C₂₇	—	3	1	no	—
62	C₂₇	—	3	18	no	—
63	C₂₇	—	3	8	no	—
64	C₃²	—	3	9	no	—
65	C₃²	—	9	4	no	—
66	C₃²	—	27	12	no	—
67	C₃²	C₃×C₃²⋊C₉	1	12	yes	—
68	C₃²	C₃³×C₉	1	1	yes	characteristic
69	C₉	—	3	18	no	—
70	C₉	C₃×C₃²⋊C₉	1	24	yes	—
71	C₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
72	C₉	C₃²×3₋¹⁺²	1	2	yes	—
73	C₃	—	3	1	no	—
74	C₃	—	3	12	no	—
75	C₃	—	9	9	no	—
76	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²×H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃×C₉	C₃⁴
socle	C₃	C₃²×C₃²⋊C₉
prime radical	H(5,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃×#737 ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624918	C₃	semidirect
3. C₃×#6903	C₃	semidirect
4. #10624702	C₃	semidirect
5. C₃×#6872	C₃	semidirect
15. C₃×C₂₇⋊C₃	C₃²	semidirect
16. C₃×C₂₇⋊C₃	C₃²	semidirect
17. C₃²×C₂₇	C₃²	semidirect
18. #6872	C₃²	semidirect
19. #6872	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624943	3
#10624918	3
C₃×#6903	3
#10624702	3
C₃×#6872	3