Subgroups — C₃×#10624761 #14554338

Order 2187.

2432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6910	C₃	1	8	yes	maximal, complemented
3	C₃²×AΣL(1,𝔽₂₇)	C₃	1	4	yes	maximal, complemented
4	#10624761	C₃	1	27	yes	maximal, direct product factor
5	C₃⁵	C₃²	1	1	yes	characteristic, complemented
6	H(5,𝔽₃)	—	3	24	no	—
7	H(5,𝔽₃)	C₃²	1	9	yes	complemented
8	C₃²×3₋¹⁺²	C₃²	1	8	yes	complemented
9	C₃²×H(3,𝔽₃)	—	3	12	no	—
10	C₃²×H(3,𝔽₃)	C₃²	1	4	yes	—
11	#6910	—	3	48	no	—
12	#6910	C₃²	1	72	yes	complemented
13	C₃×#731	—	3	24	no	—
14	C₃×AΣL(1,𝔽₂₇)	—	3	12	no	—
15	C₃×AΣL(1,𝔽₂₇)	—	3	24	no	—
16	C₃×AΣL(1,𝔽₂₇)	C₃²	1	36	yes	complemented
17	C₃⁴	—	3	9	no	—
18	C₃⁴	—	3	4	no	—
19	C₃⁴	—	3	4	no	—
20	C₃⁴	—	3	8	no	—
21	C₃⁴	—	9	9	no	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃⁴	C₃³	1	3	yes	complemented
24	C₃×3₋¹⁺²	—	3	16	no	—
25	C₃×3₋¹⁺²	—	3	16	no	—
26	C₃×3₋¹⁺²	—	3	8	no	—
27	C₃×3₋¹⁺²	—	3	16	no	—
28	C₃×3₋¹⁺²	—	3	32	no	—
29	C₃×3₋¹⁺²	C₃³	1	24	yes	complemented
30	C₃×H(3,𝔽₃)	—	3	18	no	—
31	C₃×H(3,𝔽₃)	—	3	216	no	—
32	C₃×H(3,𝔽₃)	—	3	8	no	—
33	C₃×H(3,𝔽₃)	—	3	16	no	—
34	C₃×H(3,𝔽₃)	—	3	8	no	—
35	C₃×H(3,𝔽₃)	—	3	108	no	—
36	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	12	yes	—
37	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	24	yes	—
38	C₃×H(3,𝔽₃)	C₃³	1	12	yes	—
39	C₃²×C₉	—	3	8	no	—
40	#731	—	3	216	no	—
41	AΣL(1,𝔽₂₇)	—	3	108	no	—
42	C₃³	—	3	81	no	—
43	C₃³	—	3	12	no	—
44	C₃³	—	3	12	no	—
45	C₃³	—	3	24	no	—
46	C₃³	—	3	12	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	9	36	no	—
49	C₃³	—	9	9	no	—
50	C₃³	—	9	72	no	—
51	C₃³	—	9	8	no	—
52	C₃³	—	9	1	no	—
53	C₃³	—	27	12	no	—
54	C₃³	—	27	27	no	—
55	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
56	C₃³	C₃×H(3,𝔽₃)	1	8	yes	—
57	C₃³	C₃⁴	1	1	yes	characteristic
58	3₋¹⁺²	—	3	24	no	—
59	3₋¹⁺²	—	3	48	no	—
60	3₋¹⁺²	—	9	32	no	—
61	3₋¹⁺²	—	9	16	no	—
62	H(3,𝔽₃)	—	3	162	no	—
63	H(3,𝔽₃)	—	3	24	no	—
64	H(3,𝔽₃)	—	3	48	no	—
65	H(3,𝔽₃)	—	3	24	no	—
66	H(3,𝔽₃)	C₃×H(3,𝔽₃)	1	36	yes	—
67	C₃×C₉	—	3	24	no	—
68	C₃×C₉	—	9	16	no	—
69	C₃×C₉	—	9	8	no	—
70	C₃²	—	3	12	no	—
71	C₃²	—	3	36	no	—
72	C₃²	—	9	8	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	9	36	no	—
75	C₃²	—	9	72	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	36	no	—
78	C₃²	—	9	9	no	—
79	C₃²	—	27	12	no	—
80	C₃²	—	27	24	no	—
81	C₃²	—	27	81	no	—
82	C₃²	C₃²⋊H(3,𝔽₃)	1	8	yes	—
83	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
84	C₃²	C₃²×H(3,𝔽₃)	1	4	yes	—
85	C₉	—	9	24	no	—
86	C₃	—	3	4	no	—
87	C₃	—	3	8	no	—
88	C₃	—	9	36	no	—
89	C₃	—	9	9	no	—
90	C₃	#10624761	1	3	yes	direct product factor
91	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₃)	C₃	semidirect
2. C₃×#6910	C₃	semidirect
3. C₃²×AΣL(1,𝔽₂₇)	C₃	semidirect
4. #10624761	C₃	direct (N × H)
5. C₃⁵	C₃²	semidirect
7. H(5,𝔽₃)	C₃²	semidirect
8. C₃²×3₋¹⁺²	C₃²	semidirect
12. #6910	C₃²	semidirect
16. C₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
23. C₃⁴	C₃³	semidirect
29. C₃×3₋¹⁺²	C₃³	semidirect
90. C₃	#10624761	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₃)	3
C₃×#6910	3
C₃²×AΣL(1,𝔽₂₇)	3
#10624761	3

Subgroups — C3×#10624761 #14554338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624761 #14554338