Subgroups — C₃×C₅₃×AΣL(1,𝔽₂₇) #27166239

Order 12879.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×3₋¹⁺²×C₅₃	C₃	1	2	yes	maximal, complemented
3	C₃×H(3,𝔽₃)×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×AΣL(1,𝔽₂₇)	C₃	1	9	yes	maximal, direct product factor
5	C₃³×C₅₃	—	3	2	no	—
6	C₃³×C₅₃	—	3	1	no	—
7	C₃³×C₅₃	—	3	9	no	—
8	C₃³×C₅₃	—	3	1	no	—
9	C₃³×C₅₃	C₃²	1	1	yes	characteristic
10	C₃³×C₅₃	C₃²	1	3	yes	complemented
11	3₋¹⁺²×C₅₃	—	3	4	no	—
12	3₋¹⁺²×C₅₃	C₃²	1	6	yes	complemented
13	H(3,𝔽₃)×C₅₃	—	3	2	no	—
14	H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	complemented
15	C₃×C₉×C₅₃	—	3	2	no	—
16	C₃²×C₅₃	—	3	9	no	—
17	C₃²×C₅₃	—	3	18	no	—
18	C₃²×C₅₃	—	3	9	no	—
19	C₃²×C₅₃	—	3	3	no	—
20	C₃²×C₅₃	—	3	3	no	—
21	C₃²×C₅₃	—	3	3	no	—
22	C₃²×C₅₃	—	9	3	no	—
23	C₃²×C₅₃	H(3,𝔽₃)	1	2	yes	—
24	C₃²×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₅₃	C₃³	1	1	yes	characteristic
26	C₉×C₅₃	—	3	6	no	—
27	C₃×AΣL(1,𝔽₂₇)	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃×C₅₃	—	3	9	no	—
29	C₃×C₅₃	—	3	2	no	—
30	C₃×C₅₃	—	3	1	no	—
31	C₃×C₅₃	—	9	3	no	—
32	C₃×C₅₃	AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
33	C₃×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃⁴	C₃×C₅₃	1	1	yes	characteristic, complemented
35	C₃×3₋¹⁺²	C₃×C₅₃	1	2	yes	complemented
36	C₃×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic, complemented
37	AΣL(1,𝔽₂₇)	C₃×C₅₃	1	9	yes	direct product factor
38	C₅₃	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic, direct product factor, Sylow
39	C₃³	—	3	9	no	—
40	C₃³	—	3	2	no	—
41	C₃³	—	3	1	no	—
42	C₃³	—	3	1	no	—
43	C₃³	C₃²×C₅₃	1	1	yes	characteristic
44	C₃³	C₃²×C₅₃	1	3	yes	complemented
45	3₋¹⁺²	—	3	4	no	—
46	3₋¹⁺²	C₃²×C₅₃	1	6	yes	complemented
47	H(3,𝔽₃)	—	3	2	no	—
48	H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	complemented
49	C₃×C₉	—	3	2	no	—
50	C₃²	—	3	9	no	—
51	C₃²	—	3	18	no	—
52	C₃²	—	3	9	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	3	no	—
56	C₃²	—	9	3	no	—
57	C₃²	H(3,𝔽₃)×C₅₃	1	2	yes	—
58	C₃²	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
59	C₃²	C₃³×C₅₃	1	1	yes	characteristic
60	C₉	—	3	6	no	—
61	C₃	—	3	2	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	9	no	—
64	C₃	—	9	3	no	—
65	C₃	C₅₃×AΣL(1,𝔽₂₇)	1	3	yes	direct product factor
66	C₃	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₅₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃³×C₅₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃⁴×C₅₃	C₃	semidirect
2. C₃×3₋¹⁺²×C₅₃	C₃	semidirect
3. C₃×H(3,𝔽₃)×C₅₃	C₃	semidirect
4. C₅₃×AΣL(1,𝔽₂₇)	C₃	direct (N × H)
10. C₃³×C₅₃	C₃²	semidirect
12. 3₋¹⁺²×C₅₃	C₃²	semidirect
14. H(3,𝔽₃)×C₅₃	C₃²	semidirect
27. C₃×AΣL(1,𝔽₂₇)	C₅₃	direct (N × H)
32. C₃×C₅₃	AΣL(1,𝔽₂₇)	direct (N × H)
34. C₃⁴	C₃×C₅₃	semidirect
35. C₃×3₋¹⁺²	C₃×C₅₃	semidirect
36. C₃×H(3,𝔽₃)	C₃×C₅₃	semidirect
37. AΣL(1,𝔽₂₇)	C₃×C₅₃	direct (N × H)
38. C₅₃	C₃×AΣL(1,𝔽₂₇)	direct (N × H)
44. C₃³	C₃²×C₅₃	semidirect
46. 3₋¹⁺²	C₃²×C₅₃	semidirect
48. H(3,𝔽₃)	C₃²×C₅₃	semidirect
65. C₃	C₅₃×AΣL(1,𝔽₂₇)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₅₃	3
C₃×3₋¹⁺²×C₅₃	3
C₃×H(3,𝔽₃)×C₅₃	3
C₅₃×AΣL(1,𝔽₂₇)	3
C₃×AΣL(1,𝔽₂₇)	53

Subgroups — C3×C53×AΣL(1,𝔽27) #27166239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅₃×AΣL(1,𝔽₂₇) #27166239