Subgroups — C₂⁴×AΣL(1,𝔽₂₇) #13048208

Order 1296.

1340 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×AΣL(1,𝔽₂₇)	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₃³	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	C₂⁴×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×AΣL(1,𝔽₂₇)	C₂²	1	35	yes	direct product factor
6	C₂³×C₃³	C₂×C₃	1	15	yes	complemented
7	C₂³×3₋¹⁺²	C₂×C₃	1	30	yes	complemented
8	C₂³×H(3,𝔽₃)	C₂×C₃	1	15	yes	complemented
9	C₂×AΣL(1,𝔽₂₇)	C₂³	1	15	yes	direct product factor
10	C₂⁴×C₃²	—	3	3	no	—
11	C₂⁴×C₃²	—	3	1	no	—
12	C₂⁴×C₃²	—	3	1	no	—
13	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
14	C₂⁴×C₉	—	3	2	no	—
15	C₂²×C₃³	C₂²×C₃	1	35	yes	complemented
16	C₂²×3₋¹⁺²	C₂²×C₃	1	70	yes	complemented
17	C₂²×H(3,𝔽₃)	C₂²×C₃	1	35	yes	complemented
18	AΣL(1,𝔽₂₇)	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
19	C₂³×C₃²	—	3	15	no	—
20	C₂³×C₃²	—	3	45	no	—
21	C₂³×C₃²	—	3	15	no	—
22	C₂³×C₃²	C₂×C₃²	1	15	yes	—
23	C₂³×C₉	—	3	30	no	—
24	C₂×C₃³	C₂³×C₃	1	15	yes	complemented
25	C₂×3₋¹⁺²	C₂³×C₃	1	30	yes	complemented
26	C₂×H(3,𝔽₃)	C₂³×C₃	1	15	yes	complemented
27	C₂⁴×C₃	—	3	3	no	—
28	C₂⁴×C₃	—	3	1	no	—
29	C₂⁴×C₃	—	9	1	no	—
30	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
31	C₂²×C₃²	—	3	35	no	—
32	C₂²×C₃²	—	3	105	no	—
33	C₂²×C₃²	—	3	35	no	—
34	C₂²×C₃²	C₂²×C₃²	1	35	yes	—
35	C₂²×C₉	—	3	70	no	—
36	C₃³	C₂⁴×C₃	1	1	yes	characteristic, complemented
37	3₋¹⁺²	C₂⁴×C₃	1	2	yes	complemented
38	H(3,𝔽₃)	C₂⁴×C₃	1	1	yes	characteristic, complemented
39	C₂³×C₃	—	3	15	no	—
40	C₂³×C₃	—	3	45	no	—
41	C₂³×C₃	—	9	15	no	—
42	C₂³×C₃	C₂×H(3,𝔽₃)	1	15	yes	—
43	C₂×C₃²	—	3	15	no	—
44	C₂×C₃²	—	3	45	no	—
45	C₂×C₃²	—	3	15	no	—
46	C₂×C₃²	C₂³×C₃²	1	15	yes	—
47	C₂×C₉	—	3	30	no	—
48	C₂⁴	AΣL(1,𝔽₂₇)	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₃	—	3	105	no	—
50	C₂²×C₃	—	3	35	no	—
51	C₂²×C₃	—	9	35	no	—
52	C₂²×C₃	C₂²×H(3,𝔽₃)	1	35	yes	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	1	no	—
55	C₃²	—	3	1	no	—
56	C₃²	C₂⁴×C₃²	1	1	yes	characteristic
57	C₉	—	3	2	no	—
58	C₂³	C₂×AΣL(1,𝔽₂₇)	1	15	yes	direct product factor
59	C₂×C₃	—	3	45	no	—
60	C₂×C₃	—	3	15	no	—
61	C₂×C₃	—	9	15	no	—
62	C₂×C₃	C₂³×H(3,𝔽₃)	1	15	yes	—
63	C₂²	C₂²×AΣL(1,𝔽₂₇)	1	35	yes	direct product factor
64	C₃	—	3	3	no	—
65	C₃	—	3	1	no	—
66	C₃	—	9	1	no	—
67	C₃	C₂⁴×H(3,𝔽₃)	1	1	yes	characteristic
68	C₂	C₂³×AΣL(1,𝔽₂₇)	1	15	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁴×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂⁴×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×AΣL(1,𝔽₂₇)	C₂	direct (N × H)
2. C₂⁴×C₃³	C₃	semidirect
3. C₂⁴×3₋¹⁺²	C₃	semidirect
4. C₂⁴×H(3,𝔽₃)	C₃	semidirect
5. C₂²×AΣL(1,𝔽₂₇)	C₂²	direct (N × H)
6. C₂³×C₃³	C₂×C₃	semidirect
7. C₂³×3₋¹⁺²	C₂×C₃	semidirect
8. C₂³×H(3,𝔽₃)	C₂×C₃	semidirect
9. C₂×AΣL(1,𝔽₂₇)	C₂³	direct (N × H)
15. C₂²×C₃³	C₂²×C₃	semidirect
16. C₂²×3₋¹⁺²	C₂²×C₃	semidirect
17. C₂²×H(3,𝔽₃)	C₂²×C₃	semidirect
18. AΣL(1,𝔽₂₇)	C₂⁴	direct (N × H)
24. C₂×C₃³	C₂³×C₃	semidirect
25. C₂×3₋¹⁺²	C₂³×C₃	semidirect
26. C₂×H(3,𝔽₃)	C₂³×C₃	semidirect
36. C₃³	C₂⁴×C₃	semidirect
37. 3₋¹⁺²	C₂⁴×C₃	semidirect
38. H(3,𝔽₃)	C₂⁴×C₃	semidirect
48. C₂⁴	AΣL(1,𝔽₂₇)	direct (N × H)
58. C₂³	C₂×AΣL(1,𝔽₂₇)	direct (N × H)
63. C₂²	C₂²×AΣL(1,𝔽₂₇)	direct (N × H)
68. C₂	C₂³×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
C₂³×AΣL(1,𝔽₂₇)	2
C₂⁴×C₃³	3
C₂⁴×3₋¹⁺²	3
C₂⁴×H(3,𝔽₃)	3

Subgroups — C24×AΣL(1,𝔽27) #13048208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×AΣL(1,𝔽₂₇) #13048208