Subgroups — C₈×AΣL(1,𝔽₂₇) #10619518

Order 648.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₈×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
4	C₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×AΣL(1,𝔽₂₇)	C₄	1	1	yes	characteristic
6	C₃³×C₄	C₂×C₃	1	1	yes	characteristic
7	C₄×3₋¹⁺²	C₂×C₃	1	2	yes	—
8	C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	AΣL(1,𝔽₂₇)	C₈	1	1	yes	characteristic, direct product factor, Sylow
10	C₃²×C₈	—	3	1	no	—
11	C₃²×C₈	—	3	3	no	—
12	C₃²×C₈	—	3	1	no	—
13	C₃²×C₈	C₃²	1	1	yes	characteristic
14	C₈×C₉	—	3	2	no	—
15	C₂×C₃³	C₃×C₄	1	1	yes	characteristic
16	C₂×3₋¹⁺²	C₃×C₄	1	2	yes	—
17	C₂×H(3,𝔽₃)	C₃×C₄	1	1	yes	characteristic
18	C₃²×C₄	—	3	1	no	—
19	C₃²×C₄	—	3	3	no	—
20	C₃²×C₄	—	3	1	no	—
21	C₃²×C₄	C₂×C₃²	1	1	yes	characteristic
22	C₄×C₉	—	3	2	no	—
23	C₃³	C₃×C₈	1	1	yes	characteristic, complemented
24	3₋¹⁺²	C₃×C₈	1	2	yes	complemented
25	H(3,𝔽₃)	C₃×C₈	1	1	yes	characteristic, complemented
26	C₃×C₈	—	3	3	no	—
27	C₃×C₈	—	3	1	no	—
28	C₃×C₈	—	9	1	no	—
29	C₃×C₈	H(3,𝔽₃)	1	1	yes	characteristic
30	C₂×C₃²	—	3	1	no	—
31	C₂×C₃²	—	3	3	no	—
32	C₂×C₃²	—	3	1	no	—
33	C₂×C₃²	C₃²×C₄	1	1	yes	characteristic
34	C₂×C₉	—	3	2	no	—
35	C₃×C₄	—	3	3	no	—
36	C₃×C₄	—	3	1	no	—
37	C₃×C₄	—	9	1	no	—
38	C₃×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃²	—	3	3	no	—
40	C₃²	—	3	1	no	—
41	C₃²	—	3	1	no	—
42	C₃²	C₃²×C₈	1	1	yes	characteristic
43	C₉	—	3	2	no	—
44	C₈	AΣL(1,𝔽₂₇)	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃	—	3	3	no	—
46	C₂×C₃	—	3	1	no	—
47	C₂×C₃	—	9	1	no	—
48	C₂×C₃	C₄×H(3,𝔽₃)	1	1	yes	characteristic
49	C₄	C₂×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
50	C₃	—	3	3	no	—
51	C₃	—	3	1	no	—
52	C₃	—	9	1	no	—
53	C₃	C₈×H(3,𝔽₃)	1	1	yes	characteristic
54	C₂	C₄×AΣL(1,𝔽₂₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	H(3,𝔽₃)
commutator	C₃²	C₃²×C₈
Frattini, cosocle	C₃²×C₄	C₂×C₃²
socle	C₂×C₃	C₄×H(3,𝔽₃)
prime radical	C₂×AΣL(1,𝔽₂₇)	C₄
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×C₈	C₃	semidirect
3. C₈×3₋¹⁺²	C₃	semidirect
4. C₈×H(3,𝔽₃)	C₃	semidirect
9. AΣL(1,𝔽₂₇)	C₈	direct (N × H)
23. C₃³	C₃×C₈	semidirect
24. 3₋¹⁺²	C₃×C₈	semidirect
25. H(3,𝔽₃)	C₃×C₈	semidirect
44. C₈	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
C₄×AΣL(1,𝔽₂₇)	2
C₃³×C₈	3
C₈×3₋¹⁺²	3
C₈×H(3,𝔽₃)	3

Subgroups — C8×AΣL(1,𝔽27) #10619518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×AΣL(1,𝔽₂₇) #10619518