Subgroups — C₂×A₄×ΓL(1,𝔽₈) #92737

Order 504.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×C₂²⋊ΓL(1,𝔽₈)	C₃	1	2	yes	maximal, complemented
3	C₂×C₇×A₄	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂³×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×ΓL(1,𝔽₈)	—	4	1	no	maximal
6	C₂²⋊ΓL(1,𝔽₈)	C₂×C₃	1	2	yes	complemented
7	C₇×A₄	C₂×C₃	1	1	yes	characteristic, complemented
8	C₂²×ΓL(1,𝔽₈)	—	3	1	no	—
9	C₂²×ΓL(1,𝔽₈)	—	3	1	no	—
10	C₂²×ΓL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃×A₄	—	7	1	no	maximal, Hall
12	C₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
13	C₂³×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂×C₃×C₇	—	4	1	no	—
15	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
16	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
17	C₂×ΓL(1,𝔽₈)	—	4	2	no	—
18	C₂×ΓL(1,𝔽₈)	A₄	1	1	yes	characteristic, direct product factor
19	C₃×A₄	—	7	1	no	—
20	C₂²×C₇	—	3	1	no	—
21	C₂²×C₇	—	3	1	no	—
22	C₂²×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
23	C₂³×C₃	—	7	1	no	—
24	C₂×A₄	—	7	2	no	—
25	C₂×A₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
26	C₃×C₇	—	4	1	no	—
27	ΓL(1,𝔽₈)	—	4	2	no	—
28	ΓL(1,𝔽₈)	C₂×A₄	1	1	yes	characteristic, direct product factor
29	C₂×C₃²	—	28	1	no	—
30	C₂×C₇	—	3	1	no	—
31	C₂×C₇	—	3	1	no	—
32	C₂×C₇	C₃×A₄	1	1	yes	characteristic, complemented
33	C₂²×C₃	—	7	1	no	—
34	C₂²×C₃	—	21	1	no	—
35	C₂²×C₃	—	21	1	no	—
36	A₄	—	7	2	no	—
37	A₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
38	C₃²	—	28	1	no	Sylow
39	C₂³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
40	C₇	C₂×C₃×A₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	4	1	no	—
42	C₂×C₃	—	7	1	no	—
43	C₂×C₃	—	21	1	no	—
44	C₂×C₃	—	21	1	no	—
45	C₂×C₃	—	28	2	no	—
46	C₂²	—	3	1	no	—
47	C₂²	—	3	1	no	—
48	C₂²	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
49	C₃	—	4	1	no	—
50	C₃	—	7	1	no	—
51	C₃	—	28	2	no	—
52	C₂	—	3	1	no	—
53	C₂	—	3	1	no	—
54	C₂	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	A₄×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₇	C₃²
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. A₄×ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂×C₂²⋊ΓL(1,𝔽₈)	C₃	semidirect
3. C₂×C₇×A₄	C₃	semidirect
4. C₂³×ΓL(1,𝔽₈)	C₃	semidirect
6. C₂²⋊ΓL(1,𝔽₈)	C₂×C₃	semidirect
7. C₇×A₄	C₂×C₃	semidirect
10. C₂²×ΓL(1,𝔽₈)	C₂×C₃	semidirect
13. C₂³×C₇	C₃²	semidirect
18. C₂×ΓL(1,𝔽₈)	A₄	direct (N × H)
22. C₂²×C₇	C₂×C₃²	semidirect
25. C₂×A₄	ΓL(1,𝔽₈)	direct (N × H)
28. ΓL(1,𝔽₈)	C₂×A₄	direct (N × H)
32. C₂×C₇	C₃×A₄	semidirect
37. A₄	C₂×ΓL(1,𝔽₈)	direct (N × H)
39. C₂³	C₃×ΓL(1,𝔽₈)	semidirect
40. C₇	C₂×C₃×A₄	semidirect
48. C₂²	C₂×C₃×ΓL(1,𝔽₈)	semidirect
54. C₂	A₄×ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
A₄×ΓL(1,𝔽₈)	2
C₂×C₂²⋊ΓL(1,𝔽₈)	3
C₂×C₇×A₄	3
C₂³×ΓL(1,𝔽₈)	3
C₂×C₃×ΓL(1,𝔽₈)	4
C₂×C₃×A₄	7

Subgroups — C2×A4×ΓL(1,𝔽8) #92737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×A₄×ΓL(1,𝔽₈) #92737