Subgroups — C₂×C₃×C₂²⋊ΓL(1,𝔽₈) #92770

Order 504.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₃×C₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₂²⋊ΓL(1,𝔽₈)	C₂	direct (N × H)
2. C₂³×C₃×C₇	C₃	semidirect
3. C₂×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
7. C₂²×C₃×C₇	C₂×C₃	semidirect
8. C₂²⋊ΓL(1,𝔽₈)	C₂×C₃	direct (N × H)
11. C₂³×C₇	C₃²	semidirect
14. C₂×C₃×C₇	A₄	semidirect
19. C₂²×C₇	C₂×C₃²	semidirect
20. C₂³×C₃	ΓL(1,𝔽₈)	semidirect
22. C₃×C₇	C₂×A₄	semidirect
27. C₂×C₇	C₃×A₄	semidirect
30. C₂²×C₃	C₂×ΓL(1,𝔽₈)	semidirect
33. C₂³	C₃×ΓL(1,𝔽₈)	semidirect
34. C₇	C₂×C₃×A₄	semidirect
38. C₂×C₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
41. C₂²	C₂×C₃×ΓL(1,𝔽₈)	semidirect
43. C₃	C₂×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
46. C₂	C₃×C₂²⋊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
C₃×C₂²⋊ΓL(1,𝔽₈)	2
C₂³×C₃×C₇	3
C₂×C₂²⋊ΓL(1,𝔽₈)	3
C₂×C₃×ΓL(1,𝔽₈)	4
C₂×C₃×A₄	7

Subgroups — C2×C3×C22⋊ΓL(1,𝔽8) #92770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂²⋊ΓL(1,𝔽₈) #92770