Subgroups — ΓL(1,𝔽₈)×#1044 #14523596

Order 2100.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₇×#1044	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	5	2	no	maximal
5	C₇×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
6	C₃×#1044	—	7	1	no	maximal, Hall
7	D₅×ΓL(1,𝔽₈)	—	5	2	no	—
8	D₅×ΓL(1,𝔽₈)	—	10	2	no	—
9	C₅²×C₇	C₃×C₄	1	1	yes	characteristic, complemented, Hall
10	C₃×C₅⋊D₅	—	7	1	no	—
11	C₇×AGL(1,𝔽₅)	—	5	2	no	—
12	C₅×ΓL(1,𝔽₈)	—	2	2	no	—
13	C₅×ΓL(1,𝔽₈)	AGL(1,𝔽₅)	1	2	yes	complemented
14	#1044	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
15	C₄×ΓL(1,𝔽₈)	—	25	1	no	Hall
16	C₃×C₅²	—	7	1	no	Hall
17	C₇×D₅	—	5	2	no	—
18	C₇×D₅	—	10	2	no	—
19	C₃×AGL(1,𝔽₅)	—	35	2	no	—
20	C₅⋊D₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
21	C₂×ΓL(1,𝔽₈)	—	25	1	no	—
22	C₅×C₇	—	2	2	no	—
23	C₅×C₇	C₃×AGL(1,𝔽₅)	1	2	yes	complemented
24	C₃×D₅	—	35	2	no	—
25	C₃×D₅	—	70	2	no	—
26	C₄×C₇	—	25	1	no	Hall
27	C₅²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
28	ΓL(1,𝔽₈)	#1044	1	1	yes	characteristic, direct product factor, Hall
29	AGL(1,𝔽₅)	—	5	2	no	—
30	C₃×C₅	—	7	2	no	—
31	C₃×C₅	—	14	2	no	—
32	C₂×C₇	—	25	1	no	—
33	C₃×C₄	—	175	1	no	Hall
34	D₅	—	5	2	no	—
35	D₅	—	10	2	no	—
36	C₇	C₃×#1044	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃	—	175	1	no	—
38	C₅	—	2	2	no	—
39	C₅	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	2	yes	complemented
40	C₄	—	25	1	no	Sylow
41	C₃	—	7	1	no	Sylow
42	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₇	C₃×C₄
prime radical	ΓL(1,𝔽₈)×C₅⋊D₅	C₂
cosocle	C₇×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×#1044	C₃	semidirect
3. C₅²×ΓL(1,𝔽₈)	C₄	semidirect
9. C₅²×C₇	C₃×C₄	semidirect
13. C₅×ΓL(1,𝔽₈)	AGL(1,𝔽₅)	semidirect
14. #1044	ΓL(1,𝔽₈)	direct (N × H)
23. C₅×C₇	C₃×AGL(1,𝔽₅)	semidirect
27. C₅²	C₄×ΓL(1,𝔽₈)	semidirect
28. ΓL(1,𝔽₈)	#1044	direct (N × H)
36. C₇	C₃×#1044	semidirect
39. C₅	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×C₅⋊D₅	2
C₇×#1044	3
AGL(1,𝔽₅)×ΓL(1,𝔽₈)	5
C₃×#1044	7

Subgroups — ΓL(1,𝔽8)×#1044 #14523596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#1044 #14523596