Subgroups — ΓL(1,𝔽₈)×#13091732 #44501322

Order 30660.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₆₅	C₂	1	1	yes	maximal, characteristic
2	C₇×#13091732	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×ΓL(1,𝔽₈)×C₇₃	C₄	1	1	yes	characteristic, complemented, Hall
4	ΓL(1,𝔽₈)×#64408	—	5	1	no	maximal, Hall
5	C₇×D₃₆₅	C₂×C₃	1	1	yes	characteristic
6	C₃×#13091732	—	7	1	no	maximal, Hall
7	ΓL(1,𝔽₈)×D₇₃	—	5	1	no	—
8	C₅×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×D₃₆₅	—	7	1	no	—
10	C₇×#64408	—	5	1	no	Hall
11	ΓL(1,𝔽₈)×C₇₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
12	#13091732	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×C₇₃	—	7	1	no	Hall
14	C₇×D₇₃	—	5	1	no	—
15	C₃×#64408	—	35	1	no	Hall
16	D₃₆₅	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₇×C₇₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
18	C₃×D₇₃	—	35	1	no	—
19	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	—	73	1	no	maximal, Hall
20	C₅×C₇₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
21	#64408	—	5	1	no	Hall
22	C₃×C₇₃	—	7	1	no	Hall
23	D₅×ΓL(1,𝔽₈)	—	73	1	no	—
24	D₇₃	—	5	1	no	—
25	C₇×AGL(1,𝔽₅)	—	73	1	no	Hall
26	C₅×ΓL(1,𝔽₈)	#64408	1	1	yes	characteristic, complemented, Hall
27	C₄×ΓL(1,𝔽₈)	—	365	1	no	Hall
28	C₇₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
29	C₇×D₅	—	73	1	no	—
30	C₃×AGL(1,𝔽₅)	—	511	1	no	Hall
31	C₂×ΓL(1,𝔽₈)	—	365	1	no	—
32	C₅×C₇	C₃×#64408	1	1	yes	characteristic, complemented, Hall
33	C₃×D₅	—	511	1	no	—
34	C₄×C₇	—	365	1	no	Hall
35	ΓL(1,𝔽₈)	#13091732	1	1	yes	characteristic, direct product factor, Hall
36	AGL(1,𝔽₅)	—	73	1	no	Hall
37	C₃×C₅	—	7	1	no	Hall
38	C₂×C₇	—	365	1	no	—
39	C₃×C₄	—	2555	1	no	Hall
40	D₅	—	73	1	no	—
41	C₇	C₃×#13091732	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃	—	2555	1	no	—
43	C₅	ΓL(1,𝔽₈)×#64408	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	365	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	365	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₇×C₇₃	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₃₆₅	C₂
cosocle	C₇×D₃₆₅	C₂×C₃
LCS	G ▷ C₅×C₇×C₇₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×#13091732	C₃	semidirect
3. C₅×ΓL(1,𝔽₈)×C₇₃	C₄	semidirect
8. C₅×C₇×C₇₃	C₃×C₄	semidirect
11. ΓL(1,𝔽₈)×C₇₃	AGL(1,𝔽₅)	semidirect
12. #13091732	ΓL(1,𝔽₈)	direct (N × H)
17. C₇×C₇₃	C₃×AGL(1,𝔽₅)	semidirect
20. C₅×C₇₃	C₄×ΓL(1,𝔽₈)	semidirect
26. C₅×ΓL(1,𝔽₈)	#64408	semidirect
28. C₇₃	AGL(1,𝔽₅)×ΓL(1,𝔽₈)	semidirect
32. C₅×C₇	C₃×#64408	semidirect
35. ΓL(1,𝔽₈)	#13091732	direct (N × H)
41. C₇	C₃×#13091732	semidirect
43. C₅	ΓL(1,𝔽₈)×#64408	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)×D₃₆₅	2
C₇×#13091732	3
ΓL(1,𝔽₈)×#64408	5
C₃×#13091732	7
AGL(1,𝔽₅)×ΓL(1,𝔽₈)	73

Subgroups — ΓL(1,𝔽8)×#13091732 #44501322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#13091732 #44501322