Subgroups — ΓL(1,𝔽₈)×D₅₉₇ #38736985

Order 25074.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	ΓL(1,𝔽₈)×D₁₉₉	—	3	1	no	maximal
3	C₇×D₅₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×C₁₉₉	C₂×C₃	1	1	yes	characteristic, complemented
5	ΓL(1,𝔽₈)×C₁₉₉	—	2	1	no	—
6	ΓL(1,𝔽₈)×C₁₉₉	D₃	1	1	yes	characteristic, complemented
7	C₃×D₅₉₇	—	7	1	no	maximal, Hall
8	C₇×D₁₉₉	—	3	1	no	Hall
9	C₃²×C₁₉₉	—	7	1	no	Hall
10	C₇×C₁₉₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
11	D₅₉₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
12	C₃×D₁₉₉	—	21	1	no	—
13	C₃×C₁₉₉	—	7	1	no	—
14	C₃×C₁₉₉	—	14	1	no	—
15	C₃×C₁₉₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
16	D₁₉₉	—	3	1	no	Hall
17	C₁₉₉	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
18	D₃×ΓL(1,𝔽₈)	—	199	1	no	maximal, Hall
19	C₃×ΓL(1,𝔽₈)	D₁₉₉	1	1	yes	characteristic, complemented, Hall
20	D₃×C₇	—	199	1	no	—
21	C₂×ΓL(1,𝔽₈)	—	597	1	no	—
22	C₃×C₇	C₃×D₁₉₉	1	1	yes	characteristic, complemented
23	ΓL(1,𝔽₈)	—	2	1	no	—
24	ΓL(1,𝔽₈)	D₅₉₇	1	1	yes	characteristic, direct product factor
25	C₃×D₃	—	1393	1	no	Hall
26	C₂×C₇	—	597	1	no	Hall
27	C₃²	—	7	1	no	Sylow
28	C₇	C₃×D₅₉₇	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃	—	4179	1	no	—
30	D₃	—	199	1	no	—
31	C₃	—	7	1	no	—
32	C₃	—	14	1	no	—
33	C₃	ΓL(1,𝔽₈)×D₁₉₉	1	1	yes	characteristic, complemented
34	C₂	—	597	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₉₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₉₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×ΓL(1,𝔽₈)×C₁₉₉	C₂	semidirect
3. C₇×D₅₉₇	C₃	semidirect
4. C₃×C₇×C₁₉₉	C₂×C₃	semidirect
6. ΓL(1,𝔽₈)×C₁₉₉	D₃	semidirect
10. C₇×C₁₉₉	C₃×D₃	semidirect
11. D₅₉₇	ΓL(1,𝔽₈)	direct (N × H)
15. C₃×C₁₉₉	C₂×ΓL(1,𝔽₈)	semidirect
17. C₁₉₉	D₃×ΓL(1,𝔽₈)	semidirect
19. C₃×ΓL(1,𝔽₈)	D₁₉₉	semidirect
22. C₃×C₇	C₃×D₁₉₉	semidirect
24. ΓL(1,𝔽₈)	D₅₉₇	direct (N × H)
28. C₇	C₃×D₅₉₇	semidirect
33. C₃	ΓL(1,𝔽₈)×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×C₁₉₉	2
ΓL(1,𝔽₈)×D₁₉₉	3
C₇×D₅₉₇	3
C₃×D₅₉₇	7
D₃×ΓL(1,𝔽₈)	199

Subgroups — ΓL(1,𝔽8)×D597 #38736985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×D₅₉₇ #38736985