Subgroups — #41×ΓL(1,𝔽₈) #66698

Order 336.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
3	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal, complemented
4	C₇×#41	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂²×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
6	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₄×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
8	C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×D₄	C₂×C₃	1	3	yes	complemented
10	C₂×C₄×C₇	C₂×C₃	1	3	yes	complemented
11	C₃×#41	—	7	1	no	maximal, Hall
12	C₂×ΓL(1,𝔽₈)	—	2	3	no	—
13	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
14	C₂²×C₇	C₂²×C₃	1	3	yes	—
15	C₄×C₇	C₂²×C₃	1	3	yes	—
16	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
17	C₃×Q₈	—	7	1	no	—
18	C₃×D₄	—	7	3	no	—
19	C₂×C₃×C₄	—	7	3	no	—
20	ΓL(1,𝔽₈)	#41	1	1	yes	characteristic, direct product factor, Hall
21	#41	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₇	—	2	3	no	—
23	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
24	C₂²×C₃	—	7	3	no	—
25	C₃×C₄	—	7	3	no	—
26	C₃×C₄	—	7	1	no	—
27	Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	D₄	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
29	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	3	yes	complemented
30	C₇	C₃×#41	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	7	1	no	—
32	C₂×C₃	—	14	3	no	—
33	C₂²	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
34	C₄	C₂²×ΓL(1,𝔽₈)	1	3	yes	—
35	C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
36	C₃	—	7	1	no	Sylow
37	C₂	—	2	3	no	—
38	C₂	C₂³×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂×C₇	C₂³×C₃
Frattini	C₂	C₂³×ΓL(1,𝔽₈)
Fitting	C₇×#41	C₃
hypercenter	#41	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×#41
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. Q₈×ΓL(1,𝔽₈)	C₂	semidirect
2. D₄×ΓL(1,𝔽₈)	C₂	semidirect
3. C₂×C₄×ΓL(1,𝔽₈)	C₂	semidirect
4. C₇×#41	C₃	semidirect
8. C₇×Q₈	C₂×C₃	semidirect
9. C₇×D₄	C₂×C₃	semidirect
10. C₂×C₄×C₇	C₂×C₃	semidirect
20. ΓL(1,𝔽₈)	#41	direct (N × H)
21. #41	ΓL(1,𝔽₈)	direct (N × H)
27. Q₈	C₂×ΓL(1,𝔽₈)	semidirect
28. D₄	C₂×ΓL(1,𝔽₈)	semidirect
29. C₂×C₄	C₂×ΓL(1,𝔽₈)	semidirect
30. C₇	C₃×#41	semidirect

Maximal subgroups

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

subgroup	index
Q₈×ΓL(1,𝔽₈)	2
D₄×ΓL(1,𝔽₈)	2
C₂×C₄×ΓL(1,𝔽₈)	2
C₇×#41	3
C₃×#41	7

Subgroups — #41×ΓL(1,𝔽8) #66698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×ΓL(1,𝔽₈) #66698