Subgroups — C₁₃×ΓL(1,𝔽₈)×D₄₁ #36375601

Order 22386.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×ΓL(1,𝔽₈)×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₃×D₄₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇×C₁₃×C₄₁	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₃×C₁₃×D₄₁	—	7	1	no	maximal, Hall
5	ΓL(1,𝔽₈)×D₄₁	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₃×C₁₃×C₄₁	—	7	1	no	Hall
7	C₁₃×D₄₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
8	ΓL(1,𝔽₈)×C₄₁	C₂×C₁₃	1	1	yes	characteristic, complemented, Hall
9	C₇×D₄₁	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
10	C₂×C₁₃×ΓL(1,𝔽₈)	—	41	1	no	maximal, Hall
11	C₁₃×C₄₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
12	C₇×C₄₁	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
13	C₁₃×ΓL(1,𝔽₈)	D₄₁	1	1	yes	characteristic, direct product factor, Hall
14	C₃×D₄₁	—	7	1	no	Hall
15	C₂×C₇×C₁₃	—	41	1	no	Hall
16	C₃×C₄₁	—	7	1	no	Hall
17	C₇×C₁₃	C₃×D₄₁	1	1	yes	characteristic, complemented, Hall
18	D₄₁	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃×C₁₃	—	287	1	no	Hall
20	C₂×ΓL(1,𝔽₈)	—	41	1	no	Hall
21	C₄₁	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
22	C₃×C₁₃	—	7	1	no	Hall
23	C₂×C₁₃	—	41	1	no	Hall
24	ΓL(1,𝔽₈)	C₁₃×D₄₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₇	—	41	1	no	Hall
26	C₁₃	ΓL(1,𝔽₈)×D₄₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₇	C₃×C₁₃×D₄₁	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃	—	287	1	no	Hall
29	C₃	—	7	1	no	Sylow
30	C₂	—	41	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₃	ΓL(1,𝔽₈)×D₄₁
commutator, cosocle, hypocenter	C₇×C₄₁	C₂×C₃×C₁₃
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₄₁	C₂×C₃
LCS	G ▷ C₇×C₄₁
UCS	C₁ ◁ C₁₃

Complements

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

N	H ≅ G/N	type
1. C₁₃×ΓL(1,𝔽₈)×C₄₁	C₂	semidirect
2. C₇×C₁₃×D₄₁	C₃	semidirect
3. C₇×C₁₃×C₄₁	C₂×C₃	semidirect
5. ΓL(1,𝔽₈)×D₄₁	C₁₃	direct (N × H)
7. C₁₃×D₄₁	ΓL(1,𝔽₈)	direct (N × H)
8. ΓL(1,𝔽₈)×C₄₁	C₂×C₁₃	semidirect
9. C₇×D₄₁	C₃×C₁₃	semidirect
11. C₁₃×C₄₁	C₂×ΓL(1,𝔽₈)	semidirect
12. C₇×C₄₁	C₂×C₃×C₁₃	semidirect
13. C₁₃×ΓL(1,𝔽₈)	D₄₁	direct (N × H)
17. C₇×C₁₃	C₃×D₄₁	semidirect
18. D₄₁	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
21. C₄₁	C₂×C₁₃×ΓL(1,𝔽₈)	semidirect
24. ΓL(1,𝔽₈)	C₁₃×D₄₁	direct (N × H)
26. C₁₃	ΓL(1,𝔽₈)×D₄₁	direct (N × H)
27. C₇	C₃×C₁₃×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×ΓL(1,𝔽₈)×C₄₁	2
C₇×C₁₃×D₄₁	3
C₃×C₁₃×D₄₁	7
ΓL(1,𝔽₈)×D₄₁	13
C₂×C₁₃×ΓL(1,𝔽₈)	41

Subgroups — C13×ΓL(1,𝔽8)×D41 #36375601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×ΓL(1,𝔽₈)×D₄₁ #36375601