Subgroups — ΓL(1,𝔽₈)×C₂₉×#261 #44539862

Order 31668.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₁₃×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₇×C₂₉×#261	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×D₁₃×C₂₉	C₂×C₃	1	1	yes	characteristic
5	C₃×C₂₉×#261	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₂₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	C₄×ΓL(1,𝔽₈)×C₂₉	—	13	1	no	maximal, Hall
8	C₃×D₁₃×C₂₉	—	7	1	no	—
9	C₂₉×#261	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
10	C₂×ΓL(1,𝔽₈)×C₂₉	—	13	1	no	—
11	C₃×C₁₃×C₂₉	—	7	1	no	Hall
12	ΓL(1,𝔽₈)×#261	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₄×C₇×C₂₉	—	13	1	no	Hall
14	D₁₃×C₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
15	ΓL(1,𝔽₈)×C₂₉	#261	1	1	yes	characteristic, direct product factor, Hall
16	ΓL(1,𝔽₈)×D₁₃	C₂×C₂₉	1	1	yes	characteristic
17	C₂×C₇×C₂₉	—	13	1	no	—
18	C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₇×#261	C₃×C₂₉	1	1	yes	characteristic, complemented, Hall
20	C₃×C₄×C₂₉	—	91	1	no	Hall
21	C₁₃×ΓL(1,𝔽₈)	C₄×C₂₉	1	1	yes	characteristic, complemented, Hall
22	C₇×C₂₉	C₃×#261	1	1	yes	characteristic, complemented, Hall
23	C₇×D₁₃	C₂×C₃×C₂₉	1	1	yes	characteristic
24	C₂×C₃×C₂₉	—	91	1	no	—
25	C₃×#261	—	7	1	no	Hall
26	C₄×C₂₉	—	13	1	no	Hall
27	C₇×C₁₃	C₃×C₄×C₂₉	1	1	yes	characteristic, complemented, Hall
28	C₃×C₂₉	—	7	1	no	Hall
29	C₄×ΓL(1,𝔽₈)	—	13	1	no	Hall
30	C₃×D₁₃	—	7	1	no	—
31	C₂×C₂₉	—	13	1	no	—
32	#261	ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, direct product factor, Hall
33	C₂×ΓL(1,𝔽₈)	—	13	1	no	—
34	C₃×C₁₃	—	7	1	no	Hall
35	C₂₉	ΓL(1,𝔽₈)×#261	1	1	yes	characteristic, direct product factor, Sylow
36	C₄×C₇	—	13	1	no	Hall
37	D₁₃	C₂×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic
38	ΓL(1,𝔽₈)	C₂₉×#261	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₇	—	13	1	no	—
40	C₁₃	C₄×ΓL(1,𝔽₈)×C₂₉	1	1	yes	characteristic, complemented, Sylow
41	C₃×C₄	—	91	1	no	Hall
42	C₇	C₃×C₂₉×#261	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	91	1	no	—
44	C₄	—	13	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₂₉	ΓL(1,𝔽₈)×#261
commutator, hypocenter	C₇×C₁₃	C₃×C₄×C₂₉
Frattini	C₁	G
Fitting, socle	C₇×C₁₃×C₂₉	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₁₃×C₂₉	C₂
cosocle	C₇×D₁₃	C₂×C₃×C₂₉
LCS	G ▷ C₇×C₁₃
UCS	C₁ ◁ C₂₉

Complements

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

N	H ≅ G/N	type
2. C₇×C₂₉×#261	C₃	semidirect
3. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	semidirect
6. C₇×C₁₃×C₂₉	C₃×C₄	semidirect
9. C₂₉×#261	ΓL(1,𝔽₈)	direct (N × H)
12. ΓL(1,𝔽₈)×#261	C₂₉	direct (N × H)
15. ΓL(1,𝔽₈)×C₂₉	#261	direct (N × H)
18. C₁₃×C₂₉	C₄×ΓL(1,𝔽₈)	semidirect
19. C₇×#261	C₃×C₂₉	semidirect
21. C₁₃×ΓL(1,𝔽₈)	C₄×C₂₉	semidirect
22. C₇×C₂₉	C₃×#261	semidirect
27. C₇×C₁₃	C₃×C₄×C₂₉	semidirect
32. #261	ΓL(1,𝔽₈)×C₂₉	direct (N × H)
35. C₂₉	ΓL(1,𝔽₈)×#261	direct (N × H)
38. ΓL(1,𝔽₈)	C₂₉×#261	direct (N × H)
40. C₁₃	C₄×ΓL(1,𝔽₈)×C₂₉	semidirect
42. C₇	C₃×C₂₉×#261	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)×D₁₃×C₂₉	2
C₇×C₂₉×#261	3
C₃×C₂₉×#261	7
C₄×ΓL(1,𝔽₈)×C₂₉	13
ΓL(1,𝔽₈)×#261	29

Subgroups — ΓL(1,𝔽8)×C29×#261 #44539862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂₉×#261 #44539862