Subgroups — ΓL(1,𝔽₈)⋊#13094087 #44539836

Order 31668.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₇₇	C₂	1	1	yes	maximal, characteristic
2	C₇⋊#13094087	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₇×D₃₇₇	C₂×C₃	1	1	yes	characteristic
5	C₃×#13094087	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₂₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)⋊#1181	—	13	1	no	maximal, Hall
8	C₃×D₃₇₇	—	7	1	no	—
9	#13094087	—	7	1	no	Hall
10	ΓL(1,𝔽₈)×D₂₉	—	13	1	no	—
11	C₃×C₁₃×C₂₉	—	7	1	no	Hall
12	ΓL(1,𝔽₈)⋊#261	—	29	1	no	maximal, Hall
13	C₇⋊#1181	—	13	1	no	Hall
14	D₃₇₇	AGL(1,𝔽₇)	1	1	yes	characteristic
15	ΓL(1,𝔽₈)×C₂₉	#261	1	1	yes	characteristic, complemented, Hall
16	ΓL(1,𝔽₈)×D₁₃	—	29	1	no	—
17	C₇×D₂₉	—	13	1	no	—
18	C₁₃×C₂₉	ΓL(1,𝔽₈)⋊C₄	1	1	yes	characteristic, complemented, Hall
19	C₇⋊#261	—	29	1	no	Hall
20	C₃×#1181	—	91	1	no	Hall
21	C₁₃×ΓL(1,𝔽₈)	#1181	1	1	yes	characteristic, complemented, Hall
22	C₇×C₂₉	C₃×#261	1	1	yes	characteristic, complemented, Hall
23	C₇×D₁₃	—	29	1	no	—
24	C₃×D₂₉	—	91	1	no	—
25	C₃×#261	—	203	1	no	Hall
26	#1181	—	91	1	no	Hall
27	C₇×C₁₃	C₃×#1181	1	1	yes	characteristic, complemented, Hall
28	C₃×C₂₉	—	7	1	no	Hall
29	ΓL(1,𝔽₈)⋊C₄	—	377	1	no	Hall
30	C₃×D₁₃	—	203	1	no	—
31	D₂₉	—	13	1	no	—
32	#261	—	203	1	no	Hall
33	C₂×ΓL(1,𝔽₈)	—	377	1	no	—
34	C₃×C₁₃	—	7	1	no	Hall
35	C₂₉	ΓL(1,𝔽₈)⋊#261	1	1	yes	characteristic, complemented, Sylow
36	Dic₇	—	377	1	no	Hall
37	D₁₃	—	29	1	no	—
38	ΓL(1,𝔽₈)	#13094087	1	1	yes	characteristic, complemented, Hall
39	C₂×C₇	—	377	1	no	—
40	C₁₃	ΓL(1,𝔽₈)⋊#1181	1	1	yes	characteristic, complemented, Sylow
41	C₃×C₄	—	2639	1	no	Hall
42	C₇	C₃×#13094087	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	2639	1	no	—
44	C₄	—	2639	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	—	377	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₇×C₁₃×C₂₉	C₃×C₄
prime radical	ΓL(1,𝔽₈)×D₃₇₇	C₂
cosocle	C₇×D₃₇₇	C₂×C₃
LCS	G ▷ C₇×C₁₃×C₂₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇⋊#13094087	C₃	semidirect
3. C₁₃×ΓL(1,𝔽₈)×C₂₉	C₄	semidirect
6. C₇×C₁₃×C₂₉	C₃×C₄	semidirect
15. ΓL(1,𝔽₈)×C₂₉	#261	semidirect
18. C₁₃×C₂₉	ΓL(1,𝔽₈)⋊C₄	semidirect
21. C₁₃×ΓL(1,𝔽₈)	#1181	semidirect
22. C₇×C₂₉	C₃×#261	semidirect
27. C₇×C₁₃	C₃×#1181	semidirect
35. C₂₉	ΓL(1,𝔽₈)⋊#261	semidirect
38. ΓL(1,𝔽₈)	#13094087	semidirect
40. C₁₃	ΓL(1,𝔽₈)⋊#1181	semidirect
42. C₇	C₃×#13094087	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)×D₃₇₇	2
C₇⋊#13094087	3
C₃×#13094087	7
ΓL(1,𝔽₈)⋊#1181	13
ΓL(1,𝔽₈)⋊#261	29

Subgroups — ΓL(1,𝔽8)⋊#13094087 #44539836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊#13094087 #44539836