Subgroups — ΓL(1,𝔽₈)⋊(D₁₃₉⋊C₃) #31776113

Order 17514.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×C₁₃₉⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#19154254	C₃	1	1	yes	maximal, characteristic, complemented
3	#19154253	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)⋊D₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
5	(C₁₃₉⋊C₃)⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented
6	#15668589	C₂×C₃	1	1	yes	characteristic, complemented
7	#15668588	C₂×C₃	1	1	yes	characteristic, complemented
8	C₇×C₁₃₉⋊C₃	C₂×C₃	1	1	yes	characteristic, complemented
9	ΓL(1,𝔽₈)×C₁₃₉	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₁₃₉⋊C₃	—	7	1	no	maximal, Hall
11	D₉₇₃	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₁₃₉⋊C₃	—	7	1	no	Hall
13	C₇×C₁₃₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₃×D₁₃₉	—	7	1	no	—
15	D₁₃₉⋊C₃	—	7	1	no	—
16	D₁₃₉⋊C₃	—	7	1	no	—
17	D₁₃₉⋊C₃	—	21	1	no	—
18	C₃×C₁₃₉	—	7	1	no	—
19	C₁₃₉⋊C₃	—	7	1	no	—
20	C₁₃₉⋊C₃	—	7	1	no	—
21	C₁₃₉⋊C₃	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
22	D₁₃₉	—	21	1	no	Hall
23	C₁₃₉	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
24	C₃×AGL(1,𝔽₇)	—	139	1	no	maximal, Hall
25	C₃×ΓL(1,𝔽₈)	—	139	1	no	Hall
26	C₃×D₇	—	139	1	no	—
27	AGL(1,𝔽₇)	—	139	1	no	—
28	AGL(1,𝔽₇)	—	139	1	no	—
29	AGL(1,𝔽₇)	—	417	1	no	—
30	C₃×C₇	—	139	1	no	—
31	ΓL(1,𝔽₈)	—	139	1	no	—
32	ΓL(1,𝔽₈)	—	139	1	no	—
33	ΓL(1,𝔽₈)	D₁₃₉⋊C₃	1	1	yes	characteristic, complemented
34	C₂×C₃²	—	973	1	no	Hall
35	D₇	—	417	1	no	Hall
36	C₃²	—	973	1	no	Sylow
37	C₇	C₃×D₁₃₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₃	—	2919	1	no	—
39	C₂×C₃	—	2919	1	no	—
40	C₂×C₃	—	973	1	no	—
41	C₂×C₃	—	973	1	no	—
42	C₃	—	7	1	no	—
43	C₃	—	139	1	no	—
44	C₃	—	973	1	no	—
45	C₃	—	973	1	no	—
46	C₂	—	8757	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃₉	C₂×C₃²
LCS	G ▷ C₇×C₁₃₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×C₁₃₉⋊C₃	C₂	semidirect
2. #19154254	C₃	semidirect
3. #19154253	C₃	semidirect
4. ΓL(1,𝔽₈)⋊D₁₃₉	C₃	semidirect
5. (C₁₃₉⋊C₃)⋊D₇	C₃	semidirect
6. #15668589	C₂×C₃	semidirect
7. #15668588	C₂×C₃	semidirect
8. C₇×C₁₃₉⋊C₃	C₂×C₃	semidirect
9. ΓL(1,𝔽₈)×C₁₃₉	C₂×C₃	semidirect
11. D₉₇₃	C₃²	semidirect
13. C₇×C₁₃₉	C₂×C₃²	semidirect
21. C₁₃₉⋊C₃	AGL(1,𝔽₇)	semidirect
23. C₁₃₉	C₃×AGL(1,𝔽₇)	semidirect
33. ΓL(1,𝔽₈)	D₁₃₉⋊C₃	semidirect
37. C₇	C₃×D₁₃₉⋊C₃	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)×C₁₃₉⋊C₃	2
#19154254	3
#19154253	3
ΓL(1,𝔽₈)⋊D₁₃₉	3
(C₁₃₉⋊C₃)⋊D₇	3
C₃×D₁₃₉⋊C₃	7
C₃×AGL(1,𝔽₇)	139

Subgroups — ΓL(1,𝔽8)⋊(D139⋊C3) #31776113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)⋊(D₁₃₉⋊C₃) #31776113