Subgroups — C₇×C₁₉×C₂²⋊ΓL(1,𝔽₈) #26014033

Order 11172.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇²×C₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₉×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
3	C₂×C₇²×C₁₉	—	3	1	no	—
4	C₁₉×C₂²⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₇×A₄×C₁₉	—	7	1	no	maximal
6	C₇²×C₁₉	A₄	1	1	yes	characteristic, complemented, Hall
7	C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₇×C₁₉	—	3	2	no	—
9	C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
10	C₂²×C₇×C₁₉	C₃×C₇	1	1	yes	characteristic, complemented
11	C₃×C₇×C₁₉	—	28	1	no	—
12	C₁₉×ΓL(1,𝔽₈)	—	4	1	no	—
13	C₂×C₇×C₁₉	—	3	6	no	—
14	C₂×C₇×C₁₉	—	3	1	no	—
15	C₂×C₇×C₁₉	—	3	1	no	—
16	A₄×C₁₉	—	7	1	no	Hall
17	C₂²×C₇²	C₃×C₁₉	1	1	yes	characteristic, complemented, Hall
18	C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
19	C₇×C₁₉	—	3	2	no	—
20	C₇×C₁₉	C₇×A₄	1	1	yes	characteristic, complemented
21	C₇×C₁₉	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
22	C₂×C₇²	—	3	1	no	—
23	C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₉	1	1	yes	characteristic, direct product factor
24	C₇×A₄	—	7	1	no	—
25	C₂²×C₁₉	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
26	C₃×C₁₉	—	28	1	no	Hall
27	C₇²	A₄×C₁₉	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₁₉	—	3	1	no	—
29	C₂²×C₇	—	3	2	no	—
30	C₂²×C₇	C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₂²×C₇	C₃×C₇×C₁₉	1	1	yes	characteristic, complemented
32	C₃×C₇	—	28	1	no	—
33	ΓL(1,𝔽₈)	—	4	1	no	—
34	C₁₉	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
35	C₂×C₇	—	3	6	no	—
36	C₂×C₇	—	3	1	no	—
37	C₂×C₇	—	3	1	no	—
38	A₄	—	7	1	no	Hall
39	C₇	—	3	2	no	—
40	C₇	C₇×A₄×C₁₉	1	1	yes	characteristic, complemented
41	C₇	C₁₉×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
42	C₂²	C₇×C₁₉×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	28	1	no	Sylow
44	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₇×C₁₉	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₇×C₁₉
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₁₉	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₇×C₁₉

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇²×C₁₉	C₃	semidirect
4. C₁₉×C₂²⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
6. C₇²×C₁₉	A₄	semidirect
7. C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₉	direct (N × H)
9. C₂²×C₇×C₁₉	ΓL(1,𝔽₈)	semidirect
10. C₂²×C₇×C₁₉	C₃×C₇	semidirect
17. C₂²×C₇²	C₃×C₁₉	semidirect
20. C₇×C₁₉	C₇×A₄	semidirect
21. C₇×C₁₉	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
23. C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₉	direct (N × H)
25. C₂²×C₁₉	C₇×ΓL(1,𝔽₈)	semidirect
27. C₇²	A₄×C₁₉	semidirect
30. C₂²×C₇	C₁₉×ΓL(1,𝔽₈)	semidirect
31. C₂²×C₇	C₃×C₇×C₁₉	semidirect
34. C₁₉	C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
40. C₇	C₇×A₄×C₁₉	semidirect
41. C₇	C₁₉×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
42. C₂²	C₇×C₁₉×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇²×C₁₉	3
C₇×C₁₉×ΓL(1,𝔽₈)	4
C₁₉×C₂²⋊ΓL(1,𝔽₈)	7
C₇×A₄×C₁₉	7
C₇×C₂²⋊ΓL(1,𝔽₈)	19

Subgroups — C7×C19×C22⋊ΓL(1,𝔽8) #26014033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₁₉×C₂²⋊ΓL(1,𝔽₈) #26014033