Subgroups — C₁₁×C₃₇×C₂²⋊ΓL(1,𝔽₈) #46803295

Order 34188.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₁×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁×ΓL(1,𝔽₈)×C₃₇	—	4	1	no	maximal, Hall
3	C₂×C₇×C₁₁×C₃₇	—	3	1	no	—
4	C₁₁×A₄×C₃₇	—	7	1	no	maximal, Hall
5	C₃₇×C₂²⋊ΓL(1,𝔽₈)	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₇×C₁₁×C₃₇	A₄	1	1	yes	characteristic, complemented, Hall
7	C₂²×C₁₁×C₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₁×C₃₇	—	28	1	no	Hall
9	C₂²×C₇×C₃₇	C₃×C₁₁	1	1	yes	characteristic, complemented, Hall
10	C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₁₁×C₃₇	—	3	1	no	—
12	ΓL(1,𝔽₈)×C₃₇	—	4	1	no	Hall
13	C₂×C₇×C₃₇	—	3	1	no	—
14	A₄×C₃₇	—	7	1	no	Hall
15	C₁₁×C₃₇	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
16	C₂²×C₇×C₁₁	C₃×C₃₇	1	1	yes	characteristic, complemented, Hall
17	C₇×C₃₇	C₁₁×A₄	1	1	yes	characteristic, complemented, Hall
18	C₁₁×ΓL(1,𝔽₈)	—	4	1	no	Hall
19	C₂×C₇×C₁₁	—	3	1	no	—
20	C₂²×C₃₇	C₁₁×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
21	C₁₁×A₄	—	7	1	no	Hall
22	C₃×C₃₇	—	28	1	no	Hall
23	C₂²⋊ΓL(1,𝔽₈)	C₁₁×C₃₇	1	1	yes	characteristic, direct product factor, Hall
24	C₇×C₁₁	A₄×C₃₇	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃₇	—	3	1	no	—
26	C₂²×C₁₁	ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented, Hall
27	C₃₇	C₁₁×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×C₁₁	—	28	1	no	Hall
29	C₂²×C₇	C₃×C₁₁×C₃₇	1	1	yes	characteristic, complemented, Hall
30	C₂×C₁₁	—	3	1	no	—
31	ΓL(1,𝔽₈)	—	4	1	no	Hall
32	C₂×C₇	—	3	1	no	—
33	A₄	—	7	1	no	Hall
34	C₁₁	C₃₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
35	C₇	C₁₁×A₄×C₃₇	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₁₁×ΓL(1,𝔽₈)×C₃₇	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	28	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁×C₃₇	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₁₁×C₃₇
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₁×C₃₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₁×C₃₇

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₁×C₃₇	C₃	semidirect
5. C₃₇×C₂²⋊ΓL(1,𝔽₈)	C₁₁	direct (N × H)
6. C₇×C₁₁×C₃₇	A₄	semidirect
7. C₂²×C₁₁×C₃₇	ΓL(1,𝔽₈)	semidirect
9. C₂²×C₇×C₃₇	C₃×C₁₁	semidirect
10. C₁₁×C₂²⋊ΓL(1,𝔽₈)	C₃₇	direct (N × H)
15. C₁₁×C₃₇	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
16. C₂²×C₇×C₁₁	C₃×C₃₇	semidirect
17. C₇×C₃₇	C₁₁×A₄	semidirect
20. C₂²×C₃₇	C₁₁×ΓL(1,𝔽₈)	semidirect
23. C₂²⋊ΓL(1,𝔽₈)	C₁₁×C₃₇	direct (N × H)
24. C₇×C₁₁	A₄×C₃₇	semidirect
26. C₂²×C₁₁	ΓL(1,𝔽₈)×C₃₇	semidirect
27. C₃₇	C₁₁×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
29. C₂²×C₇	C₃×C₁₁×C₃₇	semidirect
34. C₁₁	C₃₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
35. C₇	C₁₁×A₄×C₃₇	semidirect
36. C₂²	C₁₁×ΓL(1,𝔽₈)×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₁×C₃₇	3
C₁₁×ΓL(1,𝔽₈)×C₃₇	4
C₁₁×A₄×C₃₇	7
C₃₇×C₂²⋊ΓL(1,𝔽₈)	11
C₁₁×C₂²⋊ΓL(1,𝔽₈)	37

Subgroups — C11×C37×C22⋊ΓL(1,𝔽8) #46803295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×C₃₇×C₂²⋊ΓL(1,𝔽₈) #46803295