Subgroups — C₁₃×C₃₁×C₂²⋊ΓL(1,𝔽₈) #46799609

Order 33852.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₁₃×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₃×ΓL(1,𝔽₈)×C₃₁	—	4	1	no	maximal, Hall
3	C₂×C₇×C₁₃×C₃₁	—	3	1	no	—
4	A₄×C₁₃×C₃₁	—	7	1	no	maximal, Hall
5	C₇×C₁₃×C₃₁	A₄	1	1	yes	characteristic, complemented, Hall
6	C₃₁×C₂²⋊ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₁₃×C₃₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₃×C₃₁	—	28	1	no	Hall
9	C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂²×C₇×C₃₁	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
11	C₂×C₁₃×C₃₁	—	3	1	no	—
12	ΓL(1,𝔽₈)×C₃₁	—	4	1	no	Hall
13	C₂×C₇×C₃₁	—	3	1	no	—
14	C₁₃×C₃₁	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
15	A₄×C₃₁	—	7	1	no	Hall
16	C₂²×C₇×C₁₃	C₃×C₃₁	1	1	yes	characteristic, complemented, Hall
17	C₁₃×ΓL(1,𝔽₈)	—	4	1	no	Hall
18	C₇×C₃₁	A₄×C₁₃	1	1	yes	characteristic, complemented, Hall
19	C₂×C₇×C₁₃	—	3	1	no	—
20	A₄×C₁₃	—	7	1	no	Hall
21	C₂²×C₃₁	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
22	C₃×C₃₁	—	28	1	no	Hall
23	C₇×C₁₃	A₄×C₃₁	1	1	yes	characteristic, complemented, Hall
24	C₂²⋊ΓL(1,𝔽₈)	C₁₃×C₃₁	1	1	yes	characteristic, direct product factor, Hall
25	C₂×C₃₁	—	3	1	no	—
26	C₂²×C₁₃	ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, complemented, Hall
27	C₃×C₁₃	—	28	1	no	Hall
28	C₃₁	C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²×C₇	C₃×C₁₃×C₃₁	1	1	yes	characteristic, complemented, Hall
30	C₂×C₁₃	—	3	1	no	—
31	ΓL(1,𝔽₈)	—	4	1	no	Hall
32	C₂×C₇	—	3	1	no	—
33	C₁₃	C₃₁×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
34	A₄	—	7	1	no	Hall
35	C₇	A₄×C₁₃×C₃₁	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₁₃×ΓL(1,𝔽₈)×C₃₁	1	1	yes	characteristic, complemented, Sylow
37	C₃	—	28	1	no	Sylow
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃×C₃₁	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃×C₁₃×C₃₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇×C₁₃×C₃₁	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₁₃×C₃₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×C₁₃×C₃₁	C₃	semidirect
5. C₇×C₁₃×C₃₁	A₄	semidirect
6. C₃₁×C₂²⋊ΓL(1,𝔽₈)	C₁₃	direct (N × H)
7. C₂²×C₁₃×C₃₁	ΓL(1,𝔽₈)	semidirect
9. C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃₁	direct (N × H)
10. C₂²×C₇×C₃₁	C₃×C₁₃	semidirect
14. C₁₃×C₃₁	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
16. C₂²×C₇×C₁₃	C₃×C₃₁	semidirect
18. C₇×C₃₁	A₄×C₁₃	semidirect
21. C₂²×C₃₁	C₁₃×ΓL(1,𝔽₈)	semidirect
23. C₇×C₁₃	A₄×C₃₁	semidirect
24. C₂²⋊ΓL(1,𝔽₈)	C₁₃×C₃₁	direct (N × H)
26. C₂²×C₁₃	ΓL(1,𝔽₈)×C₃₁	semidirect
28. C₃₁	C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
29. C₂²×C₇	C₃×C₁₃×C₃₁	semidirect
33. C₁₃	C₃₁×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
35. C₇	A₄×C₁₃×C₃₁	semidirect
36. C₂²	C₁₃×ΓL(1,𝔽₈)×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×C₁₃×C₃₁	3
C₁₃×ΓL(1,𝔽₈)×C₃₁	4
A₄×C₁₃×C₃₁	7
C₃₁×C₂²⋊ΓL(1,𝔽₈)	13
C₁₃×C₂²⋊ΓL(1,𝔽₈)	31

Subgroups — C13×C31×C22⋊ΓL(1,𝔽8) #46799609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×C₃₁×C₂²⋊ΓL(1,𝔽₈) #46799609