Subgroups — C₃×C₄×ΓL(1,𝔽₈)×C₇₁ #31822042

Order 17892.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄×ΓL(1,𝔽₈)×C₇₁	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₄×C₇×C₇₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₈)×C₇₁	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₂×ΓL(1,𝔽₈)×C₇₁	C₂×C₃	1	3	yes	—
6	C₂×C₃×C₇×C₇₁	C₂×C₃	1	1	yes	characteristic
7	C₃²×C₄×C₇₁	—	7	1	no	maximal, Hall
8	C₄×C₇×C₇₁	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇×C₇₁	C₃×C₄	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)×C₇₁	C₃×C₄	1	3	yes	direct product factor
11	C₂×C₃²×C₇₁	—	7	1	no	—
12	C₂×C₇×C₇₁	C₂×C₃²	1	1	yes	characteristic
13	C₃×C₄×C₇₁	—	7	3	no	—
14	C₃×C₄×C₇₁	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²×C₇₁	—	7	1	no	Hall
16	C₇×C₇₁	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₇₁	—	7	3	no	—
18	C₂×C₃×C₇₁	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	C₄×C₇₁	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
20	C₃×C₄×ΓL(1,𝔽₈)	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₃×C₇₁	—	7	3	no	—
22	C₃×C₇₁	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₂×C₇₁	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
24	C₂×C₃×ΓL(1,𝔽₈)	C₂×C₇₁	1	1	yes	characteristic
25	C₃×C₄×C₇	C₃×C₇₁	1	1	yes	characteristic, complemented
26	C₄×ΓL(1,𝔽₈)	C₃×C₇₁	1	3	yes	direct product factor
27	C₇₁	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
28	C₃×ΓL(1,𝔽₈)	C₄×C₇₁	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₃×C₇	C₂×C₃×C₇₁	1	1	yes	characteristic
30	C₂×ΓL(1,𝔽₈)	C₂×C₃×C₇₁	1	3	yes	—
31	C₃²×C₄	—	7	1	no	Hall
32	C₄×C₇	C₃²×C₇₁	1	1	yes	characteristic, complemented, Hall
33	C₃×C₇	C₃×C₄×C₇₁	1	1	yes	characteristic, complemented
34	ΓL(1,𝔽₈)	C₃×C₄×C₇₁	1	3	yes	direct product factor
35	C₂×C₃²	—	7	1	no	—
36	C₂×C₇	C₂×C₃²×C₇₁	1	1	yes	characteristic
37	C₃×C₄	—	7	3	no	—
38	C₃×C₄	ΓL(1,𝔽₈)×C₇₁	1	1	yes	characteristic, direct product factor
39	C₃²	—	7	1	no	Sylow
40	C₇	C₃²×C₄×C₇₁	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	7	3	no	—
42	C₂×C₃	C₂×ΓL(1,𝔽₈)×C₇₁	1	1	yes	characteristic
43	C₄	C₃×ΓL(1,𝔽₈)×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
44	C₃	—	7	3	no	—
45	C₃	C₄×ΓL(1,𝔽₈)×C₇₁	1	1	yes	characteristic, direct product factor
46	C₂	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄×C₇₁	ΓL(1,𝔽₈)
commutator, hypocenter	C₇	C₃²×C₄×C₇₁
Frattini	C₂	C₂×C₃×ΓL(1,𝔽₈)×C₇₁
Fitting	C₃×C₄×C₇×C₇₁	C₃
socle	C₂×C₃×C₇×C₇₁	C₂×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)×C₇₁	C₂
cosocle	C₂×C₇	C₂×C₃²×C₇₁
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₄×C₇₁

Complements

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

N	H ≅ G/N	type
2. C₄×ΓL(1,𝔽₈)×C₇₁	C₃	direct (N × H)
3. C₃×C₄×C₇×C₇₁	C₃	semidirect
4. C₃×ΓL(1,𝔽₈)×C₇₁	C₄	direct (N × H)
8. C₄×C₇×C₇₁	C₃²	semidirect
9. C₃×C₇×C₇₁	C₃×C₄	semidirect
10. ΓL(1,𝔽₈)×C₇₁	C₃×C₄	direct (N × H)
14. C₃×C₄×C₇₁	ΓL(1,𝔽₈)	direct (N × H)
16. C₇×C₇₁	C₃²×C₄	semidirect
19. C₄×C₇₁	C₃×ΓL(1,𝔽₈)	direct (N × H)
20. C₃×C₄×ΓL(1,𝔽₈)	C₇₁	direct (N × H)
22. C₃×C₇₁	C₄×ΓL(1,𝔽₈)	direct (N × H)
25. C₃×C₄×C₇	C₃×C₇₁	semidirect
26. C₄×ΓL(1,𝔽₈)	C₃×C₇₁	direct (N × H)
27. C₇₁	C₃×C₄×ΓL(1,𝔽₈)	direct (N × H)
28. C₃×ΓL(1,𝔽₈)	C₄×C₇₁	direct (N × H)
32. C₄×C₇	C₃²×C₇₁	semidirect
33. C₃×C₇	C₃×C₄×C₇₁	semidirect
34. ΓL(1,𝔽₈)	C₃×C₄×C₇₁	direct (N × H)
38. C₃×C₄	ΓL(1,𝔽₈)×C₇₁	direct (N × H)
40. C₇	C₃²×C₄×C₇₁	semidirect
43. C₄	C₃×ΓL(1,𝔽₈)×C₇₁	direct (N × H)
45. C₃	C₄×ΓL(1,𝔽₈)×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₈)×C₇₁	2
C₄×ΓL(1,𝔽₈)×C₇₁	3
C₃×C₄×C₇×C₇₁	3
C₃²×C₄×C₇₁	7
C₃×C₄×ΓL(1,𝔽₈)	71

Subgroups — C3×C4×ΓL(1,𝔽8)×C71 #31822042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×ΓL(1,𝔽₈)×C₇₁ #31822042