Subgroups — C₃×C₇×AGL(1,𝔽₅)×C₇₁ #44447229

Order 29820.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×D₅×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇×AGL(1,𝔽₅)×C₇₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₃×C₅×C₇×C₇₁	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₄×C₇×C₇₁	—	5	1	no	maximal, Hall
5	C₇×D₅×C₇₁	C₂×C₃	1	1	yes	characteristic
6	C₃×AGL(1,𝔽₅)×C₇₁	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×C₇₁	—	5	1	no	—
8	C₅×C₇×C₇₁	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₃×D₅×C₇₁	C₂×C₇	1	1	yes	characteristic
10	C₄×C₇×C₇₁	—	5	1	no	Hall
11	C₃×C₇×C₇₁	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
12	AGL(1,𝔽₅)×C₇₁	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×C₇₁	C₄×C₇	1	1	yes	characteristic, complemented, Hall
14	C₂×C₇×C₇₁	—	5	1	no	—
15	C₃×C₄×C₇₁	—	5	1	no	Hall
16	D₅×C₇₁	C₂×C₃×C₇	1	1	yes	characteristic
17	C₇×C₇₁	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃×C₇₁	—	5	1	no	—
19	C₃×C₇×AGL(1,𝔽₅)	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×C₇₁	C₃×C₄×C₇	1	1	yes	characteristic, complemented, Hall
21	C₄×C₇₁	—	5	1	no	Hall
22	C₃×C₇₁	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₇×D₅	C₂×C₇₁	1	1	yes	characteristic
24	C₂×C₇₁	—	5	1	no	—
25	C₇×AGL(1,𝔽₅)	C₃×C₇₁	1	1	yes	characteristic, direct product factor, Hall
26	C₃×C₅×C₇	C₄×C₇₁	1	1	yes	characteristic, complemented, Hall
27	C₃×C₄×C₇	—	5	1	no	Hall
28	C₇₁	C₃×C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Sylow
29	C₇×D₅	C₂×C₃×C₇₁	1	1	yes	characteristic
30	C₃×AGL(1,𝔽₅)	C₇×C₇₁	1	1	yes	characteristic, direct product factor, Hall
31	C₂×C₃×C₇	—	5	1	no	—
32	C₅×C₇	C₃×C₄×C₇₁	1	1	yes	characteristic, complemented, Hall
33	C₃×D₅	C₂×C₇×C₇₁	1	1	yes	characteristic
34	C₄×C₇	—	5	1	no	Hall
35	C₃×C₇	AGL(1,𝔽₅)×C₇₁	1	1	yes	characteristic, direct product factor, Hall
36	AGL(1,𝔽₅)	C₃×C₇×C₇₁	1	1	yes	characteristic, direct product factor, Hall
37	C₃×C₅	C₄×C₇×C₇₁	1	1	yes	characteristic, complemented, Hall
38	C₂×C₇	—	5	1	no	—
39	C₃×C₄	—	5	1	no	Hall
40	D₅	C₂×C₃×C₇×C₇₁	1	1	yes	characteristic
41	C₇	C₃×AGL(1,𝔽₅)×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃	—	5	1	no	—
43	C₅	C₃×C₄×C₇×C₇₁	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	5	1	no	Sylow
45	C₃	C₇×AGL(1,𝔽₅)×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₇×C₇₁	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₃×C₄×C₇×C₇₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₇₁	C₄
prime radical	C₃×C₇×D₅×C₇₁	C₂
cosocle	D₅	C₂×C₃×C₇×C₇₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₇×C₇₁

Complements

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

N	H ≅ G/N	type
2. C₇×AGL(1,𝔽₅)×C₇₁	C₃	direct (N × H)
3. C₃×C₅×C₇×C₇₁	C₄	semidirect
6. C₃×AGL(1,𝔽₅)×C₇₁	C₇	direct (N × H)
8. C₅×C₇×C₇₁	C₃×C₄	semidirect
11. C₃×C₇×C₇₁	AGL(1,𝔽₅)	direct (N × H)
12. AGL(1,𝔽₅)×C₇₁	C₃×C₇	direct (N × H)
13. C₃×C₅×C₇₁	C₄×C₇	semidirect
17. C₇×C₇₁	C₃×AGL(1,𝔽₅)	direct (N × H)
19. C₃×C₇×AGL(1,𝔽₅)	C₇₁	direct (N × H)
20. C₅×C₇₁	C₃×C₄×C₇	semidirect
22. C₃×C₇₁	C₇×AGL(1,𝔽₅)	direct (N × H)
25. C₇×AGL(1,𝔽₅)	C₃×C₇₁	direct (N × H)
26. C₃×C₅×C₇	C₄×C₇₁	semidirect
28. C₇₁	C₃×C₇×AGL(1,𝔽₅)	direct (N × H)
30. C₃×AGL(1,𝔽₅)	C₇×C₇₁	direct (N × H)
32. C₅×C₇	C₃×C₄×C₇₁	semidirect
35. C₃×C₇	AGL(1,𝔽₅)×C₇₁	direct (N × H)
36. AGL(1,𝔽₅)	C₃×C₇×C₇₁	direct (N × H)
37. C₃×C₅	C₄×C₇×C₇₁	semidirect
41. C₇	C₃×AGL(1,𝔽₅)×C₇₁	direct (N × H)
43. C₅	C₃×C₄×C₇×C₇₁	semidirect
45. C₃	C₇×AGL(1,𝔽₅)×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
C₃×C₇×D₅×C₇₁	2
C₇×AGL(1,𝔽₅)×C₇₁	3
C₃×C₄×C₇×C₇₁	5
C₃×AGL(1,𝔽₅)×C₇₁	7
C₃×C₇×AGL(1,𝔽₅)	71

Subgroups — C3×C7×AGL(1,𝔽5)×C71 #44447229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×AGL(1,𝔽₅)×C₇₁ #44447229