Subgroups — C₇×AGL(1,𝔽₅)×C₇₃⋊C₃ #44501302

Order 30660.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₅×C₇₃⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₇×AGL(1,𝔽₅)×C₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×C₇×C₇₃⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₄×C₇×C₇₃⋊C₃	—	5	1	no	maximal, Hall
5	C₇×D₅×C₇₃	C₂×C₃	1	1	yes	characteristic
6	AGL(1,𝔽₅)×C₇₃⋊C₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₇×C₇₃⋊C₃	—	5	1	no	—
8	C₅×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	D₅×C₇₃⋊C₃	C₂×C₇	1	1	yes	characteristic
10	C₄×C₇×C₇₃	—	5	1	no	Hall
11	C₇×C₇₃⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
12	AGL(1,𝔽₅)×C₇₃	C₃×C₇	1	1	yes	characteristic, complemented, Hall
13	C₅×C₇₃⋊C₃	C₄×C₇	1	1	yes	characteristic, complemented, Hall
14	C₂×C₇×C₇₃	—	5	1	no	—
15	C₄×C₇₃⋊C₃	—	5	1	no	Hall
16	D₅×C₇₃	C₂×C₃×C₇	1	1	yes	characteristic
17	C₇×C₇₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
18	C₂×C₇₃⋊C₃	—	5	1	no	—
19	C₃×C₇×AGL(1,𝔽₅)	—	73	1	no	maximal, Hall
20	C₅×C₇₃	C₃×C₄×C₇	1	1	yes	characteristic, complemented, Hall
21	C₄×C₇₃	—	5	1	no	Hall
22	C₇₃⋊C₃	C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, direct product factor, Hall
23	C₃×C₇×D₅	—	73	1	no	—
24	C₂×C₇₃	—	5	1	no	—
25	C₇×AGL(1,𝔽₅)	C₇₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
26	C₃×C₅×C₇	—	73	1	no	Hall
27	C₃×C₄×C₇	—	365	1	no	Hall
28	C₇₃	C₃×C₇×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
29	C₇×D₅	C₂×C₇₃⋊C₃	1	1	yes	characteristic
30	C₃×AGL(1,𝔽₅)	—	73	1	no	Hall
31	C₂×C₃×C₇	—	365	1	no	—
32	C₅×C₇	C₄×C₇₃⋊C₃	1	1	yes	characteristic, complemented, Hall
33	C₃×D₅	—	73	1	no	—
34	C₄×C₇	—	5	1	no	Hall
35	C₃×C₇	—	73	1	no	Hall
36	AGL(1,𝔽₅)	C₇×C₇₃⋊C₃	1	1	yes	characteristic, direct product factor, Hall
37	C₃×C₅	—	73	1	no	Hall
38	C₂×C₇	—	5	1	no	—
39	C₃×C₄	—	365	1	no	Hall
40	D₅	C₂×C₇×C₇₃⋊C₃	1	1	yes	characteristic
41	C₇	AGL(1,𝔽₅)×C₇₃⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×C₃	—	365	1	no	—
43	C₅	C₄×C₇×C₇₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	5	1	no	Sylow
45	C₃	—	73	1	no	Sylow
46	C₂	—	5	1	no	—

	subgroup	G/H
center, hypercenter	C₇	AGL(1,𝔽₅)×C₇₃⋊C₃
commutator, hypocenter	C₅×C₇₃	C₃×C₄×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₇₃	C₃×C₄
prime radical	C₇×D₅×C₇₃⋊C₃	C₂
cosocle	D₅×C₇₃	C₂×C₃×C₇
LCS	G ▷ C₅×C₇₃
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
2. C₇×AGL(1,𝔽₅)×C₇₃	C₃	semidirect
3. C₅×C₇×C₇₃⋊C₃	C₄	semidirect
6. AGL(1,𝔽₅)×C₇₃⋊C₃	C₇	direct (N × H)
8. C₅×C₇×C₇₃	C₃×C₄	semidirect
11. C₇×C₇₃⋊C₃	AGL(1,𝔽₅)	direct (N × H)
12. AGL(1,𝔽₅)×C₇₃	C₃×C₇	semidirect
13. C₅×C₇₃⋊C₃	C₄×C₇	semidirect
17. C₇×C₇₃	C₃×AGL(1,𝔽₅)	semidirect
20. C₅×C₇₃	C₃×C₄×C₇	semidirect
22. C₇₃⋊C₃	C₇×AGL(1,𝔽₅)	direct (N × H)
25. C₇×AGL(1,𝔽₅)	C₇₃⋊C₃	direct (N × H)
28. C₇₃	C₃×C₇×AGL(1,𝔽₅)	semidirect
32. C₅×C₇	C₄×C₇₃⋊C₃	semidirect
36. AGL(1,𝔽₅)	C₇×C₇₃⋊C₃	direct (N × H)
41. C₇	AGL(1,𝔽₅)×C₇₃⋊C₃	direct (N × H)
43. C₅	C₄×C₇×C₇₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₅×C₇₃⋊C₃	2
C₇×AGL(1,𝔽₅)×C₇₃	3
C₄×C₇×C₇₃⋊C₃	5
AGL(1,𝔽₅)×C₇₃⋊C₃	7
C₃×C₇×AGL(1,𝔽₅)	73

Subgroups — C7×AGL(1,𝔽5)×C73⋊C3 #44501302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×AGL(1,𝔽₅)×C₇₃⋊C₃ #44501302