Subgroups — Dic₅×ΓL(1,𝔽₈)×C₇₃ #44501271

Order 30660.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×ΓL(1,𝔽₈)×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇×Dic₅×C₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅×ΓL(1,𝔽₈)×C₇₃	C₄	1	1	yes	characteristic, complemented, Hall
4	C₄×ΓL(1,𝔽₈)×C₇₃	—	5	1	no	maximal, Hall
5	C₂×C₅×C₇×C₇₃	C₂×C₃	1	1	yes	characteristic
6	C₃×Dic₅×C₇₃	—	7	1	no	maximal, Hall
7	C₂×ΓL(1,𝔽₈)×C₇₃	D₅	1	1	yes	characteristic
8	C₅×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₅×C₇₃	—	7	1	no	—
10	C₄×C₇×C₇₃	—	5	1	no	Hall
11	ΓL(1,𝔽₈)×C₇₃	Dic₅	1	1	yes	characteristic, direct product factor, Hall
12	Dic₅×C₇₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
13	C₃×C₅×C₇₃	—	7	1	no	Hall
14	C₂×C₇×C₇₃	C₃×D₅	1	1	yes	characteristic
15	C₃×C₄×C₇₃	—	35	1	no	Hall
16	C₂×C₅×C₇₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
17	C₇×C₇₃	C₃×Dic₅	1	1	yes	characteristic, complemented, Hall
18	C₂×C₃×C₇₃	—	7	1	no	—
19	Dic₅×ΓL(1,𝔽₈)	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×C₇₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
21	C₄×C₇₃	—	5	1	no	Hall
22	C₃×C₇₃	—	7	1	no	Hall
23	C₂×C₅×ΓL(1,𝔽₈)	C₂×C₇₃	1	1	yes	characteristic
24	C₂×C₇₃	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₇×Dic₅	C₃×C₇₃	1	1	yes	characteristic, complemented, Hall
26	C₅×ΓL(1,𝔽₈)	C₄×C₇₃	1	1	yes	characteristic, complemented, Hall
27	C₄×ΓL(1,𝔽₈)	—	5	1	no	Hall
28	C₇₃	Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₅×C₇	C₂×C₃×C₇₃	1	1	yes	characteristic
30	C₃×Dic₅	—	7	1	no	Hall
31	C₂×ΓL(1,𝔽₈)	D₅×C₇₃	1	1	yes	characteristic
32	C₅×C₇	C₃×C₄×C₇₃	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₅	—	7	1	no	—
34	C₄×C₇	—	5	1	no	Hall
35	ΓL(1,𝔽₈)	Dic₅×C₇₃	1	1	yes	characteristic, direct product factor, Hall
36	Dic₅	ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, direct product factor, Hall
37	C₃×C₅	—	7	1	no	Hall
38	C₂×C₇	C₃×D₅×C₇₃	1	1	yes	characteristic
39	C₃×C₄	—	35	1	no	Hall
40	C₂×C₅	C₂×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic
41	C₇	C₃×Dic₅×C₇₃	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃	—	7	1	no	—
43	C₅	C₄×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic, complemented, Sylow
44	C₄	—	5	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	D₅×ΓL(1,𝔽₈)×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₇₃	D₅×ΓL(1,𝔽₈)
commutator, hypocenter	C₅×C₇	C₃×C₄×C₇₃
Frattini	C₂	D₅×ΓL(1,𝔽₈)×C₇₃
Fitting, socle	C₂×C₅×C₇×C₇₃	C₂×C₃
prime radical	C₂×C₅×ΓL(1,𝔽₈)×C₇₃	C₂
cosocle	C₂×C₅×C₇	C₂×C₃×C₇₃
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₇₃

Complements

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

N	H ≅ G/N	type
2. C₇×Dic₅×C₇₃	C₃	semidirect
3. C₅×ΓL(1,𝔽₈)×C₇₃	C₄	semidirect
8. C₅×C₇×C₇₃	C₃×C₄	semidirect
11. ΓL(1,𝔽₈)×C₇₃	Dic₅	direct (N × H)
12. Dic₅×C₇₃	ΓL(1,𝔽₈)	direct (N × H)
17. C₇×C₇₃	C₃×Dic₅	semidirect
19. Dic₅×ΓL(1,𝔽₈)	C₇₃	direct (N × H)
20. C₅×C₇₃	C₄×ΓL(1,𝔽₈)	semidirect
25. C₇×Dic₅	C₃×C₇₃	semidirect
26. C₅×ΓL(1,𝔽₈)	C₄×C₇₃	semidirect
28. C₇₃	Dic₅×ΓL(1,𝔽₈)	direct (N × H)
32. C₅×C₇	C₃×C₄×C₇₃	semidirect
35. ΓL(1,𝔽₈)	Dic₅×C₇₃	direct (N × H)
36. Dic₅	ΓL(1,𝔽₈)×C₇₃	direct (N × H)
41. C₇	C₃×Dic₅×C₇₃	semidirect
43. C₅	C₄×ΓL(1,𝔽₈)×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×ΓL(1,𝔽₈)×C₇₃	2
C₇×Dic₅×C₇₃	3
C₄×ΓL(1,𝔽₈)×C₇₃	5
C₃×Dic₅×C₇₃	7
Dic₅×ΓL(1,𝔽₈)	73

Subgroups — Dic5×ΓL(1,𝔽8)×C73 #44501271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×ΓL(1,𝔽₈)×C₇₃ #44501271