Subgroups — C₅×C₇⋊C₉×D₃₇ #37552003

Order 23310.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃₇×C₇⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅×C₇×D₃₇	C₃	1	1	yes	maximal, characteristic
3	C₇⋊C₉×D₃₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₅×C₇×C₃₇	C₂×C₃	1	1	yes	characteristic
5	C₅×C₉×D₃₇	—	7	1	no	maximal, Hall
6	C₅×C₇×D₃₇	C₉	1	1	yes	characteristic, complemented, Hall
7	C₃₇×C₇⋊C₉	C₂×C₅	1	1	yes	characteristic, complemented, Hall
8	C₅×C₉×C₃₇	—	7	1	no	Hall
9	C₃×C₇×D₃₇	C₃×C₅	1	1	yes	characteristic
10	C₅×C₇×C₃₇	C₂×C₉	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅×D₃₇	ΓL(1,𝔽₈)	1	1	yes	characteristic
12	C₃×C₇×C₃₇	C₂×C₃×C₅	1	1	yes	characteristic
13	C₉×D₃₇	—	7	1	no	Hall
14	C₂×C₅×C₇⋊C₉	—	37	1	no	maximal, Hall
15	C₃×C₅×C₃₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₇×D₃₇	C₅×C₉	1	1	yes	characteristic, complemented, Hall
17	C₅×D₃₇	C₇⋊C₉	1	1	yes	characteristic, direct product factor, Hall
18	C₉×C₃₇	—	7	1	no	Hall
19	C₅×C₇⋊C₉	D₃₇	1	1	yes	characteristic, direct product factor, Hall
20	C₇×C₃₇	C₂×C₅×C₉	1	1	yes	characteristic, complemented, Hall
21	C₃×D₃₇	C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
22	C₂×C₃×C₅×C₇	—	37	1	no	—
23	C₅×C₃₇	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented, Hall
24	C₂×C₇⋊C₉	—	37	1	no	Hall
25	C₃×C₃₇	C₂×C₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
26	C₃×C₅×C₇	C₃×D₃₇	1	1	yes	characteristic
27	C₂×C₅×C₉	—	259	1	no	Hall
28	D₃₇	C₅×C₇⋊C₉	1	1	yes	characteristic, direct product factor, Hall
29	C₂×C₅×C₇	—	37	1	no	Hall
30	C₇⋊C₉	C₅×D₃₇	1	1	yes	characteristic, direct product factor, Hall
31	C₅×C₉	—	7	1	no	Hall
32	C₂×C₃×C₇	—	37	1	no	—
33	C₃₇	C₂×C₅×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
34	C₅×C₇	C₉×D₃₇	1	1	yes	characteristic, complemented, Hall
35	C₂×C₃×C₅	—	37	1	no	—
36	C₃×C₇	C₃×C₅×D₃₇	1	1	yes	characteristic
37	C₂×C₉	—	259	1	no	Hall
38	C₃×C₅	ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic
39	C₂×C₇	—	37	1	no	Hall
40	C₂×C₅	—	37	1	no	Hall
41	C₉	—	7	1	no	Sylow
42	C₇	C₅×C₉×D₃₇	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	37	1	no	—
44	C₅	C₇⋊C₉×D₃₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₃	C₅×ΓL(1,𝔽₈)×D₃₇	1	1	yes	characteristic
46	C₂	—	37	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃×C₅	ΓL(1,𝔽₈)×D₃₇
commutator, hypocenter	C₇×C₃₇	C₂×C₅×C₉
Frattini	C₃	C₅×ΓL(1,𝔽₈)×D₃₇
Fitting, socle	C₃×C₅×C₇×C₃₇	C₂×C₃
prime radical	C₃×C₅×C₇×D₃₇	C₃
cosocle	C₃×C₇×C₃₇	C₂×C₃×C₅
LCS	G ▷ C₇×C₃₇
UCS	C₁ ◁ C₃×C₅

Complements

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

N	H ≅ G/N	type
1. C₅×C₃₇×C₇⋊C₉	C₂	semidirect
3. C₇⋊C₉×D₃₇	C₅	direct (N × H)
6. C₅×C₇×D₃₇	C₉	semidirect
7. C₃₇×C₇⋊C₉	C₂×C₅	semidirect
10. C₅×C₇×C₃₇	C₂×C₉	semidirect
16. C₇×D₃₇	C₅×C₉	semidirect
17. C₅×D₃₇	C₇⋊C₉	direct (N × H)
19. C₅×C₇⋊C₉	D₃₇	direct (N × H)
20. C₇×C₃₇	C₂×C₅×C₉	semidirect
23. C₅×C₃₇	C₂×C₇⋊C₉	semidirect
28. D₃₇	C₅×C₇⋊C₉	direct (N × H)
30. C₇⋊C₉	C₅×D₃₇	direct (N × H)
33. C₃₇	C₂×C₅×C₇⋊C₉	semidirect
34. C₅×C₇	C₉×D₃₇	semidirect
42. C₇	C₅×C₉×D₃₇	semidirect
44. C₅	C₇⋊C₉×D₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₃₇×C₇⋊C₉	2
C₃×C₅×C₇×D₃₇	3
C₇⋊C₉×D₃₇	5
C₅×C₉×D₃₇	7
C₂×C₅×C₇⋊C₉	37

Subgroups — C5×C7⋊C9×D37 #37552003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₇⋊C₉×D₃₇ #37552003