Subgroups — (C₅⋊D₅)⋊(C₄₉⋊C₉) #36347608

Order 22050.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²⋊(C₄₉⋊C₉)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₄₉×C₅⋊D₅	C₃	1	1	yes	maximal, characteristic
3	C₃×C₅²×C₄₉	C₂×C₃	1	1	yes	characteristic
4	(C₅⋊D₅)⋊(C₇⋊C₉)	—	7	1	no	maximal
5	C₄₉×C₅⋊D₅	C₉	1	1	yes	characteristic, complemented, Hall
6	C₇⋊(C₅²⋊C₉)	—	7	1	no	—
7	C₃×D₅×C₄₉	—	15	2	no	—
8	C₅²×C₄₉	C₂×C₉	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)	1	1	yes	characteristic
10	C₂×C₄₉⋊C₉	—	25	1	no	maximal, Hall
11	C₃×C₅×C₄₉	—	3	2	no	—
12	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
13	D₅×C₄₉	—	15	2	no	—
14	(C₅⋊D₅)⋊C₉	—	49	1	no	Hall
15	C₄₉⋊C₉	—	25	1	no	Hall
16	C₇×C₅⋊D₅	C₇⋊C₉	1	1	yes	characteristic
17	C₂×C₃×C₄₉	—	75	1	no	—
18	C₅×C₄₉	—	3	2	no	—
19	C₅²⋊C₉	—	49	1	no	Hall
20	C₃×C₇×D₅	—	15	2	no	—
21	C₅²×C₇	C₂×C₇⋊C₉	1	1	yes	characteristic
22	C₃×C₅⋊D₅	C₄₉⋊C₃	1	1	yes	characteristic
23	C₃×C₄₉	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic
24	C₂×C₇⋊C₉	—	175	1	no	—
25	C₃×C₅×C₇	—	3	2	no	—
26	C₂×C₄₉	—	75	1	no	Hall
27	C₃×C₅²	C₂×C₄₉⋊C₃	1	1	yes	characteristic
28	C₇×D₅	—	15	2	no	—
29	C₇⋊C₉	—	175	1	no	—
30	C₅⋊D₅	C₄₉⋊C₉	1	1	yes	characteristic, complemented, Hall
31	C₄₉	(C₅⋊D₅)⋊C₉	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃×C₇	—	75	1	no	—
33	C₅×C₇	—	3	2	no	—
34	C₃×D₅	—	15	2	no	—
35	C₅²	C₂×C₄₉⋊C₉	1	1	yes	characteristic, complemented, Sylow
36	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₂×C₉	—	1225	1	no	Hall
38	C₃×C₅	—	3	2	no	—
39	C₂×C₇	—	75	1	no	—
40	D₅	—	15	2	no	—
41	C₉	—	1225	1	no	Sylow
42	C₇	(C₅⋊D₅)⋊(C₇⋊C₉)	1	1	yes	characteristic
43	C₂×C₃	—	75	1	no	—
44	C₅	—	3	2	no	—
45	C₃	(C₅⋊D₅)⋊(C₄₉⋊C₃)	1	1	yes	characteristic
46	C₂	—	75	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	(C₅⋊D₅)⋊(C₄₉⋊C₃)
commutator, hypocenter	C₅²×C₄₉	C₂×C₉
Frattini	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)
Fitting, cosocle	C₃×C₅²×C₄₉	C₂×C₃
socle	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)
prime radical	C₃×C₇×C₅⋊D₅	ΓL(1,𝔽₈)
LCS	G ▷ C₅²×C₄₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₅²⋊(C₄₉⋊C₉)	C₂	semidirect
5. C₄₉×C₅⋊D₅	C₉	semidirect
8. C₅²×C₄₉	C₂×C₉	semidirect
30. C₅⋊D₅	C₄₉⋊C₉	semidirect
31. C₄₉	(C₅⋊D₅)⋊C₉	semidirect
35. C₅²	C₂×C₄₉⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₅²⋊(C₄₉⋊C₉)	2
C₃×C₄₉×C₅⋊D₅	3
(C₅⋊D₅)⋊(C₇⋊C₉)	7
C₂×C₄₉⋊C₉	25

Subgroups — (C5⋊D5)⋊(C49⋊C9) #36347608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₅⋊D₅)⋊(C₄₉⋊C₉) #36347608