Subgroups — #49146806

Order 36900.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×(C₅⋊D₅)⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#16861523	—	3	1	no	maximal
3	C₄₁×C₅²⋊C₉	C₄	1	1	yes	characteristic, complemented, Hall
4	C₃×C₄₁×C₅⋊D₅	D₃	1	1	yes	characteristic
5	#16861523	—	9	1	no	Hall
6	C₃×C₅²×C₄₁	Dic₃	1	1	yes	characteristic
7	C₃⋊#11717963	—	15	2	no	—
8	C₄₁×C₅⋊D₅	D₉	1	1	yes	characteristic
9	Dic₃₆₉	—	25	1	no	maximal, Hall
10	C₃×D₅×C₄₁	—	15	2	no	—
11	C₅²×C₄₁	Dic₉	1	1	yes	characteristic, complemented, Hall
12	#11744634	—	41	1	no	maximal, Hall
13	#11717963	—	45	2	no	—
14	C₂×C₉×C₄₁	—	25	1	no	—
15	C₃×C₅×C₄₁	—	3	2	no	—
16	Dic₁₂₃	—	75	1	no	—
17	(C₅⋊D₅)⋊C₉	D₄₁	1	1	yes	characteristic
18	D₅×C₄₁	—	15	2	no	—
19	C₉×C₄₁	—	25	1	no	Hall
20	C₃⋊#1044	—	123	1	no	—
21	C₂×C₃×C₄₁	—	25	1	no	—
22	C₅²⋊C₉	Dic₄₁	1	1	yes	characteristic, complemented, Hall
23	C₅×C₄₁	—	3	2	no	—
24	Dic₄₁	—	225	1	no	Hall
25	C₃×C₅⋊D₅	D₁₂₃	1	1	yes	characteristic
26	C₃×C₄₁	#64480	1	1	yes	characteristic
27	#1044	—	369	1	no	Hall
28	C₂×C₄₁	—	25	1	no	—
29	C₃×C₅²	Dic₁₂₃	1	1	yes	characteristic
30	ΓL(1,𝔽₁₆)	—	615	2	no	—
31	C₅⋊D₅	D₃₆₉	1	1	yes	characteristic
32	C₄₁	#11744634	1	1	yes	characteristic, complemented, Sylow
33	Dic₉	—	1025	1	no	Hall
34	C₃×D₅	—	15	2	no	—
35	C₅²	Dic₃₆₉	1	1	yes	characteristic, complemented, Sylow
36	AGL(1,𝔽₅)	—	1845	2	no	—
37	C₂×C₉	—	25	1	no	—
38	C₃×C₅	—	3	2	no	—
39	Dic₃	—	3075	1	no	—
40	D₅	—	15	2	no	—
41	C₉	—	25	1	no	Sylow
42	C₂×C₃	—	25	1	no	—
43	C₅	—	3	2	no	—
44	C₄	—	9225	1	no	Sylow
45	C₃	#27134245	1	1	yes	characteristic
46	C₂	—	25	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	C₄₁×C₅²⋊C₉	C₄
Frattini	C₃	#27134245
Fitting, socle	C₃×C₅²×C₄₁	Dic₃
prime radical	C₃×C₄₁×C₅⋊D₅	D₃
cosocle	C₄₁×(C₅⋊D₅)⋊C₉	C₂
LCS	G ▷ C₄₁×C₅²⋊C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₄₁×C₅²⋊C₉	C₄	semidirect
11. C₅²×C₄₁	Dic₉	semidirect
22. C₅²⋊C₉	Dic₄₁	semidirect
32. C₄₁	#11744634	semidirect
35. C₅²	Dic₃₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×(C₅⋊D₅)⋊C₉	2
C₃⋊#16861523	3
Dic₃₆₉	25
#11744634	41