Subgroups — #44477635

Order 30420.

288 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊#15690148	C₂	1	1	yes	maximal, characteristic
2	C₃×#16810002	C₃	1	1	yes	maximal, characteristic
3	C₅×#13095200	C₄	1	1	yes	characteristic, complemented, Hall
4	#20258768	—	5	1	no	maximal, Hall
5	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃	1	1	yes	characteristic
6	#16810002	C₉	1	1	yes	characteristic, complemented, Hall
7	#15690148	—	5	1	no	—
8	C₃×C₅×C₁₃²	C₃×C₄	1	1	yes	characteristic
9	#63122⋊C₉	—	13	14	no	maximal
10	C₃×#10621728	—	5	1	no	—
11	C₅⋊(C₁₃⋊D₁₃)	C₂×C₉	1	1	yes	characteristic
12	#13095200	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
13	C₅⋊(D₁₃⋊C₉)	—	13	14	no	—
14	C₃×C₁₃⋊D₁₃	—	5	1	no	—
15	C₅×C₁₃²	C₄×C₉	1	1	yes	characteristic, complemented, Hall
16	C₃×#63122	—	13	14	no	—
17	#10621728	—	5	1	no	Hall
18	C₅×C₁₃⋊C₉	—	13	14	no	—
19	C₃×C₁₃²	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
20	#261⋊C₉	—	65	14	no	—
21	C₃×D₆₅	—	13	14	no	—
22	C₁₃⋊D₁₃	—	5	1	no	—
23	#63122	—	13	14	no	—
24	D₁₃⋊C₉	—	65	14	no	—
25	C₃×C₅×C₁₃	AGL(1,𝔽₁₃)	1	14	yes	—
26	C₉×AGL(1,𝔽₅)	—	169	1	no	Hall
27	C₁₃²	C₉×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
28	C₃×#261	—	65	14	no	—
29	D₆₅	—	13	14	no	—
30	C₁₃⋊C₉	—	13	14	no	—
31	C₉×D₅	—	169	1	no	—
32	C₃×D₁₃	—	65	14	no	—
33	C₅×C₁₃	#261⋊C₉	1	14	yes	complemented
34	C₃×AGL(1,𝔽₅)	—	169	1	no	—
35	#261	—	65	14	no	—
36	C₅×C₉	—	169	1	no	Hall
37	C₃×C₁₃	#63122⋊C₃	1	14	yes	—
38	C₄×C₉	—	845	1	no	Hall
39	C₃×D₅	—	169	1	no	—
40	D₁₃	—	65	14	no	—
41	AGL(1,𝔽₅)	—	169	1	no	Hall
42	C₂×C₉	—	845	1	no	—
43	C₃×C₅	#14523144	1	1	yes	characteristic
44	C₁₃	#63122⋊C₉	1	14	yes	complemented
45	C₃×C₄	—	845	1	no	—
46	D₅	—	169	1	no	—
47	C₉	—	169	1	no	Sylow
48	C₂×C₃	—	845	1	no	—
49	C₅	#20258768	1	1	yes	characteristic, complemented, Sylow
50	C₄	—	845	1	no	Sylow
51	C₃	#23778453	1	1	yes	characteristic
52	C₂	—	845	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#23778453
commutator, hypocenter	C₅×C₁₃²	C₄×C₉
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical, cosocle	C₃×C₅⋊(C₁₃⋊D₁₃)	C₂×C₃
LCS	G ▷ C₅×C₁₃²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. C₅×#13095200	C₄	semidirect
6. #16810002	C₉	semidirect
12. #13095200	AGL(1,𝔽₅)	semidirect
15. C₅×C₁₃²	C₄×C₉	semidirect
27. C₁₃²	C₉×AGL(1,𝔽₅)	semidirect
33. C₅×C₁₃	#261⋊C₉	semidirect
44. C₁₃	#63122⋊C₉	semidirect
49. C₅	#20258768	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊#15690148	2
C₃×#16810002	3
#20258768	5
#63122⋊C₉	13