Subgroups — C₄₁×#11744707 #49147075

Order 36900.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
3	C₄₁×#4380	—	5	6	no	maximal
4	C₄₁×C₃⋊(C₅⋊D₅)	—	6	2	no	—
5	C₄₁×#1043	—	9	1	no	maximal, Hall
6	C₄₁×C₃⋊D₁₅	—	5	6	no	—
7	C₃×C₅²×C₄₁	—	2	2	no	—
8	C₄₁×C₅⋊D₅	—	9	1	no	—
9	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	6	yes	complemented
10	#157×C₄₁	—	25	1	no	Hall
11	D₁₅×C₄₁	—	30	12	no	—
12	C₅²×C₄₁	#157	1	1	yes	characteristic, complemented, Hall
13	#11744707	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
14	AGL(1,𝔽₅)×C₄₁	—	45	6	no	—
15	C₃⋊D₃×C₄₁	—	25	1	no	—
16	C₃×C₅×C₄₁	—	2	12	no	—
17	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₄₁	1	1	yes	characteristic
18	D₅×C₄₁	—	45	6	no	—
19	C₃²×C₄₁	#1043	1	1	yes	characteristic, complemented, Hall
20	D₃×C₄₁	—	150	2	no	—
21	C₃²×C₅²	C₄×C₄₁	1	1	yes	characteristic, complemented, Hall
22	C₅×C₄₁	#4380	1	6	yes	complemented
23	#4380	—	5	6	no	—
24	C₄×C₄₁	—	225	1	no	Hall
25	C₃⋊(C₅⋊D₅)	—	6	2	no	—
26	C₃×C₄₁	—	2	2	no	—
27	#1043	—	9	1	no	Hall
28	C₃⋊D₁₅	—	5	6	no	—
29	C₂×C₄₁	—	225	1	no	—
30	C₃×C₅²	—	2	2	no	—
31	C₅⋊D₅	—	9	1	no	—
32	C₃²×C₅	AGL(1,𝔽₅)×C₄₁	1	6	yes	complemented
33	C₄₁	#11744707	1	1	yes	characteristic, direct product factor, Sylow
34	#157	—	25	1	no	Hall
35	D₁₅	—	30	12	no	—
36	C₅²	#157×C₄₁	1	1	yes	characteristic, complemented, Sylow
37	AGL(1,𝔽₅)	—	45	6	no	—
38	C₃⋊D₃	—	25	1	no	—
39	C₃×C₅	—	2	12	no	—
40	D₅	—	45	6	no	—
41	C₃²	C₄₁×#1043	1	1	yes	characteristic, complemented, Sylow
42	D₃	—	150	2	no	—
43	C₅	C₄₁×#4380	1	6	yes	complemented
44	C₄	—	225	1	no	Sylow
45	C₃	—	2	2	no	—
46	C₂	—	225	1	no	—

	subgroup	G/H
center, hypercenter	C₄₁	#11744707
commutator, hypocenter	C₃²×C₅²	C₄×C₄₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₄₁	C₄
prime radical	C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	C₂
cosocle	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₄₁
LCS	G ▷ C₃²×C₅²
UCS	C₁ ◁ C₄₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₄₁	C₄	semidirect
9. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
12. C₅²×C₄₁	#157	semidirect
13. #11744707	C₄₁	direct (N × H)
19. C₃²×C₄₁	#1043	semidirect
21. C₃²×C₅²	C₄×C₄₁	semidirect
22. C₅×C₄₁	#4380	semidirect
32. C₃²×C₅	AGL(1,𝔽₅)×C₄₁	semidirect
33. C₄₁	#11744707	direct (N × H)
36. C₅²	#157×C₄₁	semidirect
41. C₃²	C₄₁×#1043	semidirect
43. C₅	C₄₁×#4380	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×C₃⋊(C₃⋊(C₅⋊D₅))	2
C₄₁×#4380	5
C₄₁×#1043	9
#11744707	41

Subgroups — C41×#11744707 #49147075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11744707 #49147075