Subgroups — C₄₁×#11744708 #49147079

Order 36900.

128 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	2	no	maximal
4	C₄₁×C₃⋊(C₅⋊D₅)	—	6	2	no	—
5	C₄₁×#1044	—	9	1	no	maximal, Hall
6	C₄₁×C₃⋊D₁₅	—	5	2	no	—
7	C₄₁×C₃⋊D₁₅	—	10	2	no	—
8	C₃×C₅²×C₄₁	—	2	2	no	—
9	C₄₁×C₅⋊D₅	—	9	1	no	—
10	C₃²×C₅×C₄₁	—	2	2	no	—
11	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	2	yes	complemented
12	#157×C₄₁	—	25	1	no	Hall
13	D₁₅×C₄₁	—	30	4	no	—
14	D₁₅×C₄₁	—	30	8	no	—
15	C₅²×C₄₁	#157	1	1	yes	characteristic, complemented, Hall
16	#11744708	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
17	AGL(1,𝔽₅)×C₄₁	—	45	2	no	—
18	C₃⋊D₃×C₄₁	—	25	1	no	—
19	C₃×C₅×C₄₁	—	2	8	no	—
20	C₃×C₅×C₄₁	—	2	4	no	—
21	C₃⋊(C₃⋊(C₅⋊D₅))	C₂×C₄₁	1	1	yes	characteristic
22	D₅×C₄₁	—	45	2	no	—
23	D₅×C₄₁	—	90	2	no	—
24	C₃²×C₄₁	#1044	1	1	yes	characteristic, complemented, Hall
25	D₃×C₄₁	—	150	2	no	—
26	C₃²×C₅²	C₄×C₄₁	1	1	yes	characteristic, complemented, Hall
27	C₅×C₄₁	—	2	2	no	—
28	C₅×C₄₁	#4380	1	2	yes	complemented
29	#4380	—	5	2	no	—
30	C₄×C₄₁	—	225	1	no	Hall
31	C₃⋊(C₅⋊D₅)	—	6	2	no	—
32	C₃×C₄₁	—	2	2	no	—
33	#1044	—	9	1	no	Hall
34	C₃⋊D₁₅	—	5	2	no	—
35	C₃⋊D₁₅	—	10	2	no	—
36	C₂×C₄₁	—	225	1	no	—
37	C₃×C₅²	—	2	2	no	—
38	C₅⋊D₅	—	9	1	no	—
39	C₃²×C₅	—	2	2	no	—
40	C₃²×C₅	AGL(1,𝔽₅)×C₄₁	1	2	yes	complemented
41	C₄₁	#11744708	1	1	yes	characteristic, direct product factor, Sylow
42	#157	—	25	1	no	Hall
43	D₁₅	—	30	4	no	—
44	D₁₅	—	30	8	no	—
45	C₅²	#157×C₄₁	1	1	yes	characteristic, complemented, Sylow
46	AGL(1,𝔽₅)	—	45	2	no	—
47	C₃⋊D₃	—	25	1	no	—
48	C₃×C₅	—	2	8	no	—
49	C₃×C₅	—	2	4	no	—
50	D₅	—	45	2	no	—
51	D₅	—	90	2	no	—
52	C₃²	C₄₁×#1044	1	1	yes	characteristic, complemented, Sylow
53	D₃	—	150	2	no	—
54	C₅	—	2	2	no	—
55	C₅	C₄₁×#4380	1	2	yes	complemented
56	C₄	—	225	1	no	Sylow
57	C₃	—	2	2	no	—
58	C₂	—	225	1	no	—

	subgroup	G/H
center, hypercenter	C₄₁	#11744708
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
11. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
15. C₅²×C₄₁	#157	semidirect
16. #11744708	C₄₁	direct (N × H)
24. C₃²×C₄₁	#1044	semidirect
26. C₃²×C₅²	C₄×C₄₁	semidirect
28. C₅×C₄₁	#4380	semidirect
40. C₃²×C₅	AGL(1,𝔽₅)×C₄₁	semidirect
41. C₄₁	#11744708	direct (N × H)
45. C₅²	#157×C₄₁	semidirect
52. C₃²	C₄₁×#1044	semidirect
55. 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×#1044	9
#11744708	41

Subgroups — C41×#11744708 #49147079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#11744708 #49147079