Subgroups — #20301654⋊C₅ #49146943

Order 36900.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₅⋊(D₄₁⋊C₅)))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₄₁⋊C₅	C₄	1	1	yes	characteristic, complemented, Hall
3	#20301654	C₅	1	1	yes	maximal, characteristic, complemented
4	#13093287⋊C₅	—	5	1	no	maximal
5	C₃⋊(C₅⋊(D₄₁⋊C₅))	—	6	2	no	—
6	#11717962⋊C₅	—	9	1	no	maximal, Hall
7	C₃⋊D₆₁₅	C₂×C₅	1	1	yes	characteristic
8	C₃⋊(C₃⋊(D₄₁⋊C₅))	—	5	1	no	—
9	C₃×C₅×C₄₁⋊C₅	—	2	2	no	—
10	C₅⋊(D₄₁⋊C₅)	—	9	1	no	—
11	C₃²×C₅×C₄₁	C₄×C₅	1	1	yes	characteristic, complemented
12	C₃²×C₄₁⋊C₅	—	4	1	no	—
13	C₃²×C₄₁⋊C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	#13093287	—	5	1	no	Hall
15	D₆₁₅	—	6	2	no	—
16	C₃⋊(D₄₁⋊C₅)	—	30	2	no	—
17	C₅×C₄₁⋊C₅	#157	1	1	yes	characteristic, complemented, Hall
18	C₅×#4380	—	41	1	no	maximal, Hall
19	#11717962	—	9	1	no	—
20	#4223⋊C₅	—	45	1	no	—
21	C₃⋊D₁₂₃	—	5	1	no	—
22	C₃×C₅×C₄₁	—	2	2	no	—
23	C₃×C₄₁⋊C₅	—	2	2	no	—
24	C₃×C₄₁⋊C₅	—	4	4	no	—
25	C₅×C₃⋊D₁₅	—	41	1	no	—
26	D₂₀₅	—	9	1	no	—
27	D₄₁⋊C₅	—	45	1	no	—
28	C₃²×C₄₁	C₅×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Hall
29	D₁₂₃	—	30	2	no	—
30	C₃²×C₅²	—	41	1	no	Hall
31	C₅×C₄₁	C₅×#157	1	1	yes	characteristic, complemented
32	C₄₁⋊C₅	—	4	1	no	—
33	C₄₁⋊C₅	#4380	1	1	yes	characteristic, complemented
34	#4380	—	41	1	no	—
35	C₅×#157	—	205	1	no	—
36	#4223	—	45	1	no	Hall
37	C₅×D₁₅	—	246	2	no	—
38	C₃×C₄₁	—	2	2	no	—
39	C₅×AGL(1,𝔽₅)	—	369	1	no	Hall
40	C₃⋊D₁₅	—	41	1	no	—
41	C₅×C₃⋊D₃	—	205	1	no	—
42	D₄₁	—	45	1	no	—
43	C₃×C₅²	—	82	2	no	—
44	C₅×D₅	—	369	1	no	—
45	C₃²×C₅	—	41	1	no	—
46	C₃²×C₅	—	164	1	no	—
47	C₃²×C₅	#4223⋊C₅	1	1	yes	characteristic, complemented
48	C₄₁	C₅×#4380	1	1	yes	characteristic, complemented, Sylow
49	#157	—	205	1	no	Hall
50	D₁₅	—	246	2	no	—
51	C₅×D₃	—	1230	2	no	—
52	C₅²	—	41	1	no	Sylow
53	AGL(1,𝔽₅)	—	369	1	no	—
54	C₄×C₅	—	1845	1	no	—
55	C₃⋊D₃	—	205	1	no	—
56	C₃×C₅	—	2	2	no	—
57	C₃×C₅	—	82	2	no	—
58	C₃×C₅	—	164	4	no	—
59	C₂×C₅	—	1845	1	no	—
60	D₅	—	369	1	no	—
61	C₃²	#11717962⋊C₅	1	1	yes	characteristic, complemented, Sylow
62	D₃	—	1230	2	no	—
63	C₅	—	41	1	no	—
64	C₅	—	164	1	no	—
65	C₅	#13093287⋊C₅	1	1	yes	characteristic, complemented
66	C₄	—	1845	1	no	Sylow
67	C₃	—	2	2	no	—
68	C₂	—	1845	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅×C₄₁	C₄×C₅
prime radical	C₃⋊(C₃⋊(C₅⋊(D₄₁⋊C₅)))	C₂
cosocle	C₃⋊D₆₁₅	C₂×C₅
LCS	G ▷ C₃²×C₅×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₄₁⋊C₅	C₄	semidirect
3. #20301654	C₅	semidirect
11. C₃²×C₅×C₄₁	C₄×C₅	semidirect
13. C₃²×C₄₁⋊C₅	AGL(1,𝔽₅)	semidirect
17. C₅×C₄₁⋊C₅	#157	semidirect
28. C₃²×C₄₁	C₅×AGL(1,𝔽₅)	semidirect
31. C₅×C₄₁	C₅×#157	semidirect
33. C₄₁⋊C₅	#4380	semidirect
47. C₃²×C₅	#4223⋊C₅	semidirect
48. C₄₁	C₅×#4380	semidirect
61. C₃²	#11717962⋊C₅	semidirect
65. C₅	#13093287⋊C₅	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊(C₅⋊(D₄₁⋊C₅)))	2
#20301654	5
#13093287⋊C₅	5
#11717962⋊C₅	9
C₅×#4380	41

Subgroups — #20301654⋊C5 #49146943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #20301654⋊C₅ #49146943