Subgroups — #49147080

Order 36900.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(C₅⋊D₂₀₅))	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₄₁	C₄	1	1	yes	characteristic, complemented, Hall
3	#20301654	—	5	1	no	maximal
4	#20301653	—	5	1	no	maximal
5	C₃⋊(C₅⋊D₂₀₅)	—	6	2	no	—
6	#16861522	—	9	1	no	maximal, Hall
7	C₃⋊D₆₁₅	—	5	1	no	—
8	C₃⋊D₆₁₅	—	5	1	no	—
9	C₃⋊D₆₁₅	—	10	2	no	—
10	C₃×C₅²×C₄₁	—	2	2	no	—
11	C₅⋊D₂₀₅	—	9	1	no	—
12	C₃²×C₅×C₄₁	—	2	2	no	—
13	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
14	C₃²×C₅×C₄₁	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	#13093287	—	25	1	no	Hall
16	D₆₁₅	—	30	2	no	—
17	D₆₁₅	—	30	2	no	—
18	D₆₁₅	—	30	8	no	—
19	C₅²×C₄₁	#157	1	1	yes	characteristic, complemented, Hall
20	#11744708	—	41	1	no	maximal, Hall
21	#11717962	—	45	1	no	—
22	#11717961	—	45	1	no	—
23	C₃⋊D₁₂₃	—	25	1	no	—
24	C₃×C₅×C₄₁	—	2	8	no	—
25	C₃×C₅×C₄₁	—	2	2	no	—
26	C₃×C₅×C₄₁	—	2	2	no	—
27	C₃⋊(C₃⋊(C₅⋊D₅))	—	41	1	no	—
28	D₂₀₅	—	45	1	no	—
29	D₂₀₅	—	45	1	no	—
30	D₂₀₅	—	90	2	no	—
31	C₃²×C₄₁	#1044	1	1	yes	characteristic, complemented, Hall
32	D₁₂₃	—	150	2	no	—
33	C₃²×C₅²	#4223	1	1	yes	characteristic, complemented, Hall
34	C₅×C₄₁	—	2	2	no	—
35	C₅×C₄₁	#4380	1	1	yes	characteristic, complemented
36	C₅×C₄₁	#4380	1	1	yes	characteristic, complemented
37	#4380	—	205	1	no	—
38	#4380	—	205	1	no	—
39	#4223	—	225	1	no	Hall
40	C₃⋊(C₅⋊D₅)	—	246	2	no	—
41	C₃×C₄₁	—	2	2	no	—
42	#1044	—	369	1	no	Hall
43	C₃⋊D₁₅	—	410	2	no	—
44	C₃⋊D₁₅	—	205	1	no	—
45	C₃⋊D₁₅	—	205	1	no	—
46	D₄₁	—	225	1	no	—
47	C₃×C₅²	—	2	2	no	—
48	C₅⋊D₅	—	369	1	no	—
49	C₃²×C₅	—	2	2	no	—
50	C₃²×C₅	#11717961	1	1	yes	characteristic, complemented
51	C₃²×C₅	#11717962	1	1	yes	characteristic, complemented
52	C₄₁	#11744708	1	1	yes	characteristic, complemented, Sylow
53	#157	—	1025	1	no	Hall
54	D₁₅	—	1230	2	no	—
55	D₁₅	—	1230	2	no	—
56	D₁₅	—	1230	8	no	—
57	C₅²	#13093287	1	1	yes	characteristic, complemented, Sylow
58	AGL(1,𝔽₅)	—	1845	1	no	—
59	AGL(1,𝔽₅)	—	1845	1	no	—
60	C₃⋊D₃	—	1025	1	no	—
61	C₃×C₅	—	2	8	no	—
62	C₃×C₅	—	2	2	no	—
63	C₃×C₅	—	2	2	no	—
64	D₅	—	1845	1	no	—
65	D₅	—	1845	1	no	—
66	D₅	—	3690	2	no	—
67	C₃²	#16861522	1	1	yes	characteristic, complemented, Sylow
68	D₃	—	6150	2	no	—
69	C₅	—	2	2	no	—
70	C₅	#20301653	1	1	yes	characteristic, complemented
71	C₅	#20301654	1	1	yes	characteristic, complemented
72	C₄	—	9225	1	no	Sylow
73	C₃	—	2	2	no	—
74	C₂	—	9225	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅²×C₄₁	C₄
prime radical, cosocle	C₃⋊(C₃⋊(C₅⋊D₂₀₅))	C₂
LCS	G ▷ C₃²×C₅²×C₄₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₄₁	C₄	semidirect
13. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
14. C₃²×C₅×C₄₁	AGL(1,𝔽₅)	semidirect
19. C₅²×C₄₁	#157	semidirect
31. C₃²×C₄₁	#1044	semidirect
33. C₃²×C₅²	#4223	semidirect
35. C₅×C₄₁	#4380	semidirect
36. C₅×C₄₁	#4380	semidirect
50. C₃²×C₅	#11717961	semidirect
51. C₃²×C₅	#11717962	semidirect
52. C₄₁	#11744708	semidirect
57. C₅²	#13093287	semidirect
67. C₃²	#16861522	semidirect
70. C₅	#20301653	semidirect
71. C₅	#20301654	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⋊D₂₀₅))	2
#20301654	5
#20301653	5
#16861522	9
#11744708	41