Subgroups — C₃×#11717194 #72229677

Order 2400.

864 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊(C₅⋊D₅)	C₂	1	6	yes	maximal, complemented
2	C₃×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717194	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₅⋊D₅	C₂²	1	4	yes	—
5	C₂×C₃×C₄×C₅²	C₂²	1	3	yes	—
6	C₃×C₄⋊(C₅⋊D₅)	—	2	12	no	—
7	C₃×#4020	—	5	6	no	maximal
8	C₂×C₄⋊(C₅⋊D₅)	C₂×C₃	1	6	yes	complemented
9	C₄²×C₅²	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂²×C₃×C₅²	C₂³	1	1	yes	characteristic
11	C₂×C₃×C₅⋊D₅	—	2	12	no	—
12	C₃×C₄×C₅²	D₄	1	6	yes	complemented
13	C₂×C₃×D₂₀	—	5	36	no	—
14	C₃×C₄²×C₅	D₅	1	6	yes	complemented
15	C₂²×C₅⋊D₅	C₂²×C₃	1	4	yes	—
16	C₂×C₄×C₅²	C₂²×C₃	1	3	yes	—
17	C₄⋊(C₅⋊D₅)	—	2	12	no	—
18	#4020	—	5	6	no	—
19	C₂×C₃×C₅²	C₂×D₄	1	3	yes	—
20	C₃×C₅⋊D₅	—	4	4	no	—
21	C₂²×C₃×D₅	—	5	24	no	—
22	C₂×C₃×C₄×C₅	C₂×D₅	1	18	yes	—
23	C₃×D₂₀	—	10	72	no	—
24	C₂²×C₅²	C₂³×C₃	1	1	yes	characteristic
25	C₂×C₅⋊D₅	—	2	12	no	—
26	C₄×C₅²	C₃×D₄	1	6	yes	complemented
27	C₃×#127	—	25	1	no	Hall
28	C₂×D₂₀	—	5	36	no	—
29	C₄²×C₅	C₃×D₅	1	6	yes	complemented
30	C₃×C₅²	#127	1	1	yes	characteristic, complemented, Hall
31	C₂²×C₃×C₅	C₂²×D₅	1	6	yes	—
32	C₂×C₃×D₅	—	10	72	no	—
33	C₃×C₄×C₅	D₂₀	1	36	yes	complemented
34	C₂×C₅²	C₂×C₃×D₄	1	3	yes	—
35	C₅⋊D₅	—	4	4	no	—
36	C₂×C₃×D₄	—	25	6	no	—
37	C₃×C₄²	C₅⋊D₅	1	1	yes	characteristic, complemented
38	C₂²×D₅	—	5	24	no	—
39	C₂×C₄×C₅	C₂×C₃×D₅	1	18	yes	—
40	D₂₀	—	10	72	no	—
41	#127	—	25	1	no	Sylow
42	C₂×C₃×C₅	C₂×D₂₀	1	18	yes	—
43	C₃×D₅	—	20	24	no	—
44	C₅²	C₃×#127	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₃	—	25	4	no	—
46	C₃×D₄	—	50	12	no	—
47	C₂×C₃×C₄	C₂×C₅⋊D₅	1	3	yes	—
48	C₂²×C₅	C₂²×C₃×D₅	1	6	yes	—
49	C₂×D₅	—	10	72	no	—
50	C₄×C₅	C₃×D₂₀	1	36	yes	complemented
51	C₂×D₄	—	25	6	no	—
52	C₄²	C₃×C₅⋊D₅	1	1	yes	characteristic, complemented
53	C₃×C₅	#4020	1	6	yes	complemented
54	C₂²×C₃	—	50	12	no	—
55	C₂²×C₃	C₂²×C₅⋊D₅	1	1	yes	characteristic
56	C₃×C₄	C₄⋊(C₅⋊D₅)	1	6	yes	complemented
57	C₂×C₅	C₂×C₃×D₂₀	1	18	yes	—
58	D₅	—	20	24	no	—
59	C₂³	—	25	4	no	—
60	D₄	—	50	12	no	—
61	C₂×C₄	C₂×C₃×C₅⋊D₅	1	3	yes	—
62	C₂×C₃	—	100	4	no	—
63	C₂×C₃	C₂×C₄⋊(C₅⋊D₅)	1	3	yes	—
64	C₅	C₃×#4020	1	6	yes	complemented
65	C₂²	—	50	12	no	—
66	C₂²	C₂²×C₃×C₅⋊D₅	1	1	yes	characteristic
67	C₄	C₃×C₄⋊(C₅⋊D₅)	1	6	yes	complemented
68	C₃	#11717194	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	—	100	4	no	—
70	C₂	C₂×C₃×C₄⋊(C₅⋊D₅)	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×C₅⋊D₅
commutator, cosocle	C₂²×C₅²	C₂³×C₃
Frattini	C₂²	C₂²×C₃×C₅⋊D₅
Fitting	C₃×C₄²×C₅²	C₂
socle	C₂²×C₃×C₅²	C₂³
hypercenter	C₃×C₄²	C₅⋊D₅
hypocenter	C₅²	C₃×#127
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊(C₅⋊D₅)	C₂	semidirect
2. C₃×C₄²×C₅²	C₂	semidirect
3. #11717194	C₃	direct (N × H)
8. C₂×C₄⋊(C₅⋊D₅)	C₂×C₃	semidirect
9. C₄²×C₅²	C₂×C₃	semidirect
12. C₃×C₄×C₅²	D₄	semidirect
14. C₃×C₄²×C₅	D₅	semidirect
26. C₄×C₅²	C₃×D₄	semidirect
29. C₄²×C₅	C₃×D₅	semidirect
30. C₃×C₅²	#127	semidirect
33. C₃×C₄×C₅	D₂₀	semidirect
37. C₃×C₄²	C₅⋊D₅	semidirect
44. C₅²	C₃×#127	semidirect
50. C₄×C₅	C₃×D₂₀	semidirect
52. C₄²	C₃×C₅⋊D₅	semidirect
53. C₃×C₅	#4020	semidirect
56. C₃×C₄	C₄⋊(C₅⋊D₅)	semidirect
64. C₅	C₃×#4020	semidirect
67. C₄	C₃×C₄⋊(C₅⋊D₅)	semidirect
68. C₃	#11717194	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊(C₅⋊D₅)	2
C₃×C₄²×C₅²	2
#11717194	3
C₃×#4020	5

Subgroups — C3×#11717194 #72229677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#11717194 #72229677