Subgroups — #10597114

Order 600.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊(C₅⋊D₅)	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅²	C₂²	1	1	yes	characteristic
6	C₅²×D₃	—	2	1	no	—
7	C₃×C₅⋊D₅	—	2	1	no	—
8	#1201	—	5	6	no	maximal
9	C₂²×C₅²	—	3	1	no	—
10	C₂×C₅⋊D₅	D₃	1	1	yes	characteristic, complemented
11	#1039	—	3	1	no	—
12	C₃×C₅²	D₄	1	1	yes	characteristic, complemented, Hall
13	C₂×C₅×D₃	D₅	1	6	yes	complemented
14	C₂×C₃×D₅	—	5	6	no	—
15	Dic₁₅	—	5	6	no	—
16	C₂×C₅²	—	6	1	no	—
17	C₂×C₅²	C₂×D₃	1	1	yes	characteristic
18	C₅⋊D₅	—	2	1	no	—
19	C₂²⋊D₅	—	15	6	no	—
20	C₂×C₃×C₅	C₂×D₅	1	6	yes	—
21	C₃×D₅	—	10	6	no	—
22	C₅×D₃	—	2	6	no	—
23	C₅²	#67	1	1	yes	characteristic, complemented, Sylow
24	#67	—	25	1	no	Hall
25	C₂²×C₅	—	3	6	no	—
26	C₂×D₅	—	5	6	no	—
27	Dic₅	—	15	6	no	—
28	C₃×C₅	C₂²⋊D₅	1	6	yes	complemented
29	C₂²×C₃	—	25	1	no	—
30	C₂×D₃	C₅⋊D₅	1	1	yes	characteristic, complemented
31	Dic₃	—	25	1	no	—
32	C₂×C₅	—	6	6	no	—
33	C₂×C₅	D₃×D₅	1	6	yes	—
34	D₅	—	10	6	no	—
35	D₄	—	75	1	no	Sylow
36	C₂×C₃	—	50	1	no	—
37	C₂×C₃	C₂×C₅⋊D₅	1	1	yes	characteristic
38	D₃	—	2	1	no	—
39	C₅	#1201	1	6	yes	complemented
40	C₂²	—	3	1	no	—
41	C₂²	—	25	1	no	—
42	C₄	—	75	1	no	—
43	C₃	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic, complemented, Sylow
44	C₂	—	6	1	no	—
45	C₂	—	50	1	no	—
46	C₂	D₃×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₅⋊D₅
commutator, Fitting, socle, cosocle	C₂×C₃×C₅²	C₂²
hypocenter	C₃×C₅²	D₄
LCS	G ▷ C₂×C₃×C₅² ▷ C₃×C₅²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×D₃	C₂	semidirect
2. C₂×C₃×C₅⋊D₅	C₂	semidirect
3. C₃⋊#1039	C₂	semidirect
10. C₂×C₅⋊D₅	D₃	semidirect
12. C₃×C₅²	D₄	semidirect
13. C₂×C₅×D₃	D₅	semidirect
23. C₅²	#67	semidirect
28. C₃×C₅	C₂²⋊D₅	semidirect
30. C₂×D₃	C₅⋊D₅	semidirect
39. C₅	#1201	semidirect
43. C₃	C₂²⋊(C₅⋊D₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×D₃	2
C₂×C₃×C₅⋊D₅	2
C₃⋊#1039	2
C₂²⋊(C₅⋊D₅)	3
#1201	5