Subgroups — #14268271

Order 1800.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₃⋊Dic₁₅	C₂	1	2	yes	maximal
2	C₃²×#1039	C₂	1	1	yes	maximal, characteristic
3	#10597129	—	3	4	no	maximal
4	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	#67141	—	5	2	no	maximal
6	C₅×Dic₁₅	—	3	8	no	—
7	C₃×#1039	D₃	1	4	yes	—
8	C₃²×C₅²	Q₈	1	1	yes	characteristic, complemented, Hall
9	#6039	—	9	1	no	Hall
10	C₃⋊Dic₁₅	D₅	1	2	yes	—
11	C₅×C₃⋊Dic₃	—	5	2	no	—
12	C₃²×Dic₅	—	5	2	no	—
13	C₃²×Dic₅	—	10	2	no	—
14	C₂×C₃×C₅²	C₂×D₃	1	4	yes	—
15	#1204	—	15	8	no	—
16	#1039	C₃⋊D₃	1	1	yes	characteristic
17	C₅×Dic₅	—	9	2	no	—
18	C₂×C₃²×C₅	—	2	2	no	—
19	C₂×C₃²×C₅	C₂×D₅	1	2	yes	—
20	C₃×C₅²	Dic₆	1	4	yes	complemented
21	#634	—	25	1	no	Hall
22	Dic₁₅	—	3	8	no	—
23	C₃×Dic₅	—	5	8	no	—
24	C₃×Dic₅	—	10	8	no	—
25	C₅×Dic₃	—	15	8	no	—
26	C₂×C₅²	C₂×C₃⋊D₃	1	1	yes	characteristic
27	C₃²×C₅	—	2	2	no	—
28	C₃²×C₅	Dic₁₀	1	2	yes	complemented
29	Dic₁₀	—	45	2	no	—
30	C₃²×C₄	—	25	1	no	—
31	C₃⋊Dic₃	—	5	2	no	—
32	C₂×C₃×C₅	—	2	8	no	—
33	C₂×C₃×C₅	D₃×D₅	1	8	yes	—
34	C₅²	#634	1	1	yes	characteristic, complemented, Sylow
35	Dic₆	—	75	4	no	—
36	C₄×C₅	—	45	2	no	—
37	Dic₅	—	5	2	no	—
38	Dic₅	—	9	2	no	—
39	Dic₅	—	10	2	no	—
40	C₂×C₃²	D₅²	1	1	yes	characteristic
41	C₃×C₅	—	2	8	no	—
42	C₃×C₅	#1204	1	8	yes	complemented
43	C₃×C₄	—	25	4	no	—
44	Dic₃	—	15	8	no	—
45	C₂×C₅	—	2	2	no	—
46	C₂×C₅	D₅×C₃⋊D₃	1	2	yes	—
47	C₃²	#6039	1	1	yes	characteristic, complemented, Sylow
48	Q₈	—	225	1	no	Sylow
49	C₂×C₃	D₁₅⋊D₅	1	4	yes	—
50	C₅	—	2	2	no	—
51	C₅	#67141	1	2	yes	complemented
52	C₄	—	25	1	no	—
53	C₄	—	45	2	no	—
54	C₃	#10597129	1	4	yes	complemented
55	C₂	#11744719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#11744719
commutator, Fitting, socle, prime radical, cosocle	C₂×C₃²×C₅²	C₂²
hypocenter	C₃²×C₅²	Q₈
LCS	G ▷ C₂×C₃²×C₅² ▷ C₃²×C₅²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
8. C₃²×C₅²	Q₈	semidirect
20. C₃×C₅²	Dic₆	semidirect
28. C₃²×C₅	Dic₁₀	semidirect
34. C₅²	#634	semidirect
42. C₃×C₅	#1204	semidirect
47. C₃²	#6039	semidirect
51. C₅	#67141	semidirect
54. C₃	#10597129	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₃⋊Dic₁₅	2
C₃²×#1039	2
#10597129	3
#67141	5