Subgroups — #14268270

Order 1800.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₃⋊D₁₅	C₂	1	2	yes	maximal, complemented
2	C₃²×#1039	C₂	1	1	yes	maximal, characteristic, complemented
3	#10597128	—	3	4	no	maximal
4	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	C₅×C₃⋊D₁₅	—	2	2	no	—
6	#67140	—	5	2	no	maximal
7	C₂×C₅×D₁₅	—	3	8	no	—
8	C₃×#1039	D₃	1	4	yes	complemented
9	C₃²×C₅²	D₄	1	1	yes	characteristic, complemented, Hall
10	#6037	—	9	1	no	Hall
11	C₂×C₃⋊D₁₅	D₅	1	2	yes	complemented
12	C₂×C₅×C₃⋊D₃	—	5	2	no	—
13	C₃²×Dic₅	—	5	2	no	—
14	C₃²×Dic₅	—	10	2	no	—
15	C₂×C₃×C₅²	C₂×D₃	1	4	yes	—
16	C₅×D₁₅	—	6	8	no	—
17	#1203	—	15	8	no	—
18	C₂×C₅×D₅	—	9	2	no	—
19	#1039	C₃⋊D₃	1	1	yes	characteristic, complemented
20	C₂×C₃²×C₅	—	2	2	no	—
21	C₂×C₃²×C₅	C₂×D₅	1	2	yes	—
22	C₃⋊D₁₅	—	2	2	no	—
23	C₅×C₃⋊D₃	—	10	2	no	—
24	C₃×C₅²	D₁₂	1	4	yes	complemented
25	C₄⋊(C₃⋊D₃)	—	25	1	no	Hall
26	C₂×D₁₅	—	3	8	no	—
27	C₂×C₅×D₃	—	15	8	no	—
28	C₃×Dic₅	—	5	8	no	—
29	C₃×Dic₅	—	10	8	no	—
30	C₂×C₅²	C₂×C₃⋊D₃	1	1	yes	characteristic
31	C₅×D₅	—	18	2	no	—
32	C₃²×C₅	—	2	2	no	—
33	C₃²×C₅	C₂²⋊D₅	1	2	yes	complemented
34	C₂²⋊D₅	—	45	2	no	—
35	C₂×C₃⋊D₃	—	5	2	no	—
36	C₃²×C₄	—	25	1	no	—
37	C₂×C₃×C₅	—	2	8	no	—
38	C₂×C₃×C₅	D₃×D₅	1	8	yes	—
39	D₁₅	—	6	8	no	—
40	C₅×D₃	—	30	8	no	—
41	C₅²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented, Sylow
42	D₁₂	—	75	4	no	—
43	C₂²×C₅	—	45	2	no	—
44	C₂×D₅	—	9	2	no	—
45	Dic₅	—	5	2	no	—
46	Dic₅	—	10	2	no	—
47	C₂×C₃²	D₅²	1	1	yes	characteristic
48	C₃⋊D₃	—	10	2	no	—
49	C₃×C₅	—	2	8	no	—
50	C₃×C₅	#1203	1	8	yes	complemented
51	C₂×D₃	—	15	8	no	—
52	C₃×C₄	—	25	4	no	—
53	C₂×C₅	—	2	2	no	—
54	C₂×C₅	—	90	2	no	—
55	C₂×C₅	D₅×C₃⋊D₃	1	2	yes	—
56	D₅	—	18	2	no	—
57	C₃²	#6037	1	1	yes	characteristic, complemented, Sylow
58	D₄	—	225	1	no	Sylow
59	C₂×C₃	D₁₅⋊D₅	1	4	yes	—
60	D₃	—	30	8	no	—
61	C₅	—	2	2	no	—
62	C₅	#67140	1	2	yes	complemented
63	C₂²	—	45	2	no	—
64	C₄	—	25	1	no	—
65	C₃	#10597128	1	4	yes	complemented
66	C₂	—	90	2	no	—
67	C₂	#11744719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#11744719
commutator, Fitting, socle, cosocle	C₂×C₃²×C₅²	C₂²
hypocenter	C₃²×C₅²	D₄
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
1. C₂×C₅×C₃⋊D₁₅	C₂	semidirect
2. C₃²×#1039	C₂	semidirect
8. C₃×#1039	D₃	semidirect
9. C₃²×C₅²	D₄	semidirect
11. C₂×C₃⋊D₁₅	D₅	semidirect
19. #1039	C₃⋊D₃	semidirect
24. C₃×C₅²	D₁₂	semidirect
33. C₃²×C₅	C₂²⋊D₅	semidirect
41. C₅²	C₄⋊(C₃⋊D₃)	semidirect
50. C₃×C₅	#1203	semidirect
57. C₃²	#6037	semidirect
62. C₅	#67140	semidirect
65. C₃	#10597128	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×C₃⋊D₁₅	2
C₃²×#1039	2
#10597128	3
#67140	5