Subgroups — #14268305

Order 1800.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊#1039)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×Dic₁₅	C₂	1	2	yes	maximal
3	#10597124	—	3	2	no	maximal
4	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic
5	#67153	—	5	2	no	maximal
6	C₃⋊#1039	—	3	2	no	—
7	C₃⋊#1039	—	6	1	no	—
8	C₅×Dic₁₅	D₃	1	2	yes	—
9	C₃×C₅×Dic₅	—	3	2	no	—
10	C₃²×C₅²	Q₈	1	1	yes	characteristic, complemented, Hall
11	#6039	—	9	1	no	Hall
12	C₃⋊Dic₁₅	—	5	2	no	—
13	C₃⋊Dic₁₅	—	10	2	no	—
14	C₃×Dic₁₅	D₅	1	2	yes	—
15	C₃×C₅×Dic₃	—	5	2	no	—
16	C₂×C₃×C₅²	—	2	1	no	—
17	C₂×C₃×C₅²	C₂×D₃	1	2	yes	—
18	Dic₃₀	—	15	2	no	—
19	#1204	—	15	2	no	—
20	#1039	—	9	1	no	—
21	C₅×Dic₅	—	3	2	no	—
22	C₂×C₃²×C₅	—	2	2	no	—
23	C₂×C₃²×C₅	C₂×D₅	1	2	yes	—
24	C₃×C₅²	—	2	1	no	—
25	C₃×C₅²	Dic₆	1	2	yes	complemented
26	#627	—	25	1	no	Hall
27	C₃×C₄×C₅	—	15	2	no	—
28	Dic₁₅	—	15	2	no	—
29	Dic₁₅	—	15	2	no	—
30	Dic₁₅	—	30	4	no	—
31	Dic₁₅	—	30	2	no	—
32	Dic₁₅	—	30	4	no	—
33	Dic₁₅	D₁₅	1	2	yes	—
34	C₃×Dic₅	—	3	2	no	—
35	C₅×Dic₃	—	5	2	no	—
36	C₂×C₅²	D₃²	1	1	yes	characteristic
37	C₃²×C₅	—	2	2	no	—
38	C₃²×C₅	Dic₁₀	1	2	yes	complemented
39	Dic₁₀	—	45	2	no	—
40	C₃⋊Dic₃	—	25	1	no	—
41	C₃×Dic₃	—	5	2	no	—
42	C₂×C₃×C₅	—	2	4	no	—
43	C₂×C₃×C₅	—	2	2	no	—
44	C₂×C₃×C₅	—	2	4	no	—
45	C₂×C₃×C₅	D₃×D₅	1	2	yes	—
46	C₂×C₃×C₅	C₂×D₁₅	1	2	yes	—
47	C₅²	#627	1	1	yes	characteristic, complemented, Sylow
48	Dic₆	—	75	2	no	—
49	C₄×C₅	—	15	2	no	—
50	Dic₅	—	3	2	no	—
51	Dic₅	—	45	2	no	—
52	Dic₅	—	90	2	no	—
53	C₂×C₃²	D₅²	1	1	yes	characteristic
54	C₃×C₅	—	2	4	no	—
55	C₃×C₅	—	2	2	no	—
56	C₃×C₅	—	2	4	no	—
57	C₃×C₅	#1204	1	2	yes	complemented
58	C₃×C₅	Dic₃₀	1	2	yes	complemented
59	C₃×C₄	—	15	2	no	—
60	Dic₃	—	5	2	no	—
61	Dic₃	—	75	2	no	—
62	Dic₃	—	150	1	no	—
63	C₂×C₅	—	2	2	no	—
64	C₂×C₅	D₃×D₁₅	1	2	yes	—
65	C₃²	#6039	1	1	yes	characteristic, complemented, Sylow
66	Q₈	—	225	1	no	Sylow
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	D₅×D₁₅	1	2	yes	—
69	C₅	—	2	2	no	—
70	C₅	#67153	1	2	yes	complemented
71	C₄	—	15	2	no	—
72	C₄	—	225	1	no	—
73	C₃	—	2	1	no	—
74	C₃	#10597124	1	2	yes	complemented
75	C₂	D₁₅²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅²
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
10. C₃²×C₅²	Q₈	semidirect
25. C₃×C₅²	Dic₆	semidirect
38. C₃²×C₅	Dic₁₀	semidirect
47. C₅²	#627	semidirect
57. C₃×C₅	#1204	semidirect
58. C₃×C₅	Dic₃₀	semidirect
65. C₃²	#6039	semidirect
70. C₅	#67153	semidirect
74. C₃	#10597124	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊(C₃⋊#1039)	2
C₃×C₅×Dic₁₅	2
#10597124	3
#67153	5