Subgroups — C₂×#11744719 #14268536

Order 1800.

444 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	C₂×C₃²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#11744719	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₁₅⋊D₅	—	3	4	no	maximal
5	C₂×C₃²×C₅²	C₂²	1	1	yes	characteristic, complemented
6	C₅×C₃⋊D₁₅	C₂²	1	4	yes	complemented
7	C₃²×C₅⋊D₅	C₂²	1	2	yes	complemented
8	C₂×D₅×C₃⋊D₃	—	5	2	no	maximal
9	C₂×C₅×D₁₅	—	3	8	no	—
10	C₂×C₃×C₅⋊D₅	D₃	1	4	yes	complemented
11	D₁₅⋊D₅	—	3	16	no	—
12	C₃²×C₅²	C₂³	1	1	yes	characteristic, complemented, Hall
13	C₂×D₅²	—	9	1	no	Hall
14	C₂×C₃⋊D₁₅	D₅	1	2	yes	complemented
15	C₂×C₅×C₃⋊D₃	—	5	2	no	—
16	C₂×C₃²×D₅	—	5	2	no	—
17	C₂×C₃²×D₅	—	10	2	no	—
18	D₅×C₃⋊D₃	—	5	8	no	—
19	C₂×C₃×C₅²	C₂×D₃	1	4	yes	complemented
20	C₅×D₁₅	—	3	16	no	—
21	C₃×C₅⋊D₅	C₂×D₃	1	8	yes	complemented
22	C₂×D₃×D₅	—	15	8	no	—
23	C₂×C₅⋊D₅	C₃⋊D₃	1	1	yes	characteristic, complemented
24	C₂×C₅×D₅	—	9	2	no	—
25	D₅²	—	9	4	no	—
26	C₂×C₃²×C₅	—	2	2	no	—
27	C₂×C₃²×C₅	C₂×D₅	1	2	yes	complemented
28	C₃⋊D₁₅	C₂×D₅	1	4	yes	complemented
29	C₅×C₃⋊D₃	—	5	4	no	—
30	C₃²×D₅	—	5	4	no	—
31	C₃²×D₅	—	10	4	no	—
32	C₃×C₅²	C₂²×D₃	1	4	yes	complemented
33	C₂²×C₃⋊D₃	—	25	1	no	Hall
34	C₂×D₁₅	—	3	8	no	—
35	C₂×C₅×D₃	—	15	8	no	—
36	C₂×C₃×D₅	—	5	8	no	—
37	C₂×C₃×D₅	—	10	8	no	—
38	D₃×D₅	—	15	32	no	—
39	C₂×C₅²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
40	C₅⋊D₅	C₂×C₃⋊D₃	1	2	yes	complemented
41	C₅×D₅	—	9	4	no	—
42	C₃²×C₅	—	2	2	no	—
43	C₃²×C₅	C₂²×D₅	1	2	yes	complemented
44	C₂²×D₅	—	45	2	no	—
45	C₂²×C₃²	—	25	1	no	—
46	C₂×C₃⋊D₃	—	5	2	no	—
47	C₂×C₃⋊D₃	—	25	4	no	—
48	C₂×C₃×C₅	—	2	8	no	—
49	C₂×C₃×C₅	D₃×D₅	1	8	yes	complemented
50	D₁₅	—	3	16	no	—
51	C₃×D₅	—	5	16	no	—
52	C₃×D₅	—	10	16	no	—
53	C₅×D₃	—	15	16	no	—
54	C₅²	C₂²×C₃⋊D₃	1	1	yes	characteristic, complemented, Sylow
55	C₂²×D₃	—	75	4	no	—
56	C₂²×C₅	—	45	2	no	—
57	C₂×D₅	—	5	2	no	—
58	C₂×D₅	—	9	2	no	—
59	C₂×D₅	—	10	2	no	—
60	C₂×D₅	—	45	8	no	—
61	C₂×C₃²	—	25	2	no	—
62	C₂×C₃²	D₅²	1	1	yes	characteristic, complemented
63	C₃⋊D₃	—	5	4	no	—
64	C₃×C₅	—	2	8	no	—
65	C₃×C₅	C₂×D₃×D₅	1	8	yes	complemented
66	C₂²×C₃	—	25	4	no	—
67	C₂×D₃	—	15	8	no	—
68	C₂×D₃	—	75	16	no	—
69	C₂×C₅	—	2	2	no	—
70	C₂×C₅	—	45	4	no	—
71	C₂×C₅	D₅×C₃⋊D₃	1	2	yes	complemented
72	D₅	—	5	4	no	—
73	D₅	—	9	4	no	—
74	D₅	—	10	4	no	—
75	C₃²	C₂×D₅²	1	1	yes	characteristic, complemented, Sylow
76	C₂³	—	225	1	no	Sylow
77	C₂×C₃	—	25	8	no	—
78	C₂×C₃	D₁₅⋊D₅	1	4	yes	complemented
79	D₃	—	15	16	no	—
80	C₅	—	2	2	no	—
81	C₅	C₂×D₅×C₃⋊D₃	1	2	yes	complemented
82	C₂²	—	25	1	no	—
83	C₂²	—	45	2	no	—
84	C₂²	—	225	4	no	—
85	C₃	C₂×D₁₅⋊D₅	1	4	yes	complemented
86	C₂	—	25	2	no	—
87	C₂	—	45	4	no	—
88	C₂	#11744719	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#11744719
commutator, cosocle, hypocenter	C₃²×C₅²	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₅²	C₂²
LCS	G ▷ 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. C₂×C₃²×C₅⋊D₅	C₂	semidirect
3. #11744719	C₂	direct (N × H)
5. C₂×C₃²×C₅²	C₂²	semidirect
6. C₅×C₃⋊D₁₅	C₂²	semidirect
7. C₃²×C₅⋊D₅	C₂²	semidirect
10. C₂×C₃×C₅⋊D₅	D₃	semidirect
12. C₃²×C₅²	C₂³	semidirect
14. C₂×C₃⋊D₁₅	D₅	semidirect
19. C₂×C₃×C₅²	C₂×D₃	semidirect
21. C₃×C₅⋊D₅	C₂×D₃	semidirect
23. C₂×C₅⋊D₅	C₃⋊D₃	semidirect
27. C₂×C₃²×C₅	C₂×D₅	semidirect
28. C₃⋊D₁₅	C₂×D₅	semidirect
32. C₃×C₅²	C₂²×D₃	semidirect
39. C₂×C₅²	C₂×C₃⋊D₃	semidirect
40. C₅⋊D₅	C₂×C₃⋊D₃	semidirect
43. C₃²×C₅	C₂²×D₅	semidirect
49. C₂×C₃×C₅	D₃×D₅	semidirect
54. C₅²	C₂²×C₃⋊D₃	semidirect
62. C₂×C₃²	D₅²	semidirect
65. C₃×C₅	C₂×D₃×D₅	semidirect
71. C₂×C₅	D₅×C₃⋊D₃	semidirect
75. C₃²	C₂×D₅²	semidirect
78. C₂×C₃	D₁₅⋊D₅	semidirect
81. C₅	C₂×D₅×C₃⋊D₃	semidirect
85. C₃	C₂×D₁₅⋊D₅	semidirect
88. C₂	#11744719	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×C₅×C₃⋊D₁₅	2
C₂×C₃²×C₅⋊D₅	2
#11744719	2
C₂×D₁₅⋊D₅	3
C₂×D₅×C₃⋊D₃	5

Subgroups — C2×#11744719 #14268536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11744719 #14268536