Subgroups — C₅×D₄×A₄ #92051

Order 480.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×A₄	C₂	1	2	yes	maximal, complemented
2	C₄×C₅×A₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×D₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×C₅×A₄	—	2	2	no	—
5	C₂×C₅×A₄	C₂²	1	1	yes	characteristic
6	C₃×C₅×D₄	—	4	1	no	maximal
7	D₄×A₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂⁴×C₅	C₂×C₃	1	2	yes	complemented
9	C₂×C₅×D₄	—	3	1	no	—
10	C₂×C₅×D₄	—	3	1	no	—
11	C₂×C₅×D₄	—	3	2	no	—
12	C₂²×C₄×C₅	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₅	—	4	2	no	—
14	C₅×A₄	D₄	1	1	yes	characteristic, direct product factor
15	C₃×C₄×C₅	—	4	1	no	—
16	C₂²×A₄	C₂×C₅	1	2	yes	complemented
17	C₄×A₄	C₂×C₅	1	1	yes	characteristic, complemented
18	C₂³×C₅	—	2	2	no	—
19	C₂³×C₅	—	3	2	no	—
20	C₂³×C₅	—	3	2	no	—
21	C₂³×C₅	—	6	2	no	—
22	C₂³×C₅	C₂²×C₃	1	1	yes	characteristic
23	C₅×D₄	—	3	2	no	—
24	C₅×D₄	—	3	2	no	—
25	C₅×D₄	—	3	1	no	—
26	C₅×D₄	A₄	1	1	yes	characteristic, direct product factor
27	C₂×C₄×C₅	—	3	1	no	—
28	C₂×C₄×C₅	—	3	1	no	—
29	C₂²×D₄	C₃×C₅	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃×C₅	—	4	1	no	—
31	C₂×C₃×C₅	—	8	2	no	—
32	C₂×A₄	—	2	2	no	—
33	C₂×A₄	C₂²×C₅	1	1	yes	characteristic
34	C₃×D₄	—	4	1	no	—
35	C₂²×C₅	—	3	1	no	—
36	C₂²×C₅	—	3	1	no	—
37	C₂²×C₅	—	3	2	no	—
38	C₂²×C₅	—	6	2	no	—
39	C₂²×C₅	—	6	2	no	—
40	C₂²×C₅	—	6	2	no	—
41	C₂²×C₅	—	6	2	no	—
42	C₂²×C₅	C₃×D₄	1	1	yes	characteristic, complemented
43	C₂²×C₅	C₂×A₄	1	2	yes	complemented
44	C₄×C₅	—	3	1	no	—
45	C₄×C₅	C₂×A₄	1	1	yes	characteristic, complemented
46	C₂⁴	C₂×C₃×C₅	1	2	yes	complemented
47	C₂×D₄	—	3	1	no	—
48	C₂×D₄	—	3	1	no	—
49	C₂×D₄	—	3	2	no	—
50	C₂²×C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
51	C₃×C₅	—	4	1	no	Hall
52	C₂²×C₃	—	4	2	no	—
53	A₄	C₅×D₄	1	1	yes	characteristic, direct product factor
54	C₃×C₄	—	4	1	no	—
55	C₂×C₅	—	2	2	no	—
56	C₂×C₅	—	3	1	no	—
57	C₂×C₅	—	3	1	no	—
58	C₂×C₅	—	6	2	no	—
59	C₂×C₅	C₂²×A₄	1	1	yes	characteristic
60	C₂³	—	2	2	no	—
61	C₂³	—	3	2	no	—
62	C₂³	—	3	2	no	—
63	C₂³	—	6	2	no	—
64	C₂³	C₂²×C₃×C₅	1	1	yes	characteristic
65	D₄	—	3	2	no	—
66	D₄	—	3	2	no	—
67	D₄	—	3	1	no	—
68	D₄	C₅×A₄	1	1	yes	characteristic, direct product factor
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₃	—	4	1	no	—
72	C₂×C₃	—	8	2	no	—
73	C₅	D₄×A₄	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	3	1	no	—
75	C₂²	—	3	1	no	—
76	C₂²	—	3	2	no	—
77	C₂²	—	6	2	no	—
78	C₂²	—	6	2	no	—
79	C₂²	—	6	2	no	—
80	C₂²	—	6	2	no	—
81	C₂²	C₃×C₅×D₄	1	1	yes	characteristic, complemented
82	C₂²	C₂×C₅×A₄	1	2	yes	complemented
83	C₄	—	3	1	no	—
84	C₄	C₂×C₅×A₄	1	1	yes	characteristic, complemented
85	C₃	—	4	1	no	Sylow
86	C₂	—	2	2	no	—
87	C₂	—	3	1	no	—
88	C₂	—	3	1	no	—
89	C₂	—	6	2	no	—
90	C₂	C₂²×C₅×A₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×A₄
commutator, cosocle	C₂³	C₂²×C₃×C₅
Frattini	C₂	C₂²×C₅×A₄
Fitting	C₂²×C₅×D₄	C₃
socle	C₂³×C₅	C₂²×C₃
hypercenter	C₅×D₄	A₄
hypocenter	C₂²	C₃×C₅×D₄
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×A₄	C₂	semidirect
2. C₄×C₅×A₄	C₂	semidirect
3. C₂²×C₅×D₄	C₃	semidirect
7. D₄×A₄	C₅	direct (N × H)
8. C₂⁴×C₅	C₂×C₃	semidirect
12. C₂²×C₄×C₅	C₂×C₃	semidirect
14. C₅×A₄	D₄	direct (N × H)
16. C₂²×A₄	C₂×C₅	semidirect
17. C₄×A₄	C₂×C₅	semidirect
26. C₅×D₄	A₄	direct (N × H)
29. C₂²×D₄	C₃×C₅	semidirect
42. C₂²×C₅	C₃×D₄	semidirect
43. C₂²×C₅	C₂×A₄	semidirect
45. C₄×C₅	C₂×A₄	semidirect
46. C₂⁴	C₂×C₃×C₅	semidirect
50. C₂²×C₄	C₂×C₃×C₅	semidirect
53. A₄	C₅×D₄	direct (N × H)
68. D₄	C₅×A₄	direct (N × H)
73. C₅	D₄×A₄	direct (N × H)
81. C₂²	C₃×C₅×D₄	semidirect
82. C₂²	C₂×C₅×A₄	semidirect
84. C₄	C₂×C₅×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×A₄	2
C₄×C₅×A₄	2
C₂²×C₅×D₄	3
C₃×C₅×D₄	4
D₄×A₄	5

Subgroups — C5×D4×A4 #92051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₄×A₄ #92051