Subgroups — C₅×Dic₃×A₄ #10624074

Order 720.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×Dic₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×A₄	—	3	1	no	maximal
4	C₃×C₅×A₄	C₄	1	1	yes	characteristic, complemented
5	C₃×C₅×Dic₃	—	4	1	no	maximal
6	Dic₃×A₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₅	C₂×C₃	1	1	yes	characteristic
8	C₂×C₅×A₄	—	2	1	no	—
9	C₂×C₅×A₄	D₃	1	1	yes	characteristic
10	C₂×C₅×Dic₃	—	3	1	no	—
11	C₂×C₅×Dic₃	—	3	1	no	—
12	C₂×C₃²×C₅	—	4	1	no	—
13	C₂²×C₄×C₅	—	3	1	no	Hall
14	C₂×C₃×A₄	C₂×C₅	1	1	yes	characteristic
15	C₂²×C₃×C₅	—	3	1	no	—
16	C₂²×C₃×C₅	—	3	1	no	—
17	C₂²×C₃×C₅	C₃×C₄	1	1	yes	characteristic, complemented
18	C₅×A₄	—	2	1	no	—
19	C₅×A₄	Dic₃	1	1	yes	characteristic, direct product factor
20	C₃×C₄×C₅	—	12	1	no	—
21	C₅×Dic₃	—	3	1	no	—
22	C₅×Dic₃	A₄	1	1	yes	characteristic, direct product factor
23	C₂²×Dic₃	C₃×C₅	1	1	yes	characteristic, complemented
24	C₄×A₄	—	3	1	no	—
25	C₃²×C₅	—	4	1	no	Hall
26	C₂³×C₅	C₃×D₃	1	1	yes	characteristic
27	C₂×C₄×C₅	—	9	1	no	—
28	C₂×C₄×C₅	—	9	1	no	—
29	C₃×A₄	C₄×C₅	1	1	yes	characteristic, complemented
30	C₃×Dic₃	—	4	1	no	—
31	C₂×C₃×C₅	—	3	1	no	—
32	C₂×C₃×C₅	—	3	1	no	—
33	C₂×C₃×C₅	—	4	1	no	—
34	C₂×C₃×C₅	—	8	1	no	—
35	C₂×C₃×C₅	C₂×A₄	1	1	yes	characteristic
36	C₂³×C₃	C₂×C₃×C₅	1	1	yes	characteristic
37	C₂×A₄	—	2	1	no	—
38	C₂×A₄	C₅×D₃	1	1	yes	characteristic
39	C₂×Dic₃	—	3	1	no	—
40	C₂×Dic₃	—	3	1	no	—
41	C₂²×C₅	—	3	1	no	—
42	C₂²×C₅	—	3	1	no	—
43	C₂²×C₅	C₃×Dic₃	1	1	yes	characteristic, complemented
44	C₄×C₅	—	3	1	no	—
45	C₄×C₅	—	9	1	no	—
46	C₂×C₃²	—	4	1	no	—
47	C₂²×C₄	—	3	1	no	Sylow
48	C₃×C₅	—	4	1	no	—
49	C₃×C₅	—	8	1	no	—
50	C₃×C₅	C₄×A₄	1	1	yes	characteristic, complemented
51	C₂²×C₃	—	3	1	no	—
52	C₂²×C₃	—	3	1	no	—
53	C₂²×C₃	C₃×C₄×C₅	1	1	yes	characteristic, complemented
54	A₄	—	2	1	no	—
55	A₄	C₅×Dic₃	1	1	yes	characteristic, direct product factor
56	C₃×C₄	—	12	1	no	—
57	Dic₃	—	3	1	no	—
58	Dic₃	C₅×A₄	1	1	yes	characteristic, direct product factor
59	C₂×C₅	—	3	1	no	—
60	C₂×C₅	—	3	1	no	—
61	C₂×C₅	D₃×A₄	1	1	yes	characteristic
62	C₃²	—	4	1	no	Sylow
63	C₂³	C₃×C₅×D₃	1	1	yes	characteristic
64	C₂×C₄	—	9	1	no	—
65	C₂×C₄	—	9	1	no	—
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	4	1	no	—
69	C₂×C₃	—	8	1	no	—
70	C₂×C₃	C₂×C₅×A₄	1	1	yes	characteristic
71	C₅	Dic₃×A₄	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	—	3	1	no	—
73	C₂²	—	3	1	no	—
74	C₂²	C₃×C₅×Dic₃	1	1	yes	characteristic, complemented
75	C₄	—	3	1	no	—
76	C₄	—	9	1	no	—
77	C₃	—	4	1	no	—
78	C₃	—	8	1	no	—
79	C₃	C₄×C₅×A₄	1	1	yes	characteristic, complemented
80	C₂	—	3	1	no	—
81	C₂	—	3	1	no	—
82	C₂	C₅×D₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×A₄
commutator, hypocenter	C₂²×C₃	C₃×C₄×C₅
Frattini	C₂	C₅×D₃×A₄
Fitting, socle	C₂³×C₃×C₅	C₂×C₃
prime radical	C₂×C₃×C₅×A₄	C₂
cosocle	C₂³×C₃	C₂×C₃×C₅
LCS	G ▷ C₂²×C₃
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×Dic₃	C₃	semidirect
4. C₃×C₅×A₄	C₄	semidirect
6. Dic₃×A₄	C₅	direct (N × H)
17. C₂²×C₃×C₅	C₃×C₄	semidirect
19. C₅×A₄	Dic₃	direct (N × H)
22. C₅×Dic₃	A₄	direct (N × H)
23. C₂²×Dic₃	C₃×C₅	semidirect
29. C₃×A₄	C₄×C₅	semidirect
43. C₂²×C₅	C₃×Dic₃	semidirect
50. C₃×C₅	C₄×A₄	semidirect
53. C₂²×C₃	C₃×C₄×C₅	semidirect
55. A₄	C₅×Dic₃	direct (N × H)
58. Dic₃	C₅×A₄	direct (N × H)
71. C₅	Dic₃×A₄	direct (N × H)
74. C₂²	C₃×C₅×Dic₃	semidirect
79. C₃	C₄×C₅×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×A₄	2
C₂²×C₅×Dic₃	3
C₄×C₅×A₄	3
C₃×C₅×Dic₃	4
Dic₃×A₄	5

Subgroups — C5×Dic3×A4 #10624074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Dic₃×A₄ #10624074