Subgroups — A₄×ΓL(1,𝔽₁₆) #10624321

Order 720.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₅×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×ΓL(1,𝔽₁₆)	C₃	1	1	yes	maximal, characteristic, complemented
3	A₄×AGL(1,𝔽₅)	—	3	1	no	maximal
4	C₃×C₅×A₄	C₄	1	1	yes	characteristic, complemented
5	C₃×ΓL(1,𝔽₁₆)	—	4	1	no	maximal
6	Dic₃×A₄	—	5	1	no	maximal, Hall
7	C₂²×C₃×D₅	C₂×C₃	1	1	yes	characteristic
8	C₂×ΓL(1,𝔽₁₆)	—	3	1	no	—
9	C₂×ΓL(1,𝔽₁₆)	—	3	1	no	—
10	D₅×A₄	—	2	1	no	—
11	D₅×A₄	D₃	1	1	yes	characteristic
12	C₃²×D₅	—	4	1	no	—
13	C₂²×AGL(1,𝔽₅)	—	3	1	no	Hall
14	C₂×C₃×A₄	—	5	1	no	—
15	C₂²×C₃×C₅	C₃×C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×D₅	—	3	1	no	—
17	C₂×C₃×D₅	—	3	1	no	—
18	C₅×A₄	—	2	1	no	—
19	C₅×A₄	Dic₃	1	1	yes	characteristic, complemented
20	ΓL(1,𝔽₁₆)	—	3	1	no	—
21	ΓL(1,𝔽₁₆)	A₄	1	1	yes	characteristic, direct product factor
22	C₃×AGL(1,𝔽₅)	—	12	1	no	—
23	C₂²×Dic₃	—	5	1	no	—
24	C₄×A₄	—	15	1	no	—
25	C₃²×C₅	—	4	1	no	Hall
26	C₂²×D₅	C₃×D₃	1	1	yes	characteristic
27	C₂×AGL(1,𝔽₅)	—	9	1	no	—
28	C₂×AGL(1,𝔽₅)	—	9	1	no	—
29	C₃×A₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
30	C₃×Dic₃	—	20	1	no	—
31	C₂×C₃×C₅	—	3	1	no	—
32	C₃×D₅	—	3	1	no	—
33	C₃×D₅	—	4	1	no	—
34	C₃×D₅	—	8	1	no	—
35	C₃×D₅	C₂×A₄	1	1	yes	characteristic
36	C₂³×C₃	—	5	1	no	—
37	C₂×A₄	—	5	1	no	—
38	C₂×A₄	—	10	1	no	—
39	C₂×Dic₃	—	15	1	no	—
40	C₂×Dic₃	—	15	1	no	—
41	C₂²×C₅	C₃×Dic₃	1	1	yes	characteristic, complemented
42	C₂×D₅	—	3	1	no	—
43	C₂×D₅	—	3	1	no	—
44	AGL(1,𝔽₅)	—	3	1	no	—
45	AGL(1,𝔽₅)	—	9	1	no	—
46	C₂×C₃²	—	20	1	no	—
47	C₂²×C₄	—	15	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	—	15	1	no	—
52	C₂²×C₃	—	15	1	no	—
53	C₂²×C₃	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
54	A₄	—	2	1	no	—
55	A₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
56	C₃×C₄	—	60	1	no	—
57	Dic₃	—	5	1	no	—
58	Dic₃	—	15	1	no	—
59	C₂×C₅	—	3	1	no	—
60	D₅	—	3	1	no	—
61	D₅	D₃×A₄	1	1	yes	characteristic
62	C₃²	—	4	1	no	Sylow
63	C₂³	—	5	1	no	—
64	C₂×C₄	—	45	1	no	—
65	C₂×C₄	—	45	1	no	—
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	5	1	no	—
68	C₂×C₃	—	15	1	no	—
69	C₂×C₃	—	20	1	no	—
70	C₂×C₃	—	40	1	no	—
71	C₅	Dic₃×A₄	1	1	yes	characteristic, complemented, Sylow
72	C₂²	—	15	1	no	—
73	C₂²	—	15	1	no	—
74	C₂²	C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
75	C₄	—	15	1	no	—
76	C₄	—	45	1	no	—
77	C₃	—	4	1	no	—
78	C₃	—	8	1	no	—
79	C₃	A₄×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₂	—	3	1	no	—
81	C₂	—	5	1	no	—
82	C₂	—	15	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₂²×C₃×C₅	C₃×C₄
prime radical	C₃×D₅×A₄	C₂
cosocle	C₂²×C₃×D₅	C₂×C₃
LCS	G ▷ C₂²×C₃×C₅
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×ΓL(1,𝔽₁₆)	C₃	semidirect
4. C₃×C₅×A₄	C₄	semidirect
15. C₂²×C₃×C₅	C₃×C₄	semidirect
19. C₅×A₄	Dic₃	semidirect
21. ΓL(1,𝔽₁₆)	A₄	direct (N × H)
29. C₃×A₄	AGL(1,𝔽₅)	semidirect
41. C₂²×C₅	C₃×Dic₃	semidirect
50. C₃×C₅	C₄×A₄	semidirect
53. C₂²×C₃	C₃×AGL(1,𝔽₅)	semidirect
55. A₄	ΓL(1,𝔽₁₆)	direct (N × H)
71. C₅	Dic₃×A₄	semidirect
74. C₂²	C₃×ΓL(1,𝔽₁₆)	semidirect
79. C₃	A₄×AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×A₄	2
C₂²×ΓL(1,𝔽₁₆)	3
A₄×AGL(1,𝔽₅)	3
C₃×ΓL(1,𝔽₁₆)	4
Dic₃×A₄	5

Subgroups — A4×ΓL(1,𝔽16) #10624321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×ΓL(1,𝔽₁₆) #10624321