Subgroups — C₄²×ΓL(1,𝔽₁₆) #11752344

Order 960.

516 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×ΓL(1,𝔽₁₆)	C₂	1	6	yes	maximal
2	C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	C₄²×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
4	C₂²×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
5	C₂×C₃×C₄×D₅	C₂²	1	3	yes	—
6	C₄×ΓL(1,𝔽₁₆)	C₄	1	24	yes	direct product factor
7	C₃×C₄²×C₅	C₄	1	1	yes	characteristic, complemented
8	C₃×C₄×Dic₅	C₄	1	3	yes	complemented
9	C₄²×Dic₃	—	5	1	no	maximal, Hall
10	C₂×C₄×AGL(1,𝔽₅)	—	3	6	no	—
11	C₄²×D₅	D₃	1	1	yes	characteristic
12	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
13	C₂×ΓL(1,𝔽₁₆)	C₂×C₄	1	24	yes	—
14	C₂×C₃×C₄×C₅	C₂×C₄	1	3	yes	—
15	C₂×C₃×Dic₅	C₂×C₄	1	3	yes	—
16	C₃×C₄×D₅	C₂×C₄	1	12	yes	—
17	C₂×C₃×C₄²	—	5	1	no	—
18	C₂×C₄×Dic₃	—	5	6	no	—
19	C₂²×AGL(1,𝔽₅)	—	3	4	no	—
20	C₂×C₄×D₅	C₂×D₃	1	3	yes	—
21	C₄×AGL(1,𝔽₅)	—	3	24	no	—
22	C₄²×C₅	Dic₃	1	1	yes	characteristic, complemented
23	C₄×Dic₅	Dic₃	1	3	yes	complemented
24	C₄³	—	15	1	no	Sylow
25	C₂²×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₃×D₅	C₂²×C₄	1	6	yes	—
27	ΓL(1,𝔽₁₆)	C₄²	1	16	yes	direct product factor
28	C₃×C₄×C₅	C₄²	1	6	yes	complemented
29	C₃×Dic₅	C₄²	1	6	yes	complemented
30	C₂²×C₃×C₄	—	5	3	no	—
31	C₂²×Dic₃	—	5	4	no	—
32	C₃×C₄²	—	5	3	no	—
33	C₃×C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
34	C₄×Dic₃	—	5	24	no	—
35	C₂²×D₅	C₂²×D₃	1	1	yes	characteristic
36	C₂×AGL(1,𝔽₅)	—	3	24	no	—
37	C₂×C₄×C₅	C₂×Dic₃	1	3	yes	—
38	C₂×Dic₅	C₂×Dic₃	1	3	yes	—
39	C₄×D₅	C₄×D₃	1	12	yes	—
40	C₂×C₄²	—	5	1	no	—
41	C₂×C₄²	—	15	6	no	—
42	C₂×C₃×C₅	C₂×C₄²	1	3	yes	—
43	C₃×D₅	C₂×C₄²	1	4	yes	—
44	C₂³×C₃	—	5	1	no	—
45	C₂×C₃×C₄	—	5	3	no	—
46	C₂×C₃×C₄	—	5	12	no	—
47	C₂×C₃×C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
48	C₂×Dic₃	—	5	24	no	—
49	C₂²×C₅	C₂²×Dic₃	1	1	yes	characteristic
50	C₂×D₅	C₂×C₄×D₃	1	6	yes	—
51	AGL(1,𝔽₅)	—	3	16	no	—
52	C₄×C₅	C₄×Dic₃	1	6	yes	complemented
53	Dic₅	C₄×Dic₃	1	6	yes	complemented
54	C₂²×C₄	—	5	3	no	—
55	C₂²×C₄	—	15	4	no	—
56	C₄²	—	5	3	no	—
57	C₄²	—	15	24	no	—
58	C₄²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
59	C₃×C₅	C₄³	1	1	yes	characteristic, complemented, Hall
60	C₂²×C₃	—	5	6	no	—
61	C₂²×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₃×C₄	—	5	6	no	—
63	C₃×C₄	C₄×AGL(1,𝔽₅)	1	6	yes	complemented
64	Dic₃	—	5	16	no	—
65	C₂×C₅	C₂×C₄×Dic₃	1	3	yes	—
66	D₅	C₄²×D₃	1	4	yes	—
67	C₂³	—	5	1	no	—
68	C₂×C₄	—	5	3	no	—
69	C₂×C₄	—	5	12	no	—
70	C₂×C₄	—	15	24	no	—
71	C₂×C₄	C₂×ΓL(1,𝔽₁₆)	1	3	yes	—
72	C₂×C₃	—	5	4	no	—
73	C₂×C₃	C₂×C₄×AGL(1,𝔽₅)	1	3	yes	—
74	C₅	C₄²×Dic₃	1	1	yes	characteristic, complemented, Sylow
75	C₂²	—	5	6	no	—
76	C₂²	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
77	C₄	—	5	6	no	—
78	C₄	—	15	16	no	—
79	C₄	C₄×ΓL(1,𝔽₁₆)	1	6	yes	direct product factor
80	C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
81	C₂	—	5	4	no	—
82	C₂	C₂×C₄×ΓL(1,𝔽₁₆)	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₄³
Frattini	C₂²	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄²×C₅	C₄
socle	C₂²×C₃×C₅	C₂²×C₄
prime radical, cosocle	C₂²×C₃×D₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. C₄×ΓL(1,𝔽₁₆)	C₄	direct (N × H)
7. C₃×C₄²×C₅	C₄	semidirect
8. C₃×C₄×Dic₅	C₄	semidirect
22. C₄²×C₅	Dic₃	semidirect
23. C₄×Dic₅	Dic₃	semidirect
27. ΓL(1,𝔽₁₆)	C₄²	direct (N × H)
28. C₃×C₄×C₅	C₄²	semidirect
29. C₃×Dic₅	C₄²	semidirect
33. C₃×C₄²	AGL(1,𝔽₅)	semidirect
52. C₄×C₅	C₄×Dic₃	semidirect
53. Dic₅	C₄×Dic₃	semidirect
58. C₄²	ΓL(1,𝔽₁₆)	direct (N × H)
59. C₃×C₅	C₄³	semidirect
63. C₃×C₄	C₄×AGL(1,𝔽₅)	semidirect
74. C₅	C₄²×Dic₃	semidirect
79. C₄	C₄×ΓL(1,𝔽₁₆)	direct (N × H)
80. C₃	C₄²×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
C₂×C₄×ΓL(1,𝔽₁₆)	2
C₃×C₄²×D₅	2
C₄²×AGL(1,𝔽₅)	3
C₄²×Dic₃	5

Subgroups — C42×ΓL(1,𝔽16) #11752344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×ΓL(1,𝔽₁₆) #11752344