Subgroups — C₂×C₄×ΓL(1,𝔽₁₆) #91987

Order 480.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄	ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃×C₅	C₂×C₄²
Frattini	C₂	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₄×C₅	C₄
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²
cosocle	C₂×C₃×D₅	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄×ΓL(1,𝔽₁₆)	C₂	direct (N × H)
6. C₂×ΓL(1,𝔽₁₆)	C₄	direct (N × H)
8. C₂×C₃×C₄×C₅	C₄	semidirect
9. C₂×C₃×Dic₅	C₄	semidirect
10. C₃×C₄×D₅	C₄	semidirect
20. ΓL(1,𝔽₁₆)	C₂×C₄	direct (N × H)
21. C₃×C₄×C₅	C₂×C₄	semidirect
22. C₃×Dic₅	C₂×C₄	semidirect
29. C₂×C₄×C₅	Dic₃	semidirect
30. C₂×Dic₅	Dic₃	semidirect
31. C₄×D₅	Dic₃	semidirect
34. C₂×C₃×C₅	C₄²	semidirect
36. C₃×D₅	C₄²	semidirect
42. C₂×C₃×C₄	AGL(1,𝔽₅)	semidirect
50. C₄×C₅	C₂×Dic₃	semidirect
51. Dic₅	C₂×Dic₃	semidirect
55. C₃×C₅	C₂×C₄²	semidirect
61. C₃×C₄	C₂×AGL(1,𝔽₅)	semidirect
63. C₂×C₅	C₄×Dic₃	semidirect
65. D₅	C₄×Dic₃	semidirect
73. C₂×C₄	ΓL(1,𝔽₁₆)	direct (N × H)
76. C₂×C₃	C₄×AGL(1,𝔽₅)	semidirect
78. C₅	C₂×C₄×Dic₃	semidirect
85. C₄	C₂×ΓL(1,𝔽₁₆)	direct (N × H)
86. C₃	C₂×C₄×AGL(1,𝔽₅)	semidirect
89. C₂	C₄×ΓL(1,𝔽₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×ΓL(1,𝔽₁₆)	2
C₂×C₃×C₄×D₅	2
C₄×ΓL(1,𝔽₁₆)	2
C₂×C₄×AGL(1,𝔽₅)	3
C₂×C₄×Dic₃	5

Subgroups — C2×C4×ΓL(1,𝔽16) #91987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×ΓL(1,𝔽₁₆) #91987