Subgroups — C₃×AΓL(1,𝔽₁₆) #72241551

Order 2880.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#66297	—	3	1	no	maximal
3	AΓL(1,𝔽₁₆)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×AGL(1,𝔽₁₆)	C₄	1	1	yes	characteristic, complemented
5	C₃×#4625	—	5	1	no	maximal, Hall
6	C₃×(C₂⁴⋊C₅)⋊C₂	D₃	1	1	yes	characteristic
7	((C₂⁴⋊C₅)⋊C₂)⋊C₃	—	2	1	no	—
8	((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂×C₃	1	1	yes	characteristic
9	#66297	—	3	1	no	Hall
10	C₃×#120⋊C₃	—	5	1	no	—
11	C₃×C₂⁴⋊C₅	Dic₃	1	1	yes	characteristic, complemented
12	AGL(1,𝔽₁₆)	—	2	1	no	—
13	AGL(1,𝔽₁₆)	C₃×C₄	1	1	yes	characteristic, complemented
14	#4625	—	5	1	no	—
15	C₃×AΣL(1,𝔽₁₆)	—	15	1	no	—
16	C₃×ΓL(1,𝔽₁₆)	—	16	1	no	maximal
17	(C₂⁴⋊C₅)⋊C₂	C₃×D₃	1	1	yes	characteristic
18	C₃×C₂²⋊A₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
19	C₃×SL(2,ℤ/4)	—	20	1	no	—
20	C₃×#120	—	5	1	no	—
21	#120⋊C₃	—	5	1	no	—
22	#120⋊C₃	—	10	1	no	—
23	C₃×#100	—	15	1	no	—
24	C₃×#99	—	15	1	no	—
25	C₃²×D₅	—	16	1	no	—
26	C₂⁴⋊C₅	C₃×Dic₃	1	1	yes	characteristic, complemented
27	C₂×C₃×A₄	—	20	1	no	—
28	AΣL(1,𝔽₁₆)	—	15	1	no	Sylow
29	ΓL(1,𝔽₁₆)	—	16	1	no	—
30	C₃×AGL(1,𝔽₅)	—	48	1	no	—
31	C₂⁴×C₃	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
32	C₂²⋊A₄	—	2	1	no	—
33	C₂²⋊A₄	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
34	C₂×C₃×D₄	—	15	1	no	—
35	SL(2,ℤ/4)	—	20	1	no	—
36	C₃×M₁₆	—	30	1	no	—
37	C₃×C₂²⋊C₄	—	15	1	no	—
38	C₃×C₂²⋊C₄	—	30	1	no	—
39	C₃²×C₅	—	16	1	no	Hall
40	C₃×A₄	—	20	1	no	—
41	C₃×Dic₃	—	80	1	no	—
42	#120	—	5	1	no	—
43	#100	—	15	1	no	—
44	#99	—	15	1	no	—
45	C₃×D₅	—	16	1	no	—
46	C₃×D₅	—	16	1	no	—
47	C₃×D₅	—	32	1	no	—
48	C₂³×C₃	—	5	1	no	—
49	C₂³×C₃	—	15	1	no	—
50	C₂×A₄	—	20	1	no	—
51	C₂×A₄	—	40	1	no	—
52	C₃×D₄	—	60	1	no	—
53	C₂×C₃×C₄	—	15	1	no	—
54	C₂×C₃×C₄	—	60	1	no	—
55	C₃×C₈	—	60	1	no	—
56	AGL(1,𝔽₅)	—	48	1	no	—
57	C₂×C₃²	—	80	1	no	—
58	C₂⁴	C₃×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
59	C₂×D₄	—	15	1	no	—
60	M₁₆	—	30	1	no	—
61	C₂²⋊C₄	—	15	1	no	—
62	C₂²⋊C₄	—	30	1	no	—
63	C₃×C₅	—	16	1	no	—
64	C₃×C₅	—	16	1	no	—
65	C₃×C₅	—	32	1	no	—
66	C₂²×C₃	—	5	1	no	—
67	C₂²×C₃	—	30	1	no	—
68	C₂²×C₃	—	30	1	no	—
69	A₄	—	20	1	no	—
70	A₄	—	40	1	no	—
71	C₃×C₄	—	30	1	no	—
72	C₃×C₄	—	120	1	no	—
73	Dic₃	—	80	1	no	—
74	D₅	—	16	1	no	—
75	C₃²	—	16	1	no	Sylow
76	C₂³	—	5	1	no	—
77	C₂³	—	15	1	no	—
78	D₄	—	60	1	no	—
79	C₂×C₄	—	15	1	no	—
80	C₂×C₄	—	60	1	no	—
81	C₈	—	60	1	no	—
82	C₂×C₃	—	15	1	no	—
83	C₂×C₃	—	20	1	no	—
84	C₂×C₃	—	80	1	no	—
85	C₂×C₃	—	160	1	no	—
86	C₅	—	16	1	no	Sylow
87	C₂²	—	5	1	no	—
88	C₂²	—	30	1	no	—
89	C₂²	—	30	1	no	—
90	C₄	—	30	1	no	—
91	C₄	—	120	1	no	—
92	C₃	—	16	1	no	—
93	C₃	—	32	1	no	—
94	C₃	AΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
95	C₂	—	15	1	no	—
96	C₂	—	20	1	no	—

	subgroup	G/H
center, hypercenter	C₃	AΓL(1,𝔽₁₆)
commutator, hypocenter	AGL(1,𝔽₁₆)	C₃×C₄
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃	ΓL(1,𝔽₁₆)
prime radical	C₃×((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂
cosocle	((C₂⁴⋊C₅)⋊C₂)⋊C₃	C₂×C₃
LCS	G ▷ AGL(1,𝔽₁₆)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
3. AΓL(1,𝔽₁₆)	C₃	direct (N × H)
4. C₃×AGL(1,𝔽₁₆)	C₄	semidirect
11. C₃×C₂⁴⋊C₅	Dic₃	semidirect
13. AGL(1,𝔽₁₆)	C₃×C₄	semidirect
18. C₃×C₂²⋊A₄	AGL(1,𝔽₅)	semidirect
26. C₂⁴⋊C₅	C₃×Dic₃	semidirect
31. C₂⁴×C₃	ΓL(1,𝔽₁₆)	semidirect
33. C₂²⋊A₄	C₃×AGL(1,𝔽₅)	semidirect
58. C₂⁴	C₃×ΓL(1,𝔽₁₆)	semidirect
94. C₃	AΓL(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
C₃×((C₂⁴⋊C₅)⋊C₂)⋊C₃	2
C₃×#66297	3
AΓL(1,𝔽₁₆)	3
C₃×#4625	5
C₃×ΓL(1,𝔽₁₆)	16

Subgroups — C3×AΓL(1,𝔽16) #72241551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×AΓL(1,𝔽₁₆) #72241551