Subgroups — #91159

Order 480.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#3993	—	3	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₁₆)	—	2	1	no	—
6	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
7	C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₃⋊#168	C₄	1	1	yes	characteristic, complemented
9	C₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
10	#808	—	5	1	no	maximal, Hall
11	C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
12	C₄⋊AGL(1,𝔽₅)	D₃	1	1	yes	characteristic
13	C₈×D₅	—	3	1	no	—
14	C₂×C₃×D₅	D₄	1	1	yes	characteristic
15	ΓL(1,𝔽₁₆)	—	4	1	no	—
16	C₃×AGL(1,𝔽₅)	—	4	1	no	—
17	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×Dic₅	Q₈	1	1	yes	characteristic
19	C₃×C₄⋊C₄	—	5	1	no	—
20	C₄⋊Dic₃	—	5	1	no	—
21	C₂×C₃⋊C₈	—	5	1	no	—
22	C₂×AGL(1,𝔽₅)	—	2	1	no	—
23	C₂×AGL(1,𝔽₅)	—	6	1	no	—
24	C₄×D₅	C₂×D₃	1	1	yes	characteristic
25	C₅×C₈	—	3	1	no	—
26	#168	—	3	1	no	—
27	#106	—	15	1	no	Sylow
28	C₂×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
29	C₃×D₅	SD₁₆	1	1	yes	characteristic
30	C₃×D₅	SD₁₆	1	1	yes	characteristic
31	C₂×C₃×C₄	—	5	1	no	—
32	C₂×C₃×C₄	—	10	1	no	—
33	C₂×Dic₃	—	10	1	no	—
34	C₃⋊C₈	—	5	1	no	—
35	C₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
36	C₂×D₅	#67	1	1	yes	characteristic
37	AGL(1,𝔽₅)	—	4	1	no	—
38	AGL(1,𝔽₅)	—	12	1	no	—
39	C₄×C₅	C₄×D₃	1	1	yes	characteristic
40	Dic₅	Dic₆	1	1	yes	characteristic
41	C₂×C₈	—	15	1	no	—
42	C₄⋊C₄	—	5	1	no	—
43	C₄⋊C₄	—	15	1	no	—
44	C₃×C₅	#106	1	1	yes	characteristic, complemented, Hall
45	C₂²×C₃	—	5	1	no	—
46	C₃×C₄	—	5	1	no	—
47	C₃×C₄	—	20	1	no	—
48	C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
49	Dic₃	—	20	1	no	—
50	C₂×C₅	Dic₃⋊C₄	1	1	yes	characteristic
51	D₅	#214	1	1	yes	characteristic
52	D₅	#215	1	1	yes	characteristic
53	C₂×C₄	—	5	1	no	—
54	C₂×C₄	—	10	1	no	—
55	C₂×C₄	—	30	1	no	—
56	C₈	—	3	1	no	—
57	C₈	—	15	1	no	—
58	C₂×C₃	—	5	1	no	—
59	C₂×C₃	—	5	1	no	—
60	C₂×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₅	#808	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	5	1	no	—
63	C₄	—	5	1	no	—
64	C₄	—	20	1	no	—
65	C₄	—	60	1	no	—
66	C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₃	#3993	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	5	1	no	—
69	C₂	—	5	1	no	—
70	C₂	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	Dic₃⋊AGL(1,𝔽₅)
commutator, Fitting	C₃×C₄×C₅	C₂×C₄
Frattini, hypercenter	C₄	D₃×AGL(1,𝔽₅)
socle	C₂×C₃×C₅	C₄⋊C₄
prime radical	C₂×C₃×D₅	D₄
cosocle	C₃×C₄×D₅	C₂²
hypocenter	C₃×C₅	#106
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
8. C₃⋊#168	C₄	semidirect
9. C₅×C₃⋊C₈	C₄	semidirect
35. C₃⋊C₈	AGL(1,𝔽₅)	semidirect
44. C₃×C₅	#106	semidirect
61. C₅	#808	semidirect
67. C₃	#3993	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊ΓL(1,𝔽₁₆)	2
C₃×C₄⋊AGL(1,𝔽₅)	2
D₅×C₃⋊C₈	2
#3993	3
#808	5