Subgroups — #91221

Order 480.

80 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	D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	#3992	—	3	1	no	maximal, Hall
5	C₂×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
6	C₃×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₃⋊#170	C₄	1	2	yes	complemented
8	C₃×C₅×C₈	C₄	1	1	yes	characteristic, complemented
9	C₃×#168	C₄	1	1	yes	characteristic, complemented
10	#815	—	5	1	no	maximal, Hall
11	C₄×AGL(1,𝔽₅)	—	3	1	no	—
12	D₅⋊C₈	—	3	1	no	—
13	C₈×D₅	D₃	1	1	yes	characteristic
14	C₂×C₃×D₅	C₂×C₄	1	1	yes	characteristic
15	ΓL(1,𝔽₁₆)	—	2	2	no	—
16	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₈	—	5	1	no	—
19	C₄×Dic₃	—	5	1	no	—
20	C₂×C₃⋊C₈	—	5	1	no	—
21	C₂×AGL(1,𝔽₅)	—	3	2	no	—
22	C₄×D₅	C₂×D₃	1	1	yes	characteristic
23	#170	—	3	2	no	—
24	C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
25	#168	Dic₃	1	1	yes	characteristic, complemented
26	#97	—	15	1	no	Sylow
27	C₂×C₃×C₅	C₄²	1	1	yes	characteristic
28	C₃×D₅	M₁₆	1	2	yes	—
29	C₂×C₃×C₄	—	5	1	no	—
30	C₂×Dic₃	—	5	2	no	—
31	C₃×C₈	—	5	1	no	—
32	C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
33	C₃⋊C₈	—	5	2	no	—
34	C₂×D₅	C₄×D₃	1	1	yes	characteristic
35	AGL(1,𝔽₅)	—	6	2	no	—
36	C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
37	Dic₅	C₄×D₃	1	1	yes	characteristic
38	C₂×C₈	—	5	1	no	—
39	C₂×C₈	—	15	1	no	—
40	C₄²	—	15	1	no	—
41	C₃×C₅	#97	1	1	yes	characteristic, complemented, Hall
42	C₂²×C₃	—	5	1	no	—
43	C₃×C₄	—	5	1	no	—
44	C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
45	Dic₃	—	10	2	no	—
46	C₂×C₅	C₄×Dic₃	1	1	yes	characteristic
47	D₅	ΓL(1,𝔽₂₅)	1	2	yes	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	—	15	2	no	—
50	C₈	—	5	1	no	—
51	C₈	—	15	2	no	—
52	C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
53	C₂×C₃	—	5	2	no	—
54	C₂×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₅	#815	1	1	yes	characteristic, complemented, Sylow
56	C₂²	—	5	1	no	—
57	C₄	—	5	1	no	—
58	C₄	—	30	2	no	—
59	C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
60	C₃	#3992	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	5	2	no	—
62	C₂	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₄²
Fitting	C₃×C₅×C₈	C₄
prime radical	C₂×C₃×D₅	C₂×C₄
cosocle	C₃×C₄×D₅	C₂²
hypercenter	C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#97
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
7. C₃⋊#170	C₄	semidirect
8. C₃×C₅×C₈	C₄	semidirect
9. C₃×#168	C₄	semidirect
24. C₅×C₈	Dic₃	semidirect
25. #168	Dic₃	semidirect
32. C₃×C₈	AGL(1,𝔽₅)	semidirect
41. C₃×C₅	#97	semidirect
52. C₈	ΓL(1,𝔽₁₆)	semidirect
55. C₅	#815	semidirect
60. C₃	#3992	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
D₅⋊(C₃⋊C₈)	2
C₃×C₈×D₅	2
#3992	3
#815	5