Subgroups — #64908

Order 320.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#3994	C₂	1	1	yes	maximal, characteristic
3	D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	Q₈×D₅	—	2	1	no	—
5	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
6	C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
7	#689	—	2	1	no	—
8	Dic₂₀	C₄	1	1	yes	characteristic, complemented
9	C₈×D₅	C₂²	1	1	yes	characteristic
10	#674	—	2	1	no	—
11	#358	—	5	1	no	maximal, Sylow
12	C₂×AGL(1,𝔽₅)	—	4	1	no	—
13	C₅×Q₈	—	2	1	no	—
14	C₄×D₅	—	4	1	no	—
15	C₄×D₅	D₄	1	1	yes	characteristic
16	Dic₁₀	—	2	1	no	—
17	Dic₁₀	—	4	1	no	—
18	C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	#168	D₄	1	1	yes	characteristic
20	C₂×Q₁₆	—	5	1	no	—
21	C₂×C₁₆	—	5	1	no	—
22	#107	—	5	1	no	—
23	C₂×D₅	D₈	1	1	yes	characteristic
24	AGL(1,𝔽₅)	—	8	1	no	—
25	C₄×C₅	—	4	1	no	—
26	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	Dic₅	—	4	1	no	—
28	Dic₅	SD₁₆	1	1	yes	characteristic
29	C₂×Q₈	—	10	1	no	—
30	Q₁₆	—	5	1	no	—
31	Q₁₆	—	10	1	no	—
32	Q₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
33	C₂×C₈	—	5	1	no	—
34	C₄⋊C₄	—	10	1	no	—
35	C₁₆	—	10	1	no	—
36	C₂×C₅	#102	1	1	yes	characteristic
37	D₅	SD₃₂	1	1	yes	characteristic
38	D₅	Q₃₂	1	1	yes	characteristic
39	Q₈	—	2	1	no	—
40	Q₈	—	10	1	no	—
41	Q₈	—	20	1	no	—
42	C₂×C₄	—	5	1	no	—
43	C₂×C₄	—	20	1	no	—
44	C₂×C₄	—	20	1	no	—
45	C₈	—	5	1	no	—
46	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₅	#358	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	5	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	5	1	no	—
51	C₄	—	20	1	no	—
52	C₄	—	40	1	no	—
53	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂	—	5	1	no	—
55	C₂	—	5	1	no	—
56	C₂	#4007	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4007
commutator	C₅×C₈	C₂×C₄
Frattini	C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×Q₁₆	C₄
socle	C₂×C₅	#102
prime radical	C₂×D₅	D₈
cosocle	C₈×D₅	C₂²
hypercenter	Q₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#358
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
6. C₅×Q₁₆	C₄	semidirect
8. Dic₂₀	C₄	semidirect
32. Q₁₆	AGL(1,𝔽₅)	semidirect
47. C₅	#358	semidirect

Maximal subgroups

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

subgroup	index
D₅×Q₁₆	2
#3994	2
D₅⋊C₁₆	2
#358	5