Subgroups — #64903

Order 320.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4058	C₂	1	1	yes	maximal, characteristic
2	#3993	C₂	1	1	yes	maximal, characteristic, complemented
3	#3990	C₂	1	1	yes	maximal, characteristic, complemented
4	#711	—	2	1	no	—
5	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
6	C₅×D₈	C₄	1	1	yes	characteristic, complemented
7	#687	—	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	#360	—	5	1	no	maximal, Sylow
12	C₂×AGL(1,𝔽₅)	—	4	1	no	—
13	C₅×D₄	—	2	1	no	—
14	C₂²⋊D₅	—	4	1	no	—
15	C₂×Dic₅	—	4	1	no	—
16	C₄×D₅	D₄	1	1	yes	characteristic
17	Dic₁₀	—	2	1	no	—
18	C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	#168	D₄	1	1	yes	characteristic
20	#135	—	5	1	no	—
21	M₃₂	—	5	1	no	—
22	#106	—	5	1	no	—
23	C₂²×C₅	—	4	1	no	—
24	C₂×D₅	D₈	1	1	yes	characteristic
25	AGL(1,𝔽₅)	—	8	1	no	—
26	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	Dic₅	—	4	1	no	—
28	Dic₅	SD₁₆	1	1	yes	characteristic
29	#41	—	10	1	no	—
30	Q₁₆	—	5	1	no	—
31	SD₁₆	—	10	1	no	—
32	D₈	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	—	8	1	no	—
37	C₂×C₅	#102	1	1	yes	characteristic
38	D₅	—	2	1	no	—
39	Q₈	—	10	1	no	—
40	D₄	—	2	1	no	—
41	D₄	—	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	#360	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	4	1	no	—
49	C₂²	—	5	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	—	8	1	no	—
55	C₂	—	10	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×D₈	C₄
socle	C₂×C₅	#102
prime radical	#4058	C₂
cosocle	C₈×D₅	C₂²
hypercenter	D₈	AGL(1,𝔽₅)
hypocenter	C₅	#360
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
2. #3993	C₂	semidirect
3. #3990	C₂	semidirect
6. C₅×D₈	C₄	semidirect
8. Dic₂₀	C₄	semidirect
32. D₈	AGL(1,𝔽₅)	semidirect
47. C₅	#360	semidirect

Maximal subgroups

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

subgroup	index
#4058	2
#3993	2
#3990	2
#360	5