Subgroups — #64678

Order 320.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#168	C₂	1	2	yes	maximal
3	C₂×C₅×C₈	C₄	1	1	yes	characteristic
4	C₂×C₅×C₈	C₄	1	1	yes	characteristic
5	C₄²×C₅	C₂²	1	1	yes	characteristic
6	C₂×#168	—	2	2	no	—
7	#335	—	5	1	no	maximal, Sylow
8	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₅	D₄	1	1	yes	characteristic
10	C₂×C₄×C₅	Q₈	1	1	yes	characteristic
11	C₅×C₈	—	2	1	no	—
12	C₅×C₈	C₈	1	2	yes	complemented
13	#168	—	4	2	no	—
14	C₄⋊C₈	—	5	2	no	—
15	C₄×C₈	D₅	1	1	yes	characteristic
16	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
17	C₄×C₅	—	2	1	no	—
18	C₄×C₅	C₂×C₈	1	1	yes	characteristic
19	C₄×C₅	M₁₆	1	1	yes	characteristic
20	C₄×C₅	D₈	1	1	yes	characteristic
21	C₄×C₅	Q₁₆	1	1	yes	characteristic
22	C₂×C₈	—	10	2	no	—
23	C₂×C₈	Dic₅	1	1	yes	characteristic
24	C₂×C₈	Dic₅	1	1	yes	characteristic
25	C₄²	C₂×D₅	1	1	yes	characteristic
26	C₂×C₅	C₄⋊C₈	1	1	yes	characteristic
27	C₂×C₅	#107	1	1	yes	characteristic
28	C₂×C₅	#108	1	1	yes	characteristic
29	C₂×C₄	Dic₁₀	1	1	yes	characteristic
30	C₂×C₄	D₂₀	1	1	yes	characteristic
31	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	20	2	no	—
34	C₈	#168	1	2	yes	complemented
35	C₅	#335	1	1	yes	characteristic, complemented, Sylow
36	C₂²	C₄⋊Dic₅	1	1	yes	characteristic
37	C₄	—	2	1	no	—
38	C₄	D₄₀	1	1	yes	characteristic
39	C₄	Dic₂₀	1	1	yes	characteristic
40	C₄	C₂×#168	1	1	yes	characteristic
41	C₄	#681	1	1	yes	characteristic
42	C₂	C₄⋊#168	1	1	yes	characteristic
43	C₂	#3950	1	1	yes	characteristic
44	C₂	#3951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₀
commutator	C₄×C₅	C₂×C₈
Frattini	C₄²	C₂×D₅
Fitting	C₄×C₅×C₈	C₂
socle, prime radical	C₂²×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂²
hypercenter	C₄×C₈	D₅
hypocenter	C₅	#335
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₅×C₈	C₈	semidirect
34. C₈	#168	semidirect
35. C₅	#335	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×C₈	2
C₄⋊#168	2
#335	5