Subgroups — #64730

Order 320.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#3951	C₂	1	1	yes	maximal, characteristic, complemented
4	#709	—	2	1	no	—
5	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
6	#681	—	2	1	no	—
7	Dic₂₀	C₄	1	1	yes	characteristic
8	D₄₀	C₄	1	1	yes	characteristic
9	#677	—	2	1	no	—
10	C₅×C₁₆	—	2	1	no	—
11	#359	—	5	1	no	maximal, Sylow
12	C₂×C₄×C₅	D₄	1	1	yes	characteristic
13	C₂²⋊D₅	—	4	1	no	—
14	D₂₀	—	2	1	no	—
15	C₄×D₅	—	4	1	no	—
16	Dic₁₀	—	2	1	no	—
17	C₅×C₈	C₂×C₄	1	1	yes	characteristic
18	C₅×C₈	D₄	1	1	yes	characteristic
19	#168	—	4	1	no	—
20	#135	—	5	1	no	—
21	C₂×C₁₆	D₅	1	1	yes	characteristic, complemented
22	#108	—	5	1	no	—
23	C₂²×C₅	SD₁₆	1	1	yes	characteristic
24	C₂×D₅	—	4	1	no	—
25	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅	D₈	1	1	yes	characteristic
27	Dic₅	—	4	1	no	—
28	#41	—	10	1	no	—
29	Q₁₆	—	5	1	no	—
30	SD₁₆	—	10	1	no	—
31	D₈	—	5	1	no	—
32	M₁₆	—	10	1	no	—
33	C₂×C₈	C₂×D₅	1	1	yes	characteristic
34	C₁₆	—	2	1	no	—
35	C₂×C₅	—	2	1	no	—
36	C₂×C₅	#102	1	1	yes	characteristic
37	D₅	—	8	1	no	—
38	Q₈	—	10	1	no	—
39	D₄	—	10	1	no	—
40	D₄	—	20	1	no	—
41	C₂×C₄	—	20	1	no	—
42	C₂×C₄	D₂₀	1	1	yes	characteristic
43	C₈	—	20	1	no	—
44	C₈	C₄×D₅	1	1	yes	characteristic
45	C₈	C₂²⋊D₅	1	1	yes	characteristic
46	C₅	#359	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	20	1	no	—
48	C₂²	#677	1	1	yes	characteristic
49	C₄	—	20	1	no	—
50	C₄	D₄₀	1	1	yes	characteristic
51	C₄	#685	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	40	1	no	—
54	C₂	#3953	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₀
commutator	C₅×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	#102
prime radical	#4050	C₂
cosocle	C₂×C₅×C₈	C₂²
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#359
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₁₆	C₂	semidirect
3. #3951	C₂	semidirect
21. C₂×C₁₆	D₅	semidirect
46. C₅	#359	semidirect

Maximal subgroups

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

subgroup	index
#4050	2
C₂×C₅×C₁₆	2
#3951	2
#359	5