Subgroups — #64853

Order 320.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4020	C₂	1	1	yes	maximal, characteristic
2	#3999	C₂	1	2	yes	maximal, complemented
3	C₂×D₂₀	—	2	2	no	—
4	C₂×D₂₀	C₄	1	1	yes	characteristic, complemented
5	C₂×D₂₀	C₂²	1	1	yes	characteristic
6	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
7	C₄²×C₅	C₄	1	1	yes	characteristic, complemented
8	#353	—	5	1	no	maximal, Sylow
9	C₂²×D₅	—	2	1	no	—
10	C₂²×D₅	D₄	1	2	yes	—
11	C₂×AGL(1,𝔽₅)	—	4	2	no	—
12	C₂×C₄×C₅	—	2	1	no	—
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	D₂₀	—	4	1	no	—
15	D₂₀	—	4	1	no	—
16	D₂₀	—	4	4	no	—
17	#127	—	5	1	no	—
18	#99	—	5	2	no	—
19	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×D₅	—	2	2	no	—
21	C₂×D₅	—	4	2	no	—
22	C₂×D₅	—	4	2	no	—
23	C₂×D₅	—	4	1	no	—
24	AGL(1,𝔽₅)	—	8	2	no	—
25	C₄×C₅	—	2	2	no	—
26	C₄×C₅	—	2	1	no	—
27	C₂×D₄	—	5	1	no	—
28	C₂×D₄	—	5	1	no	—
29	C₂×D₄	—	10	2	no	—
30	C₂²⋊C₄	—	10	2	no	—
31	C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
32	C₂×C₅	—	2	1	no	—
33	C₂×C₅	#99	1	1	yes	characteristic
34	D₅	—	4	2	no	—
35	D₅	—	8	1	no	—
36	C₂³	—	5	2	no	—
37	C₂³	—	10	1	no	—
38	D₄	—	20	1	no	—
39	D₄	—	20	1	no	—
40	D₄	—	20	4	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	20	2	no	—
43	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₅	#353	1	1	yes	characteristic, complemented, Sylow
45	C₂²	—	10	2	no	—
46	C₂²	—	20	2	no	—
47	C₂²	—	20	2	no	—
48	C₂²	—	20	1	no	—
49	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	2	1	no	—
52	C₄	—	40	2	no	—
53	C₂	—	2	1	no	—
54	C₂	—	20	2	no	—
55	C₂	—	40	1	no	—
56	C₂	#3999	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#3999
commutator	C₂×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle	C₂×C₅	#99
prime radical	#4020	C₂
cosocle	C₂×D₂₀	C₂²
hypercenter	C₄²	AGL(1,𝔽₅)
hypocenter	C₅	#353
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #3999	C₂	semidirect
4. C₂×D₂₀	C₄	semidirect
7. C₄²×C₅	C₄	semidirect
31. C₄²	AGL(1,𝔽₅)	semidirect
44. C₅	#353	semidirect

Maximal subgroups

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

subgroup	index
#4020	2
#3999	2
#353	5