Subgroups — #65725

Order 320.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4053	C₂	1	1	yes	maximal, characteristic
2	#3998	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#674	C₂	1	1	yes	maximal, characteristic, complemented
4	#3990	C₂	1	2	yes	maximal, complemented
5	D₅⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#709	C₄	1	1	yes	characteristic
7	C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂×#168	C₂²	1	1	yes	characteristic
9	#676	C₄	1	2	yes	—
10	C₈×D₅	C₂²	1	2	yes	—
11	#674	C₂²	1	1	yes	characteristic
12	#674	C₂²	1	2	yes	—
13	#674	C₂²	1	1	yes	characteristic
14	#504	—	5	1	no	maximal, Sylow
15	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊D₅	C₈	1	2	yes	—
17	D₂₀	C₈	1	1	yes	characteristic
18	C₄×D₅	C₂×C₄	1	2	yes	—
19	Dic₁₀	C₈	1	1	yes	characteristic
20	C₅×C₈	C₂×C₄	1	2	yes	—
21	#168	C₂×C₄	1	1	yes	characteristic
22	#168	C₂³	1	1	yes	characteristic
23	#131	—	5	1	no	—
24	M₃₂	—	5	2	no	—
25	M₃₂	—	5	1	no	—
26	C₂×C₁₆	—	5	2	no	—
27	C₂×C₁₆	—	5	1	no	—
28	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×D₅	C₂×C₈	1	2	yes	—
30	C₄×C₅	C₂×C₈	1	1	yes	characteristic
31	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
32	Dic₅	C₂×C₈	1	2	yes	—
33	#41	—	5	1	no	—
34	M₁₆	—	5	2	no	—
35	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
36	C₂×C₈	—	5	1	no	—
37	C₂×C₈	—	5	2	no	—
38	C₁₆	—	5	1	no	—
39	C₁₆	—	5	2	no	—
40	C₁₆	—	5	1	no	—
41	C₂×C₅	—	2	1	no	—
42	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
43	D₅	—	2	2	no	—
44	Q₈	—	5	1	no	—
45	D₄	—	5	2	no	—
46	D₄	—	5	1	no	—
47	C₂×C₄	—	5	2	no	—
48	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₈	—	5	1	no	—
50	C₈	—	5	1	no	—
51	C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
52	C₅	#504	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	5	2	no	—
54	C₂²	D₅⋊C₈	1	1	yes	characteristic
55	C₄	—	5	2	no	—
56	C₄	D₅⋊C₈	1	1	yes	characteristic
57	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₂	—	2	1	no	—
59	C₂	—	10	2	no	—
60	C₂	C₂×D₅⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂²×C₈
Fitting	C₅×M₁₆	C₄
prime radical	#709	C₄
cosocle	#168	C₂³
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#504
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. #3998	C₂	semidirect
3. C₂×#674	C₂	semidirect
4. #3990	C₂	semidirect
5. D₅⋊C₁₆	C₂	semidirect
52. C₅	#504	semidirect

Maximal subgroups

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

subgroup	index
#4053	2
#3998	2
C₂×#674	2
#3990	2
D₅⋊C₁₆	2
#504	5