Subgroups — #10598486

Order 640.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	D₅×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₈×AGL(1,𝔽₅)	C₄	1	1	yes	characteristic
5	D₅⋊C₁₆	C₄	1	1	yes	characteristic
6	D₅×C₁₆	C₂²	1	1	yes	characteristic
7	#3928	C₄	1	2	yes	complemented
8	C₅×C₃₂	C₄	1	1	yes	characteristic, complemented
9	#3926	C₄	1	1	yes	characteristic, complemented
10	#1397	—	5	1	no	maximal, Sylow
11	C₄×AGL(1,𝔽₅)	C₈	1	1	yes	characteristic
12	D₅⋊C₈	C₈	1	1	yes	characteristic
13	C₈×D₅	C₂×C₄	1	1	yes	characteristic
14	#674	C₈	1	2	yes	—
15	C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
16	#672	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃₂	—	5	1	no	—
18	C₂×C₃₂	—	5	1	no	—
19	C₄×C₁₆	—	5	1	no	—
20	C₂×AGL(1,𝔽₅)	C₁₆	1	2	yes	—
21	C₄×D₅	C₂×C₈	1	1	yes	characteristic
22	#170	C₁₆	1	2	yes	—
23	C₅×C₈	C₄²	1	1	yes	characteristic
24	#168	C₂×C₈	1	1	yes	characteristic
25	C₂×C₁₆	—	5	1	no	—
26	C₂×C₁₆	—	5	1	no	—
27	C₄×C₈	—	5	1	no	—
28	C₃₂	—	5	2	no	—
29	C₃₂	—	5	1	no	—
30	C₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
31	C₂×D₅	C₂×C₁₆	1	1	yes	characteristic
32	AGL(1,𝔽₅)	—	2	2	no	—
33	C₄×C₅	C₄×C₈	1	1	yes	characteristic
34	Dic₅	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₈	—	5	1	no	—
36	C₂×C₈	—	5	1	no	—
37	C₄²	—	5	1	no	—
38	C₁₆	—	5	2	no	—
39	C₁₆	—	5	1	no	—
40	C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₂×C₅	C₄×C₁₆	1	1	yes	characteristic
42	D₅	M₆₄	1	2	yes	—
43	C₂×C₄	—	5	1	no	—
44	C₂×C₄	—	5	2	no	—
45	C₈	—	5	2	no	—
46	C₈	—	5	1	no	—
47	C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₅	#1397	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	5	1	no	—
50	C₄	—	5	1	no	—
51	C₄	—	10	2	no	—
52	C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₂	—	5	2	no	—
54	C₂	C₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₄×C₁₆
Fitting	C₅×C₃₂	C₄
prime radical	C₂×D₅	C₂×C₁₆
cosocle	D₅×C₁₆	C₂²
hypercenter	C₃₂	AGL(1,𝔽₅)
hypocenter	C₅	#1397
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #3928	C₄	semidirect
8. C₅×C₃₂	C₄	semidirect
9. #3926	C₄	semidirect
30. C₃₂	AGL(1,𝔽₅)	semidirect
48. C₅	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×AGL(1,𝔽₅)	2
D₅⋊C₃₂	2
D₅×C₃₂	2
#1397	5