Subgroups — #64898

Order 320.

56 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	#709	C₄	1	1	yes	characteristic
5	C₅×M₁₆	C₄	1	1	yes	characteristic
6	C₂×#168	C₂²	1	1	yes	characteristic
7	#676	—	2	1	no	—
8	C₈×D₅	—	2	1	no	—
9	#674	—	2	1	no	—
10	#674	—	2	1	no	—
11	#350	—	5	1	no	maximal, Sylow
12	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²⋊D₅	—	2	1	no	—
14	D₂₀	C₈	1	1	yes	characteristic
15	C₄×D₅	—	2	1	no	—
16	Dic₁₀	C₈	1	1	yes	characteristic
17	C₅×C₈	—	2	1	no	—
18	#168	D₄	1	1	yes	characteristic
19	#168	D₄	1	1	yes	characteristic
20	#131	—	5	1	no	—
21	M₃₂	—	5	1	no	—
22	C₂×C₁₆	—	5	1	no	—
23	C₂²×C₅	M₁₆	1	1	yes	characteristic
24	C₂×D₅	—	2	1	no	—
25	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅	C₂×C₈	1	1	yes	characteristic
27	Dic₅	—	2	1	no	—
28	#41	—	5	1	no	—
29	M₁₆	—	10	1	no	—
30	M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₂×C₈	—	5	1	no	—
32	C₂×C₈	—	10	1	no	—
33	C₁₆	—	10	1	no	—
34	C₁₆	—	10	1	no	—
35	C₂×C₅	—	2	1	no	—
36	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
37	D₅	—	4	1	no	—
38	Q₈	—	5	1	no	—
39	D₄	—	5	1	no	—
40	D₄	—	10	1	no	—
41	C₂×C₄	—	10	1	no	—
42	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	5	1	no	—
45	C₈	—	5	1	no	—
46	C₅	#350	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	10	1	no	—
48	C₂²	#700	1	1	yes	characteristic
49	C₄	—	10	1	no	—
50	C₄	D₅⋊C₈	1	1	yes	characteristic
51	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	20	1	no	—
54	C₂	#4003	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₈
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×M₁₆	C₄
socle	C₂×C₅	C₂²⋊C₈
prime radical	#709	C₄
cosocle	C₂×#168	C₂²
hypercenter	M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#350
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×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
46. C₅	#350	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
#350	5