Subgroups — #64895

Order 320.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4130	C₂	1	2	yes	maximal
2	C₂×#676	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
4	C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
5	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₄×AGL(1,𝔽₅)	C₄	1	4	yes	complemented
7	C₂×C₅×C₈	C₄	1	1	yes	characteristic, complemented
8	C₂×#168	C₄	1	1	yes	characteristic, complemented
9	#676	—	2	2	no	—
10	H(3,ℤ/4)	—	5	1	no	maximal, Sylow
11	C₂²×D₅	D₄	1	1	yes	characteristic
12	C₂×AGL(1,𝔽₅)	—	2	4	no	—
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₅	Q₈	1	1	yes	characteristic
15	C₄×D₅	C₂×C₄	1	2	yes	—
16	C₄×D₅	D₄	1	2	yes	—
17	C₅×C₈	—	2	1	no	—
18	#168	—	2	1	no	—
19	C₂×M₁₆	—	5	1	no	—
20	#117	—	5	2	no	—
21	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
22	C₂×D₅	—	4	1	no	—
23	C₂×D₅	C₂²⋊C₄	1	2	yes	—
24	AGL(1,𝔽₅)	—	4	4	no	—
25	C₄×C₅	C₄²	1	1	yes	characteristic
26	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	Dic₅	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄	—	5	1	no	—
29	M₁₆	—	10	2	no	—
30	C₂×C₈	—	5	1	no	—
31	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
32	C₄⋊C₄	—	10	2	no	—
33	C₂²⋊C₄	—	10	2	no	—
34	C₄²	—	5	4	no	—
35	C₂×C₅	—	2	1	no	—
36	C₂×C₅	#95	1	1	yes	characteristic
37	D₅	—	2	2	no	—
38	C₂³	—	5	1	no	—
39	C₂×C₄	—	5	1	no	—
40	C₂×C₄	—	5	2	no	—
41	C₂×C₄	—	5	2	no	—
42	C₂×C₄	—	10	4	no	—
43	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	—	10	1	no	—
46	C₅	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	5	2	no	—
48	C₂²	—	20	1	no	—
49	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄	—	5	2	no	—
51	C₄	—	20	4	no	—
52	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	10	2	no	—
56	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂×C₅	#95
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	H(3,ℤ/4)
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
6. C₄×AGL(1,𝔽₅)	C₄	semidirect
7. C₂×C₅×C₈	C₄	semidirect
8. C₂×#168	C₄	semidirect
31. C₂×C₈	AGL(1,𝔽₅)	semidirect
46. C₅	H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
#4130	2
C₂×#676	2
H(3,ℤ/4)	5