Subgroups — #64893

Order 320.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
4	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
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	C₈×D₅	—	2	2	no	—
10	#340	—	5	1	no	maximal, Sylow
11	C₂²×D₅	D₄	1	1	yes	characteristic
12	C₂×AGL(1,𝔽₅)	—	2	4	no	—
13	C₂×AGL(1,𝔽₅)	—	4	2	no	—
14	C₂×AGL(1,𝔽₅)	—	4	2	no	—
15	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₅	Q₈	1	1	yes	characteristic
17	C₄×D₅	C₂×C₄	1	2	yes	—
18	C₄×D₅	D₄	1	2	yes	—
19	C₅×C₈	—	2	1	no	—
20	#168	—	2	1	no	—
21	C₂²×C₈	—	5	1	no	—
22	C₂×C₄⋊C₄	—	5	2	no	—
23	C₂²×C₅	C₄⋊C₄	1	1	yes	characteristic
24	C₂×D₅	C₂²⋊C₄	1	2	yes	—
25	C₂×D₅	D₈	1	1	yes	characteristic
26	C₂×D₅	SD₁₆	1	2	yes	—
27	C₂×D₅	Q₁₆	1	1	yes	characteristic
28	AGL(1,𝔽₅)	—	4	4	no	—
29	C₄×C₅	C₄²	1	1	yes	characteristic
30	C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	Dic₅	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	10	2	no	—
34	C₂×C₈	—	5	1	no	—
35	C₂×C₈	—	10	2	no	—
36	C₂×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
37	C₄⋊C₄	—	5	4	no	—
38	C₄⋊C₄	—	10	2	no	—
39	C₂×C₅	#95	1	1	yes	characteristic
40	C₂×C₅	#106	1	1	yes	characteristic
41	C₂×C₅	#107	1	1	yes	characteristic
42	D₅	#102	1	2	yes	—
43	D₅	#103	1	2	yes	—
44	C₂³	—	5	1	no	—
45	C₂×C₄	—	5	1	no	—
46	C₂×C₄	—	5	2	no	—
47	C₂×C₄	—	5	2	no	—
48	C₂×C₄	—	10	4	no	—
49	C₂×C₄	—	20	2	no	—
50	C₂×C₄	—	20	2	no	—
51	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	10	1	no	—
54	C₅	#340	1	1	yes	characteristic, complemented, Sylow
55	C₂²	—	5	2	no	—
56	C₂²	—	5	1	no	—
57	C₂²	—	5	1	no	—
58	C₂²	—	5	2	no	—
59	C₂²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄	—	5	2	no	—
61	C₄	—	20	4	no	—
62	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂	—	5	2	no	—
65	C₂	—	5	2	no	—
66	C₂	#3993	1	1	yes	characteristic
67	C₂	#3994	1	1	yes	characteristic
68	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#340
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×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
36. C₂×C₈	AGL(1,𝔽₅)	semidirect
54. C₅	#340	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊AGL(1,𝔽₅)	2
C₂×C₈×D₅	2
#340	5