Subgroups — #4130

Order 160.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₄	C₂×AGL(1,𝔽₅)
commutator, socle	C₂×C₅	C₂²×C₄
Frattini	C₂	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅	C₄
prime radical	C₂²×D₅	C₂²
cosocle	C₂×D₅	C₂³
hypercenter	C₂×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#117
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊AGL(1,𝔽₅)	C₂	semidirect
4. C₄×AGL(1,𝔽₅)	C₂	semidirect
7. C₂×C₄×C₅	C₄	semidirect
8. C₂×Dic₅	C₄	semidirect
9. C₄×D₅	C₄	semidirect
10. C₄×D₅	C₄	semidirect
38. C₂×C₄	AGL(1,𝔽₅)	semidirect
39. C₅	#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅	2
C₂²⋊AGL(1,𝔽₅)	2
C₄⋊AGL(1,𝔽₅)	2
C₄×AGL(1,𝔽₅)	2
#117	5