Subgroups — #65739

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	D₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	#4009	C₂	1	2	yes	maximal
4	#3993	C₂	1	1	yes	maximal, characteristic
5	#3992	C₂	1	1	yes	maximal, characteristic
6	Q₈×D₅	C₂²	1	2	yes	—
7	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
8	C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₄×AGL(1,𝔽₅)	—	2	2	no	—
10	C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
11	D₅⋊C₈	C₂²	1	1	yes	characteristic
12	C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
13	#689	C₄	1	2	yes	complemented
14	Dic₂₀	C₄	1	1	yes	characteristic, complemented
15	C₈×D₅	C₂²	1	1	yes	characteristic
16	#441	—	5	1	no	maximal, Sylow
17	C₂×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×AGL(1,𝔽₅)	D₄	1	2	yes	—
19	C₅×Q₈	C₂×C₄	1	2	yes	—
20	C₄×D₅	—	2	2	no	—
21	C₄×D₅	C₂³	1	1	yes	characteristic
22	Dic₁₀	—	2	2	no	—
23	Dic₁₀	C₂×C₄	1	2	yes	—
24	#170	—	2	1	no	—
25	C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	#168	C₂×C₄	1	1	yes	characteristic
27	C₂×Q₁₆	—	5	1	no	—
28	C₄×Q₈	—	5	2	no	—
29	#106	—	5	1	no	—
30	#103	—	5	2	no	—
31	#97	—	5	1	no	—
32	C₂×D₅	C₂×D₄	1	1	yes	characteristic
33	AGL(1,𝔽₅)	—	2	2	no	—
34	AGL(1,𝔽₅)	—	4	2	no	—
35	C₄×C₅	—	2	2	no	—
36	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
37	Dic₅	—	2	2	no	—
38	Dic₅	#41	1	1	yes	characteristic
39	C₂×Q₈	—	5	2	no	—
40	Q₁₆	—	5	2	no	—
41	Q₁₆	—	5	1	no	—
42	Q₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
43	C₂×C₈	—	5	1	no	—
44	C₂×C₈	—	5	1	no	—
45	C₄⋊C₄	—	5	2	no	—
46	C₄⋊C₄	—	10	2	no	—
47	C₄²	—	5	1	no	—
48	C₄²	—	10	2	no	—
49	C₂×C₅	C₄×D₄	1	1	yes	characteristic
50	D₅	#137	1	2	yes	—
51	Q₈	—	5	2	no	—
52	Q₈	—	10	2	no	—
53	Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
54	C₂×C₄	—	5	1	no	—
55	C₂×C₄	—	5	2	no	—
56	C₂×C₄	—	10	2	no	—
57	C₂×C₄	—	10	2	no	—
58	C₈	—	5	1	no	—
59	C₈	—	10	1	no	—
60	C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₅	#441	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	5	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	5	1	no	—
65	C₄	—	10	2	no	—
66	C₄	—	10	2	no	—
67	C₄	—	20	2	no	—
68	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂	—	5	2	no	—
70	C₂	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×Q₁₆	C₄
socle	C₂×C₅	C₄×D₄
prime radical	C₂×D₅	C₂×D₄
cosocle	C₄×D₅	C₂³
hypercenter	Q₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#441
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. C₅×Q₁₆	C₄	semidirect
13. #689	C₄	semidirect
14. Dic₂₀	C₄	semidirect
42. Q₁₆	AGL(1,𝔽₅)	semidirect
61. C₅	#441	semidirect

Maximal subgroups

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

subgroup	index
Q₈×AGL(1,𝔽₅)	2
D₅×Q₁₆	2
#4009	2
#3993	2
#3992	2
#441	5