Subgroups — #65005

Order 320.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₀	C₂	1	2	yes	maximal
3	C₅×#97	C₂	1	1	yes	maximal, characteristic
4	#3949	C₂	1	1	yes	maximal, characteristic
5	#3948	C₂	1	2	yes	maximal
6	C₂×Dic₁₀	C₂²	1	2	yes	—
7	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅	C₂²	1	2	yes	—
11	#683	—	2	2	no	—
12	C₄×Dic₅	—	2	2	no	—
13	Dic₂₀	C₄	1	4	yes	complemented
14	#441	—	5	1	no	maximal, Sylow
15	C₂×C₄×C₅	D₄	1	2	yes	—
16	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×Dic₅	—	2	2	no	—
18	C₂×Dic₅	—	2	2	no	—
19	Dic₁₀	—	2	2	no	—
20	Dic₁₀	C₂×C₄	1	4	yes	—
21	C₅×C₈	—	2	1	no	—
22	C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂×Q₁₆	—	5	1	no	—
24	C₄×Q₈	—	5	2	no	—
25	#106	—	5	1	no	—
26	#103	—	5	2	no	—
27	#97	D₅	1	1	yes	characteristic
28	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅	—	2	2	no	—
30	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₅	#41	1	1	yes	characteristic
32	Dic₅	—	2	4	no	—
33	Dic₅	—	4	2	no	—
34	C₂×Q₈	—	5	2	no	—
35	Q₁₆	—	5	4	no	—
36	C₂×C₈	C₂×D₅	1	1	yes	characteristic
37	C₂×C₈	C₂×D₅	1	1	yes	characteristic
38	C₄⋊C₄	—	5	2	no	—
39	C₄⋊C₄	—	10	2	no	—
40	C₄²	—	10	2	no	—
41	C₄²	C₂×D₅	1	1	yes	characteristic
42	C₂×C₅	C₄×D₄	1	1	yes	characteristic
43	C₂×C₅	#137	1	2	yes	—
44	Q₈	—	5	4	no	—
45	Q₈	—	10	2	no	—
46	C₂×C₄	—	10	2	no	—
47	C₂×C₄	—	10	2	no	—
48	C₂×C₄	D₂₀	1	2	yes	—
49	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×D₅	1	2	yes	—
52	C₅	#441	1	1	yes	characteristic, complemented, Sylow
53	C₂²	C₂×D₂₀	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	—	10	4	no	—
56	C₄	—	20	2	no	—
57	C₄	C₂×C₄×D₅	1	1	yes	characteristic
58	C₄	#709	1	1	yes	characteristic
59	C₂	C₄×D₂₀	1	1	yes	characteristic
60	C₂	#4055	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#97	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#97	D₅
hypocenter	C₅	#441
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
13. Dic₂₀	C₄	semidirect
52. C₅	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀	2
C₄×Dic₁₀	2
C₅×#97	2
#3949	2
#3948	2
#441	5