Subgroups — #64873

Order 320.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4034	C₂	1	1	yes	maximal, characteristic
2	#4004	C₂	1	1	yes	maximal, characteristic
3	#4004	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₀	—	2	1	no	—
5	C₂×#170	—	2	1	no	—
6	C₂×#170	—	2	1	no	—
7	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₄⋊Dic₅	C₄	1	1	yes	characteristic
9	#683	—	2	1	no	—
10	C₄×Dic₅	C₂²	1	1	yes	characteristic
11	#332	—	5	1	no	maximal, Sylow
12	C₂×C₄×C₅	—	2	1	no	—
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₅	—	2	1	no	—
15	C₂×Dic₅	D₄	1	1	yes	characteristic
16	C₂×Dic₅	D₄	1	1	yes	characteristic
17	Dic₁₀	—	4	1	no	—
18	Dic₁₀	—	4	1	no	—
19	#170	—	4	1	no	—
20	#170	—	4	1	no	—
21	C₄⋊Q₈	—	5	1	no	—
22	C₄⋊C₈	—	5	1	no	—
23	C₄⋊C₈	—	5	1	no	—
24	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₅	—	2	1	no	—
26	C₄×C₅	—	4	1	no	—
27	Dic₅	—	4	1	no	—
28	Dic₅	D₈	1	1	yes	characteristic
29	Dic₅	SD₁₆	1	1	yes	characteristic
30	Dic₅	SD₁₆	1	1	yes	characteristic
31	Dic₅	Q₁₆	1	1	yes	characteristic
32	C₂×Q₈	—	10	1	no	—
33	C₂×C₈	—	10	1	no	—
34	C₂×C₈	—	10	1	no	—
35	C₄⋊C₄	—	5	1	no	—
36	C₄⋊C₄	—	10	1	no	—
37	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
38	C₄²	—	5	1	no	—
39	C₂×C₅	#101	1	1	yes	characteristic
40	C₂×C₅	#102	1	1	yes	characteristic
41	C₂×C₅	#103	1	1	yes	characteristic
42	Q₈	—	20	1	no	—
43	Q₈	—	20	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	5	1	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₄	—	10	1	no	—
48	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₈	—	20	1	no	—
50	C₈	—	20	1	no	—
51	C₅	#332	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	5	1	no	—
56	C₄	—	5	1	no	—
57	C₄	—	5	1	no	—
58	C₄	—	5	1	no	—
59	C₄	—	20	1	no	—
60	C₂	#4005	1	1	yes	characteristic
61	C₂	#4007	1	1	yes	characteristic
62	C₂	#4009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊C₄	C₄
socle, prime radical	C₂²×C₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂²
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#332
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
51. C₅	#332	semidirect

Maximal subgroups

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

subgroup	index
#4034	2
#4004	2
#332	5