Subgroups — #64913

Order 320.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#170	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
4	C₂²×Dic₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	C₂²	1	1	yes	characteristic
6	C₂×#170	—	2	2	no	—
7	C₄⋊Dic₅	—	2	2	no	—
8	#324	—	5	1	no	maximal, Sylow
9	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄×C₅	—	2	1	no	—
11	C₂×C₄×C₅	—	2	1	no	—
12	C₂×C₄×C₅	C₈	1	2	yes	complemented
13	C₂×Dic₅	—	2	1	no	—
14	C₂×Dic₅	—	4	1	no	—
15	C₂×Dic₅	—	4	1	no	—
16	C₂×Dic₅	D₄	1	2	yes	—
17	#170	—	4	2	no	—
18	C₂×C₄⋊C₄	—	5	1	no	—
19	C₂²⋊C₈	—	5	2	no	—
20	C₂²×C₅	—	2	1	no	—
21	C₂²×C₅	—	2	1	no	—
22	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
24	C₂²×C₅	M₁₆	1	1	yes	characteristic
25	C₄×C₅	—	2	2	no	—
26	Dic₅	—	2	2	no	—
27	Dic₅	—	4	1	no	—
28	C₂²×C₄	—	5	1	no	—
29	C₂²×C₄	—	5	1	no	—
30	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₂×C₈	—	10	2	no	—
32	C₄⋊C₄	—	10	2	no	—
33	C₂×C₅	—	2	1	no	—
34	C₂×C₅	—	2	1	no	—
35	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×C₅	#99	1	1	yes	characteristic
37	C₂×C₅	#101	1	1	yes	characteristic
38	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	C₂×C₄	—	2	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	5	2	no	—
42	C₂×C₄	—	10	1	no	—
43	C₂×C₄	—	20	1	no	—
44	C₂×C₄	—	20	1	no	—
45	C₂×C₄	#170	1	2	yes	complemented
46	C₈	—	20	2	no	—
47	C₅	#324	1	1	yes	characteristic, complemented, Sylow
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂×#170	1	1	yes	characteristic
51	C₂²	#704	1	1	yes	characteristic
52	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	10	2	no	—
55	C₄	—	20	1	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#3999	1	1	yes	characteristic
59	C₂	#4005	1	1	yes	characteristic
60	C₂	C₂²⋊#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₈
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂²
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#324
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
12. C₂×C₄×C₅	C₈	semidirect
45. C₂×C₄	#170	semidirect
47. C₅	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₅	2
C₂²⋊#170	2
#324	5