Subgroups — #64916

Order 320.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4075	C₂	1	1	yes	maximal, characteristic
2	#4013	C₂	1	1	yes	maximal, characteristic
3	#4011	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₅	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
6	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
7	#704	—	2	1	no	—
8	C₂²⋊Dic₅	C₄	1	1	yes	characteristic, complemented
9	#685	—	2	1	no	—
10	#683	—	2	1	no	—
11	#352	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂²×D₅	D₄	1	1	yes	characteristic
14	C₂×AGL(1,𝔽₅)	—	4	1	no	—
15	C₂×C₄×C₅	—	2	1	no	—
16	C₂×C₄×C₅	—	4	1	no	—
17	C₂²⋊D₅	—	4	1	no	—
18	C₂×Dic₅	—	2	1	no	—
19	C₂×Dic₅	D₄	1	1	yes	characteristic
20	#170	—	4	1	no	—
21	#123	—	5	1	no	—
22	#100	—	5	1	no	—
23	#99	—	5	1	no	—
24	C₂²×C₅	—	2	1	no	—
25	C₂²×C₅	—	4	1	no	—
26	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₅	—	2	1	no	—
28	C₂×D₅	—	4	1	no	—
29	AGL(1,𝔽₅)	—	8	1	no	—
30	C₄×C₅	—	4	1	no	—
31	Dic₅	—	2	1	no	—
32	Dic₅	—	4	1	no	—
33	C₂×D₄	—	5	1	no	—
34	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
35	M₁₆	—	10	1	no	—
36	C₄⋊C₄	—	10	1	no	—
37	C₂²⋊C₄	—	5	1	no	—
38	C₂²⋊C₄	—	10	1	no	—
39	C₂²⋊C₄	—	10	1	no	—
40	C₂×C₅	—	2	1	no	—
41	C₂×C₅	—	4	1	no	—
42	C₂×C₅	#99	1	1	yes	characteristic
43	D₅	—	4	1	no	—
44	C₂³	—	5	1	no	—
45	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	D₄	—	20	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	4	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	10	1	no	—
51	C₂×C₄	—	20	1	no	—
52	C₈	—	20	1	no	—
53	C₅	#352	1	1	yes	characteristic, complemented, Sylow
54	C₂²	—	2	1	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	10	1	no	—
57	C₂²	—	20	1	no	—
58	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₄	—	4	1	no	—
60	C₄	—	10	1	no	—
61	C₄	—	20	1	no	—
62	C₄	—	40	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	20	1	no	—
66	C₂	#4011	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4011
commutator	C₂³×C₅	C₂×C₄
Frattini	C₂³	C₂×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂×C₅	#99
prime radical, cosocle	C₂×C₂²⋊D₅	C₂²
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	#352
LCS	G ▷ C₂³×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. C₂²×C₄×C₅	C₄	semidirect
8. C₂²⋊Dic₅	C₄	semidirect
34. C₂²×C₄	AGL(1,𝔽₅)	semidirect
53. C₅	#352	semidirect

Maximal subgroups

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

subgroup	index
#4075	2
#4013	2
#4011	2
#352	5