Subgroups — #64863

Order 320.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4130	C₂	1	1	yes	maximal, characteristic
2	D₅⋊M₁₆	C₂	1	1	yes	maximal, characteristic
3	#4018	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
5	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
6	#704	—	2	1	no	—
7	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
8	C₄×AGL(1,𝔽₅)	C₄	1	2	yes	complemented
9	#700	—	2	1	no	—
10	D₅⋊C₈	C₄	1	2	yes	complemented
11	C₄²×C₅	C₄	1	1	yes	characteristic, complemented
12	#685	—	2	1	no	—
13	#683	—	2	1	no	—
14	C₄×Dic₅	C₄	1	1	yes	characteristic, complemented
15	H(3,ℤ/4)	—	5	1	no	maximal, Sylow
16	C₂²×D₅	D₄	1	1	yes	characteristic
17	C₂×AGL(1,𝔽₅)	—	2	2	no	—
18	C₂×C₄×C₅	—	2	1	no	—
19	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	—	2	1	no	—
21	C₂×Dic₅	D₄	1	1	yes	characteristic
22	C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₄×D₅	D₄	1	1	yes	characteristic
25	C₄×D₅	Q₈	1	1	yes	characteristic
26	#170	—	2	2	no	—
27	C₂×M₁₆	—	5	1	no	—
28	#117	—	5	1	no	—
29	#117	—	5	1	no	—
30	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×D₅	—	4	1	no	—
32	C₂×D₅	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×D₅	C₄⋊C₄	1	1	yes	characteristic
34	AGL(1,𝔽₅)	—	4	2	no	—
35	C₄×C₅	—	4	1	no	—
36	C₄×C₅	C₄²	1	1	yes	characteristic
37	C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
38	Dic₅	—	4	1	no	—
39	Dic₅	C₂²⋊C₄	1	1	yes	characteristic
40	Dic₅	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄	—	5	1	no	—
42	M₁₆	—	10	1	no	—
43	M₁₆	—	10	1	no	—
44	C₂×C₈	—	5	2	no	—
45	C₄⋊C₄	—	10	1	no	—
46	C₄⋊C₄	—	10	1	no	—
47	C₂²⋊C₄	—	10	1	no	—
48	C₂²⋊C₄	—	10	1	no	—
49	C₄²	—	5	2	no	—
50	C₄²	—	5	1	no	—
51	C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
52	C₂×C₅	—	2	1	no	—
53	C₂×C₅	#95	1	1	yes	characteristic
54	D₅	—	2	1	no	—
55	D₅	—	2	1	no	—
56	C₂³	—	5	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	5	1	no	—
59	C₂×C₄	—	5	1	no	—
60	C₂×C₄	—	5	1	no	—
61	C₂×C₄	—	5	1	no	—
62	C₂×C₄	—	5	1	no	—
63	C₂×C₄	—	10	2	no	—
64	C₂×C₄	—	10	1	no	—
65	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₈	—	10	2	no	—
67	C₅	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	5	1	no	—
69	C₂²	—	5	1	no	—
70	C₂²	—	20	1	no	—
71	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	5	1	no	—
74	C₄	—	5	1	no	—
75	C₄	—	20	1	no	—
76	C₄	—	20	2	no	—
77	C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₂	—	2	1	no	—
80	C₂	—	10	1	no	—
81	C₂	—	10	1	no	—
82	C₂	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₄²
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle	C₂×C₅	#95
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₄²	AGL(1,𝔽₅)
hypocenter	C₅	H(3,ℤ/4)
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
8. C₄×AGL(1,𝔽₅)	C₄	semidirect
10. D₅⋊C₈	C₄	semidirect
11. C₄²×C₅	C₄	semidirect
14. C₄×Dic₅	C₄	semidirect
51. C₄²	AGL(1,𝔽₅)	semidirect
67. C₅	H(3,ℤ/4)	semidirect

Maximal subgroups

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

subgroup	index
#4130	2
D₅⋊M₁₆	2
#4018	2
H(3,ℤ/4)	5