Subgroups — #64869

Order 320.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4043	C₂	1	1	yes	maximal, characteristic
2	(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#4003	C₂	1	1	yes	maximal, characteristic
4	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
5	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂×Dic₁₀	C₄	1	1	yes	characteristic, complemented
7	C₂×#170	—	2	1	no	—
8	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
9	#685	—	2	1	no	—
10	C₄⋊Dic₅	—	2	1	no	—
11	#328	—	5	1	no	maximal, Sylow
12	C₂²×D₅	D₄	1	1	yes	characteristic
13	C₂×AGL(1,𝔽₅)	—	2	2	no	—
14	C₂×AGL(1,𝔽₅)	—	4	1	no	—
15	C₂×AGL(1,𝔽₅)	—	4	1	no	—
16	C₂×C₄×C₅	—	2	1	no	—
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅	—	2	1	no	—
19	C₂×Dic₅	D₄	1	1	yes	characteristic
20	C₄×D₅	—	4	1	no	—
21	Dic₁₀	—	4	1	no	—
22	#170	—	4	1	no	—
23	C₂²⋊Q₈	—	5	1	no	—
24	C₂²⋊C₈	—	5	1	no	—
25	#95	—	5	1	no	—
26	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₅	—	2	1	no	—
28	C₂×D₅	—	2	1	no	—
29	C₂×D₅	SD₁₆	1	1	yes	characteristic
30	C₂×D₅	Q₁₆	1	1	yes	characteristic
31	AGL(1,𝔽₅)	—	4	2	no	—
32	C₄×C₅	—	2	1	no	—
33	C₄×C₅	—	4	1	no	—
34	Dic₅	—	2	1	no	—
35	Dic₅	—	4	1	no	—
36	C₂×Q₈	—	5	1	no	—
37	C₂²×C₄	—	5	1	no	—
38	C₂²×C₄	—	10	1	no	—
39	C₂×C₈	—	10	1	no	—
40	C₄⋊C₄	—	10	1	no	—
41	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	C₂²⋊C₄	—	10	1	no	—
43	C₂×C₅	#99	1	1	yes	characteristic
44	C₂×C₅	#103	1	1	yes	characteristic
45	C₂×C₅	#104	1	1	yes	characteristic
46	D₅	—	2	1	no	—
47	D₅	—	2	1	no	—
48	C₂³	—	5	1	no	—
49	Q₈	—	20	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	—	10	2	no	—
53	C₂×C₄	—	10	1	no	—
54	C₂×C₄	—	20	1	no	—
55	C₂×C₄	—	20	1	no	—
56	C₂×C₄	—	20	1	no	—
57	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₈	—	20	1	no	—
59	C₅	#328	1	1	yes	characteristic, complemented, Sylow
60	C₂²	—	5	1	no	—
61	C₂²	—	5	1	no	—
62	C₂²	—	10	1	no	—
63	C₂²	—	10	1	no	—
64	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	10	1	no	—
68	C₄	—	20	2	no	—
69	C₄	—	20	1	no	—
70	C₂	—	10	1	no	—
71	C₂	—	10	1	no	—
72	C₂	#3999	1	1	yes	characteristic
73	C₂	#4008	1	1	yes	characteristic
74	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	C₂²×C₅	C₂²⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₂×C₄×D₅	C₂²
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#328
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
6. C₂×Dic₁₀	C₄	semidirect
8. C₅×C₄⋊C₄	C₄	semidirect
41. C₄⋊C₄	AGL(1,𝔽₅)	semidirect
59. C₅	#328	semidirect

Maximal subgroups

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

subgroup	index
#4043	2
(C₂×AGL(1,𝔽₅))⋊C₄	2
#4003	2
#328	5