Subgroups — #64871

Order 320.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4039	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₀	C₄	1	1	yes	characteristic
5	C₂×C₄×D₅	—	2	1	no	—
6	C₂×#170	—	2	1	no	—
7	C₂×#170	—	2	1	no	—
8	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
9	#685	—	2	1	no	—
10	C₄×Dic₅	C₂²	1	1	yes	characteristic
11	D₄⋊C₈	—	5	1	no	maximal, Sylow
12	C₂²×D₅	—	2	1	no	—
13	C₂×C₄×C₅	—	2	1	no	—
14	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₅	D₄	1	1	yes	characteristic
16	C₂×Dic₅	D₄	1	1	yes	characteristic
17	D₂₀	—	2	1	no	—
18	D₂₀	C₈	1	2	yes	complemented
19	C₄×D₅	—	4	1	no	—
20	C₄×D₅	—	4	1	no	—
21	#170	—	2	2	no	—
22	#170	—	4	1	no	—
23	C₄×D₄	—	5	1	no	—
24	C₄⋊C₈	—	5	1	no	—
25	C₄×C₈	—	5	1	no	—
26	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₅	—	2	2	no	—
28	C₂×D₅	—	4	1	no	—
29	C₂×D₅	—	4	1	no	—
30	C₄×C₅	—	4	1	no	—
31	C₄×C₅	C₂×C₈	1	1	yes	characteristic
32	C₄×C₅	M₁₆	1	1	yes	characteristic
33	Dic₅	—	2	1	no	—
34	Dic₅	D₈	1	1	yes	characteristic
35	Dic₅	SD₁₆	1	1	yes	characteristic
36	C₂×D₄	—	5	1	no	—
37	C₂²×C₄	—	10	1	no	—
38	C₂×C₈	—	10	1	no	—
39	C₂×C₈	—	10	1	no	—
40	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
41	C₂²⋊C₄	—	10	1	no	—
42	C₄²	—	5	1	no	—
43	C₂×C₅	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₅	#102	1	1	yes	characteristic
45	C₂×C₅	#104	1	1	yes	characteristic
46	D₅	—	4	2	no	—
47	C₂³	—	10	1	no	—
48	D₄	—	5	2	no	—
49	D₄	—	10	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	5	1	no	—
52	C₂×C₄	—	5	1	no	—
53	C₂×C₄	—	20	1	no	—
54	C₂×C₄	—	20	1	no	—
55	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
56	C₈	—	10	2	no	—
57	C₈	—	20	1	no	—
58	C₅	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
59	C₂²	—	10	2	no	—
60	C₂²	—	20	1	no	—
61	C₂²	—	20	1	no	—
62	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₄	—	4	1	no	—
64	C₄	—	5	1	no	—
65	C₄	—	5	1	no	—
66	C₄	—	10	1	no	—
67	C₄	D₅⋊C₈	1	1	yes	characteristic
68	C₄	#700	1	1	yes	characteristic
69	C₂	—	20	2	no	—
70	C₂	#4003	1	1	yes	characteristic
71	C₂	#4007	1	1	yes	characteristic
72	C₂	#4010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₈
Frattini	C₂×C₄	C₂×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊C₄	C₄
socle	C₂²×C₅	C₂²⋊C₄
prime radical	C₂×D₂₀	C₄
cosocle	C₄×Dic₅	C₂²
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	D₄⋊C₈
LCS	G ▷ C₄×C₅ ▷ C₂×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
2. C₄⋊#170	C₂	semidirect
3. C₄×#170	C₂	semidirect
18. D₂₀	C₈	semidirect
58. C₅	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#4039	2
C₄⋊#170	2
C₄×#170	2
D₄⋊C₈	5