Subgroups — #64870

Order 320.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4036	C₂	1	1	yes	maximal, characteristic
2	#4002	C₂	1	1	yes	maximal, characteristic
3	#4002	C₂	1	1	yes	maximal, characteristic
4	C₂×#170	—	2	1	no	—
5	C₂×#170	—	2	1	no	—
6	C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₄⋊Dic₅	—	2	1	no	—
8	C₄⋊Dic₅	C₄	1	1	yes	characteristic
9	#683	—	2	1	no	—
10	C₄×Dic₅	C₂²	1	1	yes	characteristic
11	#330	—	5	1	no	maximal, Sylow
12	C₂×C₄×C₅	—	2	1	no	—
13	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₅	—	2	1	no	—
15	C₂×Dic₅	D₄	1	1	yes	characteristic
16	C₂×Dic₅	D₄	1	1	yes	characteristic
17	#170	—	2	2	no	—
18	#170	—	2	2	no	—
19	#125	—	5	1	no	—
20	#97	—	5	1	no	—
21	#97	—	5	1	no	—
22	C₂²×C₅	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅	—	2	1	no	—
24	C₄×C₅	—	4	1	no	—
25	Dic₅	—	2	1	no	—
26	Dic₅	—	2	1	no	—
27	Dic₅	—	4	1	no	—
28	C₂×C₈	—	10	1	no	—
29	C₂×C₈	—	10	1	no	—
30	C₄⋊C₄	—	5	1	no	—
31	C₄⋊C₄	—	10	1	no	—
32	C₄⋊C₄	—	10	1	no	—
33	C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
34	C₄²	—	5	1	no	—
35	C₂×C₅	#101	1	1	yes	characteristic
36	C₂×C₅	#104	1	1	yes	characteristic
37	C₂×C₅	#104	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	—	5	1	no	—
40	C₂×C₄	—	5	1	no	—
41	C₂×C₄	—	10	1	no	—
42	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
43	C₈	—	10	2	no	—
44	C₈	—	10	2	no	—
45	C₅	#330	1	1	yes	characteristic, complemented, Sylow
46	C₂²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	1	no	—
49	C₄	—	10	1	no	—
50	C₄	—	10	1	no	—
51	C₄	—	20	1	no	—
52	C₂	#4005	1	1	yes	characteristic
53	C₂	#4008	1	1	yes	characteristic
54	C₂	#4010	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, prime radical	C₂²×C₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂²
hypercenter	C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#330
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
45. C₅	#330	semidirect

Maximal subgroups

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

subgroup	index
#4036	2
#4002	2
#330	5