Subgroups — #65356

Order 320.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4020	C₂	1	1	yes	maximal, characteristic
3	#3941	C₂	1	4	yes	maximal, complemented
4	#3935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₀	—	2	2	no	—
6	C₂×D₂₀	C₂²	1	2	yes	—
7	C₅×C₄⋊C₄	—	2	2	no	—
8	C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#168	C₂²	1	2	yes	—
11	#490	—	5	1	no	maximal, Sylow
12	C₂²×D₅	—	2	2	no	—
13	C₂×C₄×C₅	—	2	2	no	—
14	C₂×C₄×C₅	D₄	1	2	yes	—
15	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
16	D₂₀	—	2	4	no	—
17	D₂₀	—	4	4	no	—
18	#168	—	2	2	no	—
19	#127	—	5	1	no	—
20	#125	D₅	1	1	yes	characteristic, complemented
21	#102	—	5	4	no	—
22	#97	—	5	1	no	—
23	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
24	C₂×D₅	—	4	2	no	—
25	C₂×D₅	—	4	4	no	—
26	C₄×C₅	—	2	2	no	—
27	C₄×C₅	—	4	2	no	—
28	C₄×C₅	#41	1	2	yes	—
29	C₂×D₄	—	5	2	no	—
30	C₂×D₄	—	10	2	no	—
31	C₂×C₈	—	5	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	C₂×D₅	1	2	yes	—
34	C₄²	C₂×D₅	1	1	yes	characteristic
35	C₂×C₅	#124	1	1	yes	characteristic
36	C₂×C₅	AGL(1,ℤ/8)	1	2	yes	—
37	D₅	—	8	2	no	—
38	C₂³	—	10	2	no	—
39	D₄	—	10	4	no	—
40	D₄	—	20	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂²⋊D₅	1	2	yes	—
43	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
44	C₈	—	10	2	no	—
45	C₅	#490	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	20	2	no	—
47	C₂²	—	20	4	no	—
48	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#713	1	2	yes	—
52	C₂	—	40	2	no	—
53	C₂	#4093	1	1	yes	characteristic
54	C₂	#4095	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#125	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	#4020	C₂
hypercenter	#125	D₅
hypocenter	C₅	#490
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
3. #3941	C₂	semidirect
4. #3935	C₂	semidirect
20. #125	D₅	semidirect
45. C₅	#490	semidirect

Maximal subgroups

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

subgroup	index
C₅×#125	2
#4020	2
#3941	2
#3935	2
#490	5