Subgroups — #11717428

Order 800.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#713	C₂	1	1	yes	maximal, characteristic
2	#87849	C₂	1	3	yes	maximal, complemented
3	D₅⋊#168	C₂	1	3	yes	maximal, complemented
4	C₅²×Q₈	C₄	1	1	yes	characteristic
5	C₅×D₂₀	C₄	1	3	yes	—
6	C₄×C₅×D₅	C₂²	1	3	yes	—
7	#6032	C₂²	1	1	yes	characteristic
8	#6032	C₂²	1	3	yes	—
9	#4133	—	5	1	no	maximal
10	#4094	—	5	1	no	maximal
11	C₂×C₅×D₅	C₂×C₄	1	3	yes	—
12	C₄×C₅²	C₂×C₄	1	3	yes	—
13	C₅×Dic₅	C₂³	1	1	yes	characteristic
14	C₅×#41	—	5	1	no	—
15	#713	D₅	1	1	yes	characteristic
16	#700	—	5	3	no	—
17	D₅⋊C₈	—	5	3	no	—
18	#681	—	5	3	no	—
19	C₂×#168	—	5	3	no	—
20	C₂×C₅²	C₂²×C₄	1	1	yes	characteristic
21	C₅×D₅	—	2	3	no	—
22	C₅×Q₈	—	4	1	no	—
23	C₅×Q₈	Dic₅	1	1	yes	characteristic
24	C₅×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₅×D₄	—	5	3	no	—
26	C₂×C₄×C₅	—	5	3	no	—
27	D₂₀	Dic₅	1	3	yes	—
28	C₄×D₅	C₂×D₅	1	3	yes	—
29	#170	—	5	1	no	—
30	#170	—	5	3	no	—
31	#168	—	5	1	no	—
32	#168	—	5	3	no	—
33	#131	—	25	1	no	Sylow
34	C₅²	#131	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₅	—	5	3	no	—
36	C₂×D₅	C₂×Dic₅	1	3	yes	—
37	C₄×C₅	—	4	3	no	—
38	C₄×C₅	—	5	1	no	—
39	C₄×C₅	C₂×Dic₅	1	3	yes	—
40	C₄×C₅	C₂×AGL(1,𝔽₅)	1	3	yes	—
41	Dic₅	C₂²×D₅	1	1	yes	characteristic
42	#41	—	5	1	no	—
43	M₁₆	—	25	3	no	—
44	C₂×C₈	—	25	3	no	—
45	C₂×C₅	—	4	1	no	—
46	C₂×C₅	—	10	3	no	—
47	C₂×C₅	C₂²×Dic₅	1	1	yes	characteristic
48	C₂×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
49	D₅	—	2	3	no	—
50	Q₈	#1042	1	1	yes	characteristic
51	D₄	—	5	3	no	—
52	C₂×C₄	—	5	3	no	—
53	C₈	—	25	1	no	—
54	C₈	—	25	3	no	—
55	C₅	—	4	1	no	—
56	C₅	#4094	1	1	yes	characteristic, complemented
57	C₅	#4133	1	1	yes	characteristic, complemented
58	C₂²	—	5	3	no	—
59	C₄	—	5	1	no	—
60	C₄	C₂×#1042	1	3	yes	—
61	C₂	—	10	3	no	—
62	C₂	C₂²×#1042	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#1042
commutator, socle	C₂×C₅²	C₂²×C₄
Fitting	C₅²×Q₈	C₄
prime radical	C₅×#713	C₂
cosocle	C₅×Dic₅	C₂³
hypercenter	Q₈	#1042
hypocenter	C₅²	#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
2. #87849	C₂	semidirect
3. D₅⋊#168	C₂	semidirect
34. C₅²	#131	semidirect
56. C₅	#4094	semidirect
57. C₅	#4133	semidirect

Maximal subgroups

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

subgroup	index
C₅×#713	2
#87849	2
D₅⋊#168	2
#4133	5
#4094	5