Subgroups — C₅×#4022 #11717110

Order 800.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#685	C₂	1	3	yes	maximal
3	C₅×#683	C₂	1	3	yes	maximal, complemented
4	C₂²×C₅×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₅²	C₂²	1	3	yes	—
6	C₂×C₅×Dic₅	C₂²	1	3	yes	—
7	C₅×#126	—	5	1	no	maximal
8	#4022	C₅	1	1	yes	maximal, characteristic, direct product factor
9	C₂²×C₅²	C₂³	1	1	yes	characteristic
10	C₂×C₅×D₅	—	2	3	no	—
11	C₄×C₅²	—	2	3	no	—
12	C₅×Dic₅	—	2	3	no	—
13	C₅×C₄⋊C₄	—	5	3	no	—
14	C₅×C₂²⋊C₄	—	5	3	no	—
15	C₄²×C₅	—	2	2	no	—
16	C₄²×C₅	D₅	1	1	yes	characteristic, complemented
17	C₄²×C₅	C₂×C₅	1	1	yes	characteristic, complemented
18	#685	C₂×C₅	1	3	yes	—
19	#683	C₂×C₅	1	3	yes	complemented
20	C₂×C₅²	#41	1	3	yes	—
21	C₅×D₅	—	4	1	no	—
22	C₂³×C₅	—	5	1	no	—
23	C₂²×D₅	C₂²×C₅	1	1	yes	characteristic
24	C₂×C₄×C₅	—	2	6	no	—
25	C₂×C₄×C₅	—	5	3	no	—
26	C₂×C₄×C₅	C₂×D₅	1	3	yes	—
27	C₂×C₄×C₅	C₂²×C₅	1	3	yes	—
28	C₂×Dic₅	C₂²×C₅	1	3	yes	—
29	#126	—	5	1	no	Sylow
30	C₅²	#126	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₅	—	2	2	no	—
32	C₂²×C₅	—	10	3	no	—
33	C₂²×C₅	C₂²×D₅	1	1	yes	characteristic
34	C₂²×C₅	C₂³×C₅	1	1	yes	characteristic
35	C₂×D₅	—	2	3	no	—
36	C₄×C₅	—	2	12	no	—
37	C₄×C₅	—	2	3	no	—
38	C₄×C₅	—	2	3	no	—
39	C₄×C₅	—	10	3	no	—
40	Dic₅	—	2	3	no	—
41	C₄⋊C₄	—	5	3	no	—
42	C₂²⋊C₄	—	5	3	no	—
43	C₄²	C₅×D₅	1	1	yes	characteristic, complemented
44	C₂×C₅	—	2	6	no	—
45	C₂×C₅	—	20	1	no	—
46	C₂×C₅	#709	1	3	yes	—
47	C₂×C₅	C₅×#41	1	3	yes	—
48	D₅	—	4	1	no	—
49	C₂³	—	5	1	no	—
50	C₂×C₄	—	5	3	no	—
51	C₂×C₄	C₂×C₅×D₅	1	3	yes	—
52	C₅	—	2	2	no	—
53	C₅	#4022	1	1	yes	characteristic, direct product factor
54	C₅	C₅×#126	1	1	yes	characteristic, complemented
55	C₂²	—	10	3	no	—
56	C₂²	C₂²×C₅×D₅	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	10	3	no	—
59	C₂	—	20	1	no	—
60	C₂	C₅×#709	1	3	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂³×C₅
Frattini	C₂²	C₂²×C₅×D₅
Fitting	C₄²×C₅²	C₂
socle	C₂²×C₅²	C₂³
prime radical	C₂²×C₅×D₅	C₂²
hypercenter	C₄²×C₅	D₅
hypocenter	C₅	C₅×#126
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅

Complements

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

N	H ≅ G/N	type
1. C₄²×C₅²	C₂	semidirect
3. C₅×#683	C₂	semidirect
8. #4022	C₅	direct (N × H)
16. C₄²×C₅	D₅	semidirect
17. C₄²×C₅	C₂×C₅	semidirect
19. #683	C₂×C₅	semidirect
30. C₅²	#126	semidirect
43. C₄²	C₅×D₅	semidirect
53. C₅	#4022	direct (N × H)
54. C₅	C₅×#126	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅²	2
C₅×#685	2
C₅×#683	2
C₅×#126	5
#4022	5

Subgroups — C5×#4022 #11717110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#4022 #11717110