Subgroups — #13102799

Order 1600.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅²	C₂	1	1	yes	maximal, characteristic
2	#11716698	C₂	1	6	yes	maximal
3	C₂²×C₄×C₅²	C₂²	1	3	yes	—
4	C₂²×#1039	C₂²	1	4	yes	—
5	C₄²×C₅²	C₄	1	4	yes	complemented
6	#65226	—	5	6	no	maximal
7	C₂³×C₅²	C₂³	1	1	yes	characteristic
8	C₂×C₄×C₅²	—	2	6	no	—
9	C₂×C₄×C₅²	C₂×C₄	1	6	yes	—
10	C₂×#1039	—	2	12	no	—
11	C₂×C₄²×C₅	D₅	1	6	yes	—
12	#3963	—	5	36	no	—
13	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
14	C₂²×C₅²	#41	1	6	yes	—
15	C₄×C₅²	—	2	6	no	—
16	#1039	—	4	4	no	—
17	C₂²×C₄×C₅	C₂×D₅	1	18	yes	—
18	C₂²×Dic₅	—	5	24	no	—
19	C₄²×C₅	Dic₅	1	24	yes	complemented
20	#383	—	25	1	no	Sylow
21	C₂×C₅²	#117	1	3	yes	—
22	C₂×C₅²	#126	1	4	yes	—
23	C₂³×C₅	C₂²×D₅	1	6	yes	—
24	C₂×C₄×C₅	—	2	36	no	—
25	C₂×C₄×C₅	C₂×Dic₅	1	36	yes	—
26	C₂×Dic₅	—	10	72	no	—
27	C₂×C₄²	C₅⋊D₅	1	1	yes	characteristic
28	#95	—	25	6	no	—
29	C₅²	#383	1	1	yes	characteristic, complemented, Sylow
30	C₂²×C₅	#709	1	36	yes	—
31	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
32	C₄×C₅	—	2	36	no	—
33	Dic₅	—	20	24	no	—
34	C₂²×C₄	—	25	4	no	—
35	C₂²×C₄	C₂×C₅⋊D₅	1	3	yes	—
36	C₄²	#1039	1	4	yes	complemented
37	C₂×C₅	#4022	1	24	yes	—
38	C₂×C₅	#4072	1	18	yes	—
39	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
40	C₂×C₄	—	2	6	no	—
41	C₂×C₄	—	50	12	no	—
42	C₂×C₄	C₂×#1039	1	6	yes	—
43	C₅	#65226	1	6	yes	complemented
44	C₂²	#87900	1	6	yes	—
45	C₂²	C₂²×#1039	1	1	yes	characteristic
46	C₄	—	2	6	no	—
47	C₄	—	100	4	no	—
48	C₂	#11717196	1	4	yes	—
49	C₂	#11717246	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₄²×C₅²	C₂
socle, prime radical, cosocle	C₂³×C₅²	C₂³
hypercenter	C₂×C₄²	C₅⋊D₅
hypocenter	C₅²	#383
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. C₄²×C₅²	C₄	semidirect
19. C₄²×C₅	Dic₅	semidirect
29. C₅²	#383	semidirect
36. C₄²	#1039	semidirect
43. C₅	#65226	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅²	2
#11716698	2
#65226	5