Subgroups — #13104702

Order 1600.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	#11716874	C₂	1	6	yes	maximal
3	C₂²×#1043	C₂²	1	4	yes	—
4	C₂×C₄×C₅⋊D₅	C₂²	1	3	yes	—
5	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
6	C₄×#1039	C₄	1	3	yes	complemented
7	#65690	—	5	6	no	maximal
8	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
9	C₂×#1043	—	2	12	no	—
10	C₂×C₄×C₅²	C₂×C₄	1	3	yes	—
11	C₂×#1039	C₂×C₄	1	3	yes	—
12	C₄×C₅⋊D₅	—	2	6	no	—
13	C₄²×D₅	—	5	6	no	—
14	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	36	no	—
15	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₅⋊D₅	#41	1	6	yes	—
17	#1043	—	4	4	no	—
18	C₄×C₅²	—	2	3	no	—
19	#1039	—	2	3	no	—
20	C₂²×AGL(1,𝔽₅)	—	5	24	no	—
21	C₂×C₄×D₅	—	5	18	no	—
22	C₄²×C₅	AGL(1,𝔽₅)	1	6	yes	complemented
23	C₄×Dic₅	—	5	18	no	—
24	#383	—	25	1	no	Sylow
25	C₂×C₅²	#117	1	3	yes	—
26	C₅⋊D₅	#126	1	4	yes	—
27	C₂²×D₅	—	5	6	no	—
28	C₂×AGL(1,𝔽₅)	—	10	72	no	—
29	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	18	yes	—
30	C₂×Dic₅	—	5	18	no	—
31	C₄×D₅	—	10	36	no	—
32	C₂×C₄²	—	25	1	no	—
33	#95	—	25	6	no	—
34	C₅²	#383	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	6	yes	—
36	C₂×D₅	—	5	36	no	—
37	AGL(1,𝔽₅)	—	20	24	no	—
38	C₄×C₅	—	2	18	no	—
39	Dic₅	—	10	18	no	—
40	C₂²×C₄	—	25	3	no	—
41	C₂²×C₄	—	25	4	no	—
42	C₄²	—	25	3	no	—
43	C₄²	#1043	1	1	yes	characteristic, complemented
44	C₂×C₅	#4130	1	18	yes	—
45	D₅	—	5	24	no	—
46	C₂³	—	25	1	no	—
47	C₂×C₄	—	25	3	no	—
48	C₂×C₄	—	50	6	no	—
49	C₂×C₄	—	50	12	no	—
50	C₂×C₄	C₂×#1043	1	3	yes	—
51	C₅	#65690	1	6	yes	complemented
52	C₂²	—	25	6	no	—
53	C₂²	C₂²×#1043	1	1	yes	characteristic
54	C₄	—	2	3	no	—
55	C₄	—	50	3	no	—
56	C₄	—	100	4	no	—
57	C₂	—	25	4	no	—
58	C₂	#11717438	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1043
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical, cosocle	C₂²×C₅⋊D₅	C₂³
hypercenter	C₄²	#1043
hypocenter	C₅²	#383
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄²×C₅²	C₄	semidirect
6. C₄×#1039	C₄	semidirect
22. C₄²×C₅	AGL(1,𝔽₅)	semidirect
34. C₅²	#383	semidirect
43. C₄²	#1043	semidirect
51. C₅	#65690	semidirect

Maximal subgroups

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

subgroup	index
C₄²×C₅⋊D₅	2
#11716874	2
#65690	5