Subgroups — C₂³×#6057 #13107920

Order 1600.

1204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1044	C₂	1	3	yes	maximal
2	C₂²×#6057	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₅⋊D₅	C₂²	1	1	yes	characteristic
4	C₂²×#1044	C₂²	1	42	yes	—
5	C₂×#6057	C₂²	1	112	yes	direct product factor
6	C₂³×C₅²	Q₈	1	1	yes	characteristic, complemented
7	C₂²×C₅⋊D₅	Q₈	1	7	yes	complemented
8	C₂²×C₅⋊D₅	C₂³	1	7	yes	—
9	C₂×#1044	C₂³	1	84	yes	—
10	#6057	C₂³	1	64	yes	direct product factor
11	C₂³×AGL(1,𝔽₅)	—	10	3	no	—
12	C₂²×C₅²	C₂×Q₈	1	7	yes	complemented
13	C₂×C₅⋊D₅	C₂×Q₈	1	21	yes	complemented
14	C₂×C₅⋊D₅	C₂⁴	1	7	yes	—
15	#1044	C₂⁴	1	24	yes	—
16	C₂³×D₅	—	10	3	no	—
17	C₂²×AGL(1,𝔽₅)	—	10	42	no	—
18	C₂³×Q₈	—	25	1	no	maximal, Sylow
19	C₂×C₅²	C₂²×Q₈	1	7	yes	complemented
20	C₅⋊D₅	C₂²×Q₈	1	7	yes	complemented
21	C₅⋊D₅	C₂⁵	1	1	yes	characteristic
22	C₂³×C₅	—	2	3	no	—
23	C₂²×D₅	—	10	21	no	—
24	C₂²×D₅	—	10	21	no	—
25	C₂×AGL(1,𝔽₅)	—	10	84	no	—
26	C₂²×Q₈	—	25	28	no	—
27	C₂³×C₄	—	25	3	no	—
28	C₅²	C₂³×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₅	—	2	21	no	—
30	C₂×D₅	—	10	63	no	—
31	C₂×D₅	—	10	21	no	—
32	AGL(1,𝔽₅)	—	10	24	no	—
33	C₂⁴	—	25	1	no	—
34	C₂×Q₈	—	25	112	no	—
35	C₂²×C₄	—	25	42	no	—
36	C₂×C₅	—	2	21	no	—
37	D₅	—	10	21	no	—
38	D₅	—	10	3	no	—
39	C₂³	—	25	7	no	—
40	C₂³	—	25	7	no	—
41	C₂³	#6057	1	1	yes	characteristic, direct product factor
42	Q₈	—	25	64	no	—
43	C₂×C₄	—	25	84	no	—
44	C₅	—	2	3	no	—
45	C₂²	—	25	7	no	—
46	C₂²	—	25	21	no	—
47	C₂²	C₂×#6057	1	7	yes	direct product factor
48	C₄	—	25	24	no	—
49	C₂	—	25	1	no	—
50	C₂	—	25	7	no	—
51	C₂	C₂²×#6057	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#6057
commutator, cosocle	C₅⋊D₅	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂³×C₅²	Q₈
prime radical	C₂³×C₅⋊D₅	C₂²
hypocenter	C₅²	C₂³×Q₈
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×#6057	C₂	direct (N × H)
5. C₂×#6057	C₂²	direct (N × H)
6. C₂³×C₅²	Q₈	semidirect
7. C₂²×C₅⋊D₅	Q₈	semidirect
10. #6057	C₂³	direct (N × H)
12. C₂²×C₅²	C₂×Q₈	semidirect
13. C₂×C₅⋊D₅	C₂×Q₈	semidirect
19. C₂×C₅²	C₂²×Q₈	semidirect
20. C₅⋊D₅	C₂²×Q₈	semidirect
28. C₅²	C₂³×Q₈	semidirect
41. C₂³	#6057	direct (N × H)
47. C₂²	C₂×#6057	direct (N × H)
51. C₂	C₂²×#6057	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1044	2
C₂²×#6057	2
C₂³×Q₈	25

Subgroups — C23×#6057 #13107920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6057 #13107920