Subgroups — C₄×C₅²×A₄ #11925669

Order 1200.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×A₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×A₄	C₄	1	1	yes	characteristic, direct product factor
4	C₃×C₄×C₅²	—	4	1	no	maximal
5	C₄×C₅×A₄	C₅	1	6	yes	maximal, direct product factor
6	C₂³×C₅²	C₂×C₃	1	1	yes	characteristic
7	C₂×C₄×C₅²	—	3	1	no	—
8	C₂×C₄×C₅²	—	3	1	no	—
9	C₂×C₃×C₅²	—	4	1	no	—
10	C₂×C₅×A₄	C₂×C₅	1	6	yes	—
11	C₂²×C₅²	—	3	1	no	—
12	C₂²×C₅²	—	3	1	no	—
13	C₂²×C₅²	C₃×C₄	1	1	yes	characteristic, complemented
14	C₄×C₅²	—	3	1	no	—
15	C₄×C₅²	A₄	1	1	yes	characteristic, direct product factor
16	C₂²×C₄×C₅	C₃×C₅	1	6	yes	complemented
17	C₃×C₅²	—	4	1	no	Hall
18	C₅×A₄	C₄×C₅	1	6	yes	direct product factor
19	C₃×C₄×C₅	—	4	6	no	—
20	C₂×C₅²	—	3	1	no	—
21	C₂×C₅²	—	3	1	no	—
22	C₂×C₅²	C₂×A₄	1	1	yes	characteristic
23	C₄×A₄	C₅²	1	1	yes	characteristic, direct product factor, Hall
24	C₂³×C₅	C₂×C₃×C₅	1	6	yes	—
25	C₂×C₄×C₅	—	3	6	no	—
26	C₂×C₄×C₅	—	3	6	no	—
27	C₂×C₃×C₅	—	4	6	no	—
28	C₅²	C₄×A₄	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×A₄	C₂×C₅²	1	1	yes	characteristic
30	C₂²×C₅	—	3	6	no	—
31	C₂²×C₅	—	3	6	no	—
32	C₂²×C₅	C₃×C₄×C₅	1	6	yes	complemented
33	C₄×C₅	—	3	6	no	—
34	C₄×C₅	C₅×A₄	1	6	yes	direct product factor
35	C₂²×C₄	C₃×C₅²	1	1	yes	characteristic, complemented, Sylow
36	C₃×C₅	—	4	6	no	—
37	A₄	C₄×C₅²	1	1	yes	characteristic, direct product factor
38	C₃×C₄	—	4	1	no	—
39	C₂×C₅	—	3	6	no	—
40	C₂×C₅	—	3	6	no	—
41	C₂×C₅	C₂×C₅×A₄	1	6	yes	—
42	C₂³	C₂×C₃×C₅²	1	1	yes	characteristic
43	C₂×C₄	—	3	1	no	—
44	C₂×C₄	—	3	1	no	—
45	C₂×C₃	—	4	1	no	—
46	C₅	C₄×C₅×A₄	1	6	yes	direct product factor
47	C₂²	—	3	1	no	—
48	C₂²	—	3	1	no	—
49	C₂²	C₃×C₄×C₅²	1	1	yes	characteristic, complemented
50	C₄	—	3	1	no	—
51	C₄	C₅²×A₄	1	1	yes	characteristic, direct product factor
52	C₃	—	4	1	no	Sylow
53	C₂	—	3	1	no	—
54	C₂	—	3	1	no	—
55	C₂	C₂×C₅²×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄×C₅²	A₄
commutator, hypocenter	C₂²	C₃×C₄×C₅²
Frattini	C₂	C₂×C₅²×A₄
Fitting	C₂²×C₄×C₅²	C₃
socle	C₂³×C₅²	C₂×C₃
prime radical	C₂×C₅²×A₄	C₂
cosocle	C₂³	C₂×C₃×C₅²
LCS	G ▷ C₂²
UCS	C₁ ◁ C₄×C₅²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅²	C₃	semidirect
3. C₅²×A₄	C₄	direct (N × H)
5. C₄×C₅×A₄	C₅	direct (N × H)
13. C₂²×C₅²	C₃×C₄	semidirect
15. C₄×C₅²	A₄	direct (N × H)
16. C₂²×C₄×C₅	C₃×C₅	semidirect
18. C₅×A₄	C₄×C₅	direct (N × H)
23. C₄×A₄	C₅²	direct (N × H)
28. C₅²	C₄×A₄	direct (N × H)
32. C₂²×C₅	C₃×C₄×C₅	semidirect
34. C₄×C₅	C₅×A₄	direct (N × H)
35. C₂²×C₄	C₃×C₅²	semidirect
37. A₄	C₄×C₅²	direct (N × H)
46. C₅	C₄×C₅×A₄	direct (N × H)
49. C₂²	C₃×C₄×C₅²	semidirect
51. C₄	C₅²×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²×A₄	2
C₂²×C₄×C₅²	3
C₃×C₄×C₅²	4
C₄×C₅×A₄	5

Subgroups — C4×C52×A4 #11925669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅²×A₄ #11925669