Subgroups — C₂²×D₅×H(3,𝔽₃) #11762000

Order 1080.

352 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅×H(3,𝔽₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×D₅	C₃	1	4	yes	maximal, complemented
4	C₂×C₅×H(3,𝔽₃)	C₂²	1	3	yes	complemented
5	D₅×H(3,𝔽₃)	C₂²	1	4	yes	direct product factor
6	C₂³×H(3,𝔽₃)	—	5	1	no	maximal, Hall
7	C₂²×C₃²×C₅	C₂×C₃	1	4	yes	complemented
8	C₂×C₃²×D₅	C₂×C₃	1	24	yes	complemented
9	C₅×H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Hall
10	C₂²×C₃×D₅	—	3	4	no	—
11	C₂²×C₃×D₅	C₃²	1	1	yes	characteristic
12	C₂²×H(3,𝔽₃)	—	5	6	no	—
13	C₂²×H(3,𝔽₃)	D₅	1	1	yes	characteristic, direct product factor
14	C₂×C₃²×C₅	C₂²×C₃	1	12	yes	complemented
15	C₃²×D₅	C₂²×C₃	1	16	yes	complemented
16	C₂³×C₃²	—	5	4	no	—
17	C₂²×C₃×C₅	—	3	4	no	—
18	C₂²×C₃×C₅	C₂×C₃²	1	1	yes	characteristic
19	C₂×C₃×D₅	—	3	24	no	—
20	C₂×C₃×D₅	C₂×C₃²	1	6	yes	—
21	C₂×H(3,𝔽₃)	—	5	4	no	—
22	C₂×H(3,𝔽₃)	C₂×D₅	1	3	yes	direct product factor
23	C₃²×C₅	C₂³×C₃	1	4	yes	complemented
24	C₂²×D₅	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Hall
25	C₂²×C₃²	—	5	24	no	—
26	C₂²×C₃²	C₃×D₅	1	4	yes	complemented
27	C₂×C₃×C₅	—	3	12	no	—
28	C₂×C₃×C₅	C₂²×C₃²	1	3	yes	—
29	C₃×D₅	—	3	16	no	—
30	C₃×D₅	C₂²×C₃²	1	4	yes	—
31	H(3,𝔽₃)	C₂²×D₅	1	1	yes	characteristic, direct product factor, Sylow
32	C₂³×C₃	—	5	1	no	—
33	C₂³×C₃	—	15	4	no	—
34	C₂²×C₅	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
35	C₂×D₅	C₂×H(3,𝔽₃)	1	6	yes	direct product factor
36	C₂×C₃²	—	5	16	no	—
37	C₂×C₃²	C₂×C₃×D₅	1	12	yes	complemented
38	C₃×C₅	—	3	4	no	—
39	C₃×C₅	C₂³×C₃²	1	1	yes	characteristic
40	C₂²×C₃	—	3	4	no	—
41	C₂²×C₃	—	5	6	no	—
42	C₂²×C₃	—	15	24	no	—
43	C₂²×C₃	C₃²×D₅	1	1	yes	characteristic
44	C₂×C₅	C₂²×H(3,𝔽₃)	1	3	yes	complemented
45	D₅	C₂²×H(3,𝔽₃)	1	4	yes	direct product factor
46	C₃²	C₂²×C₃×D₅	1	4	yes	complemented
47	C₂³	—	5	1	no	Sylow
48	C₂×C₃	—	3	12	no	—
49	C₂×C₃	—	5	4	no	—
50	C₂×C₃	—	15	16	no	—
51	C₂×C₃	C₂×C₃²×D₅	1	3	yes	—
52	C₅	C₂³×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
53	C₂²	—	5	6	no	—
54	C₂²	D₅×H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
55	C₃	—	3	4	no	—
56	C₃	C₂²×C₃²×D₅	1	1	yes	characteristic
57	C₂	—	5	4	no	—
58	C₂	C₂×D₅×H(3,𝔽₃)	1	3	yes	direct product factor

	subgroup	G/H
center	C₂²×C₃	C₃²×D₅
commutator, cosocle	C₃×C₅	C₂³×C₃²
Frattini	C₃	C₂²×C₃²×D₅
Fitting	C₂²×C₅×H(3,𝔽₃)	C₂
socle	C₂²×C₃×C₅	C₂×C₃²
hypercenter	C₂²×H(3,𝔽₃)	D₅
hypocenter	C₅	C₂³×H(3,𝔽₃)
LCS	G ▷ C₃×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×H(3,𝔽₃)	C₂	semidirect
2. C₂×D₅×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂²×C₃²×D₅	C₃	semidirect
4. C₂×C₅×H(3,𝔽₃)	C₂²	semidirect
5. D₅×H(3,𝔽₃)	C₂²	direct (N × H)
7. C₂²×C₃²×C₅	C₂×C₃	semidirect
8. C₂×C₃²×D₅	C₂×C₃	semidirect
9. C₅×H(3,𝔽₃)	C₂³	semidirect
13. C₂²×H(3,𝔽₃)	D₅	direct (N × H)
14. C₂×C₃²×C₅	C₂²×C₃	semidirect
15. C₃²×D₅	C₂²×C₃	semidirect
22. C₂×H(3,𝔽₃)	C₂×D₅	direct (N × H)
23. C₃²×C₅	C₂³×C₃	semidirect
24. C₂²×D₅	H(3,𝔽₃)	direct (N × H)
26. C₂²×C₃²	C₃×D₅	semidirect
31. H(3,𝔽₃)	C₂²×D₅	direct (N × H)
34. C₂²×C₅	C₂×H(3,𝔽₃)	semidirect
35. C₂×D₅	C₂×H(3,𝔽₃)	direct (N × H)
37. C₂×C₃²	C₂×C₃×D₅	semidirect
44. C₂×C₅	C₂²×H(3,𝔽₃)	semidirect
45. D₅	C₂²×H(3,𝔽₃)	direct (N × H)
46. C₃²	C₂²×C₃×D₅	semidirect
52. C₅	C₂³×H(3,𝔽₃)	semidirect
54. C₂²	D₅×H(3,𝔽₃)	direct (N × H)
58. C₂	C₂×D₅×H(3,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×H(3,𝔽₃)	2
C₂×D₅×H(3,𝔽₃)	2
C₂²×C₃²×D₅	3
C₂³×H(3,𝔽₃)	5

Subgroups — C22×D5×H(3,𝔽3) #11762000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₅×H(3,𝔽₃) #11762000