Subgroups — C₂²×C₅×#272 #11762011

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×C₅×#272	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×C₅×D₃	—	3	4	no	maximal
4	C₂×C₅×H(3,𝔽₃)	C₂²	1	3	yes	complemented
5	C₅×#272	C₂²	1	4	yes	direct product factor
6	C₂²×#272	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃²×C₅	D₃	1	4	yes	complemented
8	C₂×C₃×C₅×D₃	—	3	24	no	—
9	C₅×H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Hall
10	C₂³×C₃×C₅	—	9	1	no	—
11	C₂²×C₅×D₃	—	3	4	no	—
12	C₂²×H(3,𝔽₃)	C₂×C₅	1	1	yes	characteristic, complemented
13	C₂×#272	C₂×C₅	1	6	yes	direct product factor
14	C₂×C₃²×C₅	C₂×D₃	1	12	yes	complemented
15	C₃×C₅×D₃	—	3	16	no	—
16	C₂²×C₃×D₃	—	3	4	no	—
17	C₂²×C₃×C₅	—	3	4	no	—
18	C₂²×C₃×C₅	—	9	6	no	—
19	C₂²×C₃×C₅	C₃⋊D₃	1	1	yes	characteristic
20	C₂×C₅×D₃	—	3	24	no	—
21	C₂×H(3,𝔽₃)	C₂²×C₅	1	3	yes	complemented
22	#272	C₂²×C₅	1	4	yes	direct product factor
23	C₃²×C₅	C₂²×D₃	1	4	yes	complemented
24	C₂³×C₅	—	9	1	no	Hall
25	C₂²×C₃²	C₅×D₃	1	4	yes	complemented
26	C₂×C₃×D₃	—	3	24	no	—
27	C₂×C₃×C₅	—	3	12	no	—
28	C₂×C₃×C₅	—	9	4	no	—
29	C₂×C₃×C₅	C₂×C₃⋊D₃	1	3	yes	—
30	C₅×D₃	—	3	16	no	—
31	H(3,𝔽₃)	C₂³×C₅	1	1	yes	characteristic, complemented, Sylow
32	C₂³×C₃	—	9	1	no	—
33	C₂²×D₃	—	3	4	no	—
34	C₂²×C₅	—	9	6	no	—
35	C₂²×C₅	#272	1	1	yes	characteristic, direct product factor
36	C₂×C₃²	C₂×C₅×D₃	1	12	yes	complemented
37	C₃×D₃	—	3	16	no	—
38	C₃×C₅	—	3	4	no	—
39	C₃×C₅	C₂²×C₃⋊D₃	1	1	yes	characteristic
40	C₂²×C₃	—	3	4	no	—
41	C₂²×C₃	—	9	6	no	—
42	C₂²×C₃	C₅×C₃⋊D₃	1	1	yes	characteristic
43	C₂×D₃	—	3	24	no	—
44	C₂×C₅	—	9	4	no	—
45	C₂×C₅	C₂×#272	1	3	yes	direct product factor
46	C₃²	C₂²×C₅×D₃	1	4	yes	complemented
47	C₂³	—	9	1	no	Sylow
48	C₂×C₃	—	3	12	no	—
49	C₂×C₃	—	9	4	no	—
50	C₂×C₃	C₂×C₅×C₃⋊D₃	1	3	yes	—
51	D₃	—	3	16	no	—
52	C₅	C₂²×#272	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	9	6	no	—
54	C₂²	C₅×#272	1	1	yes	characteristic, direct product factor
55	C₃	—	3	4	no	—
56	C₃	C₂²×C₅×C₃⋊D₃	1	1	yes	characteristic
57	C₂	—	9	4	no	—
58	C₂	C₂×C₅×#272	1	3	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃×C₅	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂³×C₅
Frattini	C₃	C₂²×C₅×C₃⋊D₃
Fitting	C₂²×C₅×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃×C₅

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×C₅×#272	C₂	direct (N × H)
4. C₂×C₅×H(3,𝔽₃)	C₂²	semidirect
5. C₅×#272	C₂²	direct (N × H)
6. C₂²×#272	C₅	direct (N × H)
7. C₂²×C₃²×C₅	D₃	semidirect
9. C₅×H(3,𝔽₃)	C₂³	semidirect
12. C₂²×H(3,𝔽₃)	C₂×C₅	semidirect
13. C₂×#272	C₂×C₅	direct (N × H)
14. C₂×C₃²×C₅	C₂×D₃	semidirect
21. C₂×H(3,𝔽₃)	C₂²×C₅	semidirect
22. #272	C₂²×C₅	direct (N × H)
23. C₃²×C₅	C₂²×D₃	semidirect
25. C₂²×C₃²	C₅×D₃	semidirect
31. H(3,𝔽₃)	C₂³×C₅	semidirect
35. C₂²×C₅	#272	direct (N × H)
36. C₂×C₃²	C₂×C₅×D₃	semidirect
45. C₂×C₅	C₂×#272	direct (N × H)
46. C₃²	C₂²×C₅×D₃	semidirect
52. C₅	C₂²×#272	direct (N × H)
54. C₂²	C₅×#272	direct (N × H)
58. C₂	C₂×C₅×#272	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×C₅×#272	2
C₂²×C₃×C₅×D₃	3
C₂²×#272	5

Subgroups — C22×C5×#272 #11762011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#272 #11762011