Subgroups — C₂²⋊H(5,𝔽₃) #11758101

Order 972.

642 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	A₄×H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
4	H(5,𝔽₃)	—	4	1	no	maximal, Sylow
5	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	4	yes	complemented
7	C₃²×A₄	C₃²	1	36	yes	complemented
8	C₂²×H(3,𝔽₃)	C₃²	1	9	yes	complemented
9	C₂²⋊H(3,𝔽₃)	C₃²	1	72	yes	complemented
10	C₂²⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
11	C₃×H(3,𝔽₃)	—	4	12	no	—
12	C₃×H(3,𝔽₃)	—	4	27	no	—
13	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic, complemented
14	C₂×C₃³	—	3	4	no	—
15	C₂×H(3,𝔽₃)	—	3	9	no	—
16	C₂²×C₃²	—	9	4	no	—
17	C₂²×C₃²	C₃³	1	1	yes	characteristic
18	C₂²×C₃²	C₃³	1	12	yes	—
19	C₃×A₄	—	9	36	no	—
20	C₃×A₄	C₃³	1	27	yes	—
21	C₃³	—	4	36	no	—
22	C₃³	C₃×A₄	1	4	yes	complemented
23	H(3,𝔽₃)	—	4	9	no	—
24	H(3,𝔽₃)	—	4	72	no	—
25	H(3,𝔽₃)	C₃×A₄	1	9	yes	complemented
26	C₂×C₃²	—	3	1	no	—
27	C₂×C₃²	—	3	12	no	—
28	C₂×C₃²	—	9	12	no	—
29	C₂²×C₃	—	3	1	no	—
30	C₂²×C₃	—	3	12	no	—
31	C₂²×C₃	C₃⁴	1	1	yes	characteristic
32	A₄	—	3	27	no	—
33	C₃²	—	4	27	no	—
34	C₃²	—	9	4	no	—
35	C₃²	—	36	36	no	—
36	C₃²	C₃²×A₄	1	1	yes	characteristic
37	C₃²	C₃²×A₄	1	12	yes	—
38	C₂×C₃	—	3	3	no	—
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	9	12	no	—
41	C₂²	H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
42	C₃	—	3	1	no	—
43	C₃	—	3	12	no	—
44	C₃	—	12	27	no	—
45	C₃	C₃³×A₄	1	1	yes	characteristic
46	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃	C₃³×A₄
commutator, socle, cosocle	C₂²×C₃	C₃⁴
Fitting	C₂²×C₃×H(3,𝔽₃)	C₃
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	H(5,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
3. A₄×H(3,𝔽₃)	C₃	semidirect
6. C₂²×C₃³	C₃²	semidirect
7. C₃²×A₄	C₃²	semidirect
8. C₂²×H(3,𝔽₃)	C₃²	semidirect
9. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
10. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
13. C₃×H(3,𝔽₃)	A₄	semidirect
22. C₃³	C₃×A₄	semidirect
25. H(3,𝔽₃)	C₃×A₄	semidirect
41. C₂²	H(5,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(3,𝔽₃)	3
C₃×C₂²⋊H(3,𝔽₃)	3
A₄×H(3,𝔽₃)	3
H(5,𝔽₃)	4

Subgroups — C22⋊H(5,𝔽3) #11758101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊H(5,𝔽₃) #11758101