Subgroups — C₃²×C₃²⋊(C₁₂₇⋊C₃) #44525141

Order 30861.

900 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₁₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
2	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	36	yes	maximal, direct product factor
3	C₃⁴×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₂₇⋊C₃	—	3	27	no	—
5	C₃²×C₁₂₇⋊C₃	C₃²	1	36	yes	complemented
6	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	81	yes	direct product factor
7	C₃³×C₁₂₇	—	3	9	no	—
8	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
9	C₃³×C₁₂₇	C₃²	1	12	yes	complemented
10	C₃²×C₁₂₇	—	3	36	no	—
11	C₃²×C₁₂₇	H(3,𝔽₃)	1	9	yes	complemented
12	C₃²×C₁₂₇	C₃³	1	4	yes	—
13	C₃²×C₁₂₇	C₃³	1	9	yes	complemented
14	C₃×C₁₂₇⋊C₃	—	3	108	no	—
15	C₃×C₁₂₇⋊C₃	C₃³	1	27	yes	complemented
16	C₃×C₁₂₇	—	3	9	no	—
17	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	12	yes	complemented
18	C₃×C₁₂₇	C₃⁴	1	1	yes	characteristic
19	C₁₂₇⋊C₃	—	3	27	no	—
20	C₃²×H(3,𝔽₃)	—	127	1	no	maximal, Sylow
21	C₁₂₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
22	C₃⁴	—	127	3	no	—
23	C₃⁴	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
24	C₃×H(3,𝔽₃)	—	127	36	no	—
25	C₃³	—	3	9	no	—
26	C₃³	—	381	27	no	—
27	C₃³	—	127	36	no	—
28	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
29	C₃³	C₃×C₁₂₇⋊C₃	1	12	yes	complemented
30	H(3,𝔽₃)	—	127	81	no	—
31	C₃²	—	3	36	no	—
32	C₃²	—	381	108	no	—
33	C₃²	—	127	27	no	—
34	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	9	yes	direct product factor
35	C₃²	C₃²×C₁₂₇⋊C₃	1	4	yes	—
36	C₃²	C₃²×C₁₂₇⋊C₃	1	9	yes	complemented
37	C₃	—	3	9	no	—
38	C₃	—	381	27	no	—
39	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	12	yes	direct product factor
40	C₃	C₃³×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃×C₁₂₇⋊C₃
commutator, cosocle	C₃×C₁₂₇	C₃⁴
Frattini	C₃	C₃³×C₁₂₇⋊C₃
Fitting	C₃⁴×C₁₂₇	C₃
socle	C₃³×C₁₂₇	C₃²
hypercenter	C₃⁴	C₁₂₇⋊C₃
hypocenter	C₁₂₇	C₃²×H(3,𝔽₃)
LCS	G ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
1. C₃³×C₁₂₇⋊C₃	C₃	semidirect
2. C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	direct (N × H)
3. C₃⁴×C₁₂₇	C₃	semidirect
5. C₃²×C₁₂₇⋊C₃	C₃²	semidirect
6. C₃²⋊(C₁₂₇⋊C₃)	C₃²	direct (N × H)
9. C₃³×C₁₂₇	C₃²	semidirect
11. C₃²×C₁₂₇	H(3,𝔽₃)	semidirect
13. C₃²×C₁₂₇	C₃³	semidirect
15. C₃×C₁₂₇⋊C₃	C₃³	semidirect
17. C₃×C₁₂₇	C₃×H(3,𝔽₃)	semidirect
21. C₁₂₇	C₃²×H(3,𝔽₃)	semidirect
23. C₃⁴	C₁₂₇⋊C₃	semidirect
29. C₃³	C₃×C₁₂₇⋊C₃	semidirect
34. C₃²	C₃²⋊(C₁₂₇⋊C₃)	direct (N × H)
36. C₃²	C₃²×C₁₂₇⋊C₃	semidirect
39. C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₁₂₇⋊C₃	3
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
C₃⁴×C₁₂₇	3
C₃²×H(3,𝔽₃)	127

Subgroups — C32×C32⋊(C127⋊C3) #44525141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₃²⋊(C₁₂₇⋊C₃) #44525141