Subgroups — #44525122

Order 30861.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	3	yes	maximal, complemented
2	#23779848	C₃	1	9	yes	maximal, complemented
3	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
5	C₃²⋊(C₁₂₇⋊C₃)	—	3	6	no	—
6	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	9	yes	complemented
7	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
8	3₋¹⁺²×C₁₂₇	—	3	3	no	—
9	C₃×C₉×C₁₂₇	C₃²	1	3	yes	complemented
10	C₃²×C₁₂₇	—	9	1	no	—
11	C₃²×C₁₂₇	H(3,𝔽₃)	1	3	yes	—
12	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
13	C₃×C₁₂₇⋊C₃	—	3	9	no	—
14	C₃×C₁₂₇⋊C₃	—	9	9	no	—
15	C₉×C₁₂₇	—	3	3	no	—
16	C₃×C₁₂₇	—	3	3	no	—
17	C₃×C₁₂₇	—	3	1	no	—
18	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
19	C₁₂₇⋊C₃	—	9	9	no	—
20	#6911	—	127	1	no	maximal, Sylow
21	C₁₂₇	#6911	1	1	yes	characteristic, complemented, Sylow
22	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
23	C₃×H(3,𝔽₃)	—	127	3	no	—
24	#732	—	127	9	no	—
25	C₃³	—	381	3	no	—
26	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
27	3₋¹⁺²	—	3	3	no	—
28	H(3,𝔽₃)	—	381	6	no	—
29	H(3,𝔽₃)	—	127	9	no	—
30	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
31	C₃²	—	9	1	no	—
32	C₃²	—	1143	9	no	—
33	C₃²	—	381	9	no	—
34	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	3	yes	—
35	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
36	C₉	—	3	3	no	—
37	C₃	—	3	3	no	—
38	C₃	—	3	1	no	—
39	C₃	—	1143	9	no	—
40	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊(C₁₂₇⋊C₃)
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	C₃×3₋¹⁺²×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6911
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	semidirect
2. #23779848	C₃	semidirect
3. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
6. C₃²⋊(C₁₂₇⋊C₃)	C₃²	semidirect
9. C₃×C₉×C₁₂₇	C₃²	semidirect
21. C₁₂₇	#6911	semidirect
22. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect
30. C₃×C₉	C₃×C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
#23779848	3
C₃×3₋¹⁺²×C₁₂₇	3
#6911	127