Subgroups — #44525120

Order 30861.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812182	C₃	1	1	yes	maximal, characteristic, complemented
2	H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	#23779848	C₃	1	2	yes	maximal, complemented
4	#23779847	C₃	1	6	yes	maximal, complemented
5	#732×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²×C₁₂₇⋊C₃	—	3	3	no	—
7	C₃²×C₁₂₇⋊C₃	C₃²	1	1	yes	characteristic
8	#16812182	—	3	3	no	—
9	C₃²⋊(C₁₂₇⋊C₃)	—	3	6	no	—
10	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	6	yes	complemented
11	C₃×#11765917	C₃²	1	2	yes	complemented
12	H(3,𝔽₃)×C₁₂₇	C₃²	1	3	yes	complemented
13	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic, complemented
14	C₃²×C₁₂₇	—	3	3	no	—
15	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
16	C₃×C₁₂₇⋊C₃	—	3	6	no	—
17	C₃×C₁₂₇⋊C₃	—	9	1	no	—
18	C₃×C₁₂₇⋊C₃	—	9	9	no	—
19	C₃×C₁₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
20	C₉×C₁₂₇	—	3	1	no	—
21	#11765917	—	3	2	no	—
22	C₃×C₁₂₇	—	3	1	no	—
23	C₃×C₁₂₇	—	9	3	no	—
24	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₁₂₇⋊C₃	—	3	3	no	—
26	C₁₂₇⋊C₃	—	9	6	no	—
27	#6911	—	127	1	no	maximal, Sylow
28	C₁₂₇	#6911	1	1	yes	characteristic, complemented, Sylow
29	C₃×3₋¹⁺²	—	127	1	no	—
30	C₃×H(3,𝔽₃)	—	127	3	no	—
31	#732	—	127	6	no	—
32	#732	—	127	2	no	—
33	#732	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
34	C₃³	—	381	3	no	—
35	C₃³	—	127	1	no	—
36	3₋¹⁺²	—	381	3	no	—
37	H(3,𝔽₃)	—	381	6	no	—
38	H(3,𝔽₃)	—	127	6	no	—
39	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
40	C₃×C₉	—	127	2	no	—
41	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
42	C₃²	—	3	3	no	—
43	C₃²	—	1143	1	no	—
44	C₃²	—	1143	9	no	—
45	C₃²	—	381	6	no	—
46	C₃²	—	127	3	no	—
47	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	381	2	no	—
50	C₃	—	3	1	no	—
51	C₃	—	9	3	no	—
52	C₃	—	1143	6	no	—
53	C₃	—	381	3	no	—
54	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)×C₁₂₇⋊C₃
commutator, cosocle	C₃²×C₁₂₇	C₃³
Frattini	C₃²	C₃²×C₁₂₇⋊C₃
Fitting	#732×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	#732	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6911
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. C₃×#16812182	C₃	semidirect
2. H(3,𝔽₃)×C₁₂₇⋊C₃	C₃	semidirect
3. #23779848	C₃	semidirect
4. #23779847	C₃	semidirect
5. #732×C₁₂₇	C₃	semidirect
10. C₃²⋊(C₁₂₇⋊C₃)	C₃²	semidirect
11. C₃×#11765917	C₃²	semidirect
12. H(3,𝔽₃)×C₁₂₇	C₃²	semidirect
13. C₃×C₉×C₁₂₇	C₃²	semidirect
28. C₁₂₇	#6911	semidirect
33. #732	C₁₂₇⋊C₃	semidirect
39. H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	semidirect
41. 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×#16812182	3
H(3,𝔽₃)×C₁₂₇⋊C₃	3
#23779848	3
#23779847	3
#732×C₁₂₇	3
#6911	127