Subgroups — #44525135

Order 30861.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23779860	C₃	1	3	yes	maximal, complemented
2	#23779850	C₃	1	2	yes	maximal, complemented
3	#23779848	C₃	1	1	yes	maximal, characteristic, complemented
4	#23779845	C₃	1	3	yes	maximal, complemented
5	#23779842	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
7	#16812184	C₃²	1	6	yes	—
8	#16812183	—	3	3	no	—
9	#16812182	—	3	3	no	—
10	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	3	yes	—
11	C₉×C₁₂₇⋊C₃	—	3	3	no	—
12	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₁₂₇	—	3	3	no	—
14	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
16	C₃×C₉×C₁₂₇	C₃²	1	1	yes	characteristic
17	C₃²×C₁₂₇	—	3	1	no	—
18	C₃²×C₁₂₇	—	9	1	no	—
19	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
20	C₃×C₁₂₇⋊C₃	—	3	3	no	—
21	C₉×C₁₂₇	—	3	1	no	—
22	C₉×C₁₂₇	—	3	1	no	—
23	C₉×C₁₂₇	H(3,𝔽₃)	1	3	yes	—
24	#11765917	—	3	6	no	—
25	C₃×C₁₂₇	—	3	1	no	—
26	C₃×C₁₂₇	—	9	1	no	—
27	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₁₂₇⋊C₃	—	9	3	no	—
29	#6913	—	127	1	no	maximal, Sylow
30	C₁₂₇	#6913	1	1	yes	characteristic, complemented, Sylow
31	#737	—	127	3	no	—
32	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
33	#733	—	127	2	no	—
34	#732	—	127	1	no	—
35	#731	—	127	3	no	—
36	AΣL(1,𝔽₂₇)	—	127	3	no	—
37	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
38	3₋¹⁺²	—	3	3	no	—
39	3₋¹⁺²	—	381	3	no	—
40	3₋¹⁺²	—	381	3	no	—
41	3₋¹⁺²	—	127	6	no	—
42	H(3,𝔽₃)	—	127	3	no	—
43	C₃×C₉	—	381	3	no	—
44	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
45	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
46	C₃×C₉	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
47	C₃²	—	3	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	381	3	no	—
50	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
51	C₉	—	3	1	no	—
52	C₉	—	3	1	no	—
53	C₉	—	381	6	no	—
54	C₉	C₃²⋊(C₁₂₇⋊C₃)	1	3	yes	—
55	C₃	—	3	1	no	—
56	C₃	—	9	1	no	—
57	C₃	—	1143	3	no	—
58	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	#6913
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #23779860	C₃	semidirect
2. #23779850	C₃	semidirect
3. #23779848	C₃	semidirect
4. #23779845	C₃	semidirect
5. #23779842	C₃	semidirect
6. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
30. C₁₂₇	#6913	semidirect
32. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#23779860	3
#23779850	3
#23779848	3
#23779845	3
#23779842	3
C₃×3₋¹⁺²×C₁₂₇	3
#6913	127