Subgroups — #44525117

Order 30861.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812184	C₃	1	2	yes	maximal, complemented
2	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	#23779845	C₃	1	9	yes	maximal, complemented
4	C₃×3₋¹⁺²×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
6	#16812184	—	3	2	no	—
7	#16812184	—	3	2	no	—
8	#16812184	C₃²	1	6	yes	complemented
9	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
10	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	3	yes	—
11	C₃×#11765917	—	3	2	no	—
12	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
13	3₋¹⁺²×C₁₂₇	—	3	2	no	—
14	3₋¹⁺²×C₁₂₇	—	3	1	no	—
15	C₃×C₉×C₁₂₇	C₃²	1	3	yes	complemented
16	C₃²×C₁₂₇	—	9	1	no	—
17	C₃²×C₁₂₇	H(3,𝔽₃)	1	2	yes	—
18	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
20	C₃×C₁₂₇⋊C₃	—	3	3	no	—
21	C₃×C₁₂₇⋊C₃	—	9	3	no	—
22	C₉×C₁₂₇	—	3	3	no	—
23	#11765917	—	3	6	no	—
24	C₃×C₁₂₇	—	3	1	no	—
25	C₃×C₁₂₇	—	3	2	no	—
26	C₃×C₁₂₇	—	3	1	no	—
27	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	C₁₂₇⋊C₃	—	9	3	no	—
29	#6910	—	127	1	no	maximal, Sylow
30	C₁₂₇	#6910	1	1	yes	characteristic, complemented, Sylow
31	C₃×3₋¹⁺²	—	127	2	no	—
32	C₃×3₋¹⁺²	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
33	C₃×H(3,𝔽₃)	—	127	1	no	—
34	#731	—	127	9	no	—
35	C₃³	—	381	1	no	—
36	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
37	3₋¹⁺²	—	3	2	no	—
38	3₋¹⁺²	—	3	1	no	—
39	3₋¹⁺²	—	381	2	no	—
40	3₋¹⁺²	—	381	2	no	—
41	3₋¹⁺²	—	127	6	no	—
42	H(3,𝔽₃)	—	381	2	no	—
43	H(3,𝔽₃)	—	127	3	no	—
44	C₃×C₉	—	381	2	no	—
45	C₃×C₉	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
46	C₃²	—	9	1	no	—
47	C₃²	—	1143	3	no	—
48	C₃²	—	381	3	no	—
49	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	2	yes	—
50	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
51	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
52	C₉	—	3	3	no	—
53	C₉	—	381	6	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	3	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)
prime radical	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃
hypercenter	C₃×3₋¹⁺²	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6910
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×#16812184	C₃	semidirect
3. #23779845	C₃	semidirect
4. C₃×3₋¹⁺²×C₁₂₇	C₃	semidirect
8. #16812184	C₃²	semidirect
15. C₃×C₉×C₁₂₇	C₃²	semidirect
30. C₁₂₇	#6910	semidirect
32. C₃×3₋¹⁺²	C₁₂₇⋊C₃	semidirect
45. 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×#16812184	3
C₃×C₃²⋊(C₁₂₇⋊C₃)	3
#23779845	3
C₃×3₋¹⁺²×C₁₂₇	3
#6910	127