Subgroups — #44525121

Order 30861.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812184	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#23779847	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₁₂₇⋊C₃	—	3	2	no	—
6	#16812184	—	3	2	no	—
7	#16812184	—	3	1	no	—
8	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
9	C₃²⋊(C₁₂₇⋊C₃)	—	3	2	no	—
10	C₃²⋊(C₁₂₇⋊C₃)	C₃²	1	6	yes	complemented
11	C₃×#11765917	C₃²	1	3	yes	complemented
12	C₃³×C₁₂₇	—	3	1	no	—
13	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
14	H(3,𝔽₃)×C₁₂₇	—	3	2	no	—
15	H(3,𝔽₃)×C₁₂₇	C₃²	1	3	yes	complemented
16	C₃²×C₁₂₇	—	3	3	no	—
17	C₃²×C₁₂₇	—	9	1	no	—
18	C₃²×C₁₂₇	—	9	3	no	—
19	C₃²×C₁₂₇	H(3,𝔽₃)	1	2	yes	—
20	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₁₂₇	C₃³	1	1	yes	characteristic
22	C₃×C₁₂₇⋊C₃	—	3	6	no	—
23	C₃×C₁₂₇⋊C₃	—	9	6	no	—
24	#11765917	—	3	3	no	—
25	C₃×C₁₂₇	—	3	2	no	—
26	C₃×C₁₂₇	—	3	1	no	—
27	C₃×C₁₂₇	—	3	1	no	—
28	C₃×C₁₂₇	—	9	3	no	—
29	C₃×C₁₂₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
30	C₁₂₇⋊C₃	—	9	6	no	—
31	#6911	—	127	1	no	maximal, Sylow
32	C₁₂₇	#6911	1	1	yes	characteristic, complemented, Sylow
33	C₃×3₋¹⁺²	—	127	1	no	—
34	C₃×H(3,𝔽₃)	—	127	2	no	—
35	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
36	#732	—	127	9	no	—
37	C₃³	—	3	1	no	—
38	C₃³	—	381	2	no	—
39	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
40	3₋¹⁺²	—	381	2	no	—
41	3₋¹⁺²	—	381	1	no	—
42	H(3,𝔽₃)	—	3	2	no	—
43	H(3,𝔽₃)	—	381	2	no	—
44	H(3,𝔽₃)	—	381	2	no	—
45	H(3,𝔽₃)	—	127	6	no	—
46	H(3,𝔽₃)	C₃×C₁₂₇⋊C₃	1	3	yes	complemented
47	C₃×C₉	—	127	3	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	9	1	no	—
50	C₃²	—	9	3	no	—
51	C₃²	—	1143	6	no	—
52	C₃²	—	381	6	no	—
53	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	2	yes	—
54	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
55	C₃²	C₃²×C₁₂₇⋊C₃	1	1	yes	characteristic
56	C₉	—	381	3	no	—
57	C₃	—	3	1	no	—
58	C₃	—	3	2	no	—
59	C₃	—	3	1	no	—
60	C₃	—	9	3	no	—
61	C₃	—	1143	6	no	—
62	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×H(3,𝔽₃)×C₁₂₇	C₃
socle	C₃×C₁₂₇	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	C₁₂₇⋊C₃
hypocenter	C₁₂₇	#6911
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#16812184	C₃	semidirect
2. C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	semidirect
3. #23779847	C₃	semidirect
4. C₃×H(3,𝔽₃)×C₁₂₇	C₃	semidirect
10. C₃²⋊(C₁₂₇⋊C₃)	C₃²	semidirect
11. C₃×#11765917	C₃²	semidirect
15. H(3,𝔽₃)×C₁₂₇	C₃²	semidirect
32. C₁₂₇	#6911	semidirect
35. C₃×H(3,𝔽₃)	C₁₂₇⋊C₃	semidirect
46. H(3,𝔽₃)	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
#23779847	3
C₃×H(3,𝔽₃)×C₁₂₇	3
#6911	127