Subgroups — #44524951

Order 30861.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊#11765917	C₃	1	2	yes	maximal, complemented
3	C₃²×C₉×C₁₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₁₂₇⋊C₃	—	3	1	no	—
5	C₃²⋊(C₁₂₇⋊C₃)	—	3	1	no	—
6	C₃²⋊(C₁₂₇⋊C₃)	—	3	1	no	—
7	C₃²⋊(C₁₂₇⋊C₃)	C₉	1	3	yes	complemented
8	C₃×#11765917	—	3	2	no	—
9	C₃³×C₁₂₇	C₃²	1	1	yes	characteristic
10	C₃×C₉×C₁₂₇	—	3	1	no	—
11	C₃×C₉×C₁₂₇	—	3	3	no	—
12	C₃²×C₁₂₇	—	3	1	no	—
13	C₃²×C₁₂₇	—	3	1	no	—
14	C₃²×C₁₂₇	—	3	1	no	—
15	C₃²×C₁₂₇	C₃×C₉	1	1	yes	characteristic
16	C₃²×C₁₂₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
18	C₃²×C₁₂₇	3₋¹⁺²	1	1	yes	characteristic
19	C₃×C₁₂₇⋊C₃	—	3	3	no	—
20	C₃×C₁₂₇⋊C₃	—	9	1	no	—
21	C₃×C₁₂₇⋊C₃	—	9	1	no	—
22	C₃×C₁₂₇⋊C₃	—	9	1	no	—
23	C₉×C₁₂₇	—	3	3	no	—
24	#11765917	—	9	2	no	—
25	C₃×C₁₂₇	—	3	1	no	—
26	C₃×C₁₂₇	—	3	1	no	—
27	C₃×C₁₂₇	—	3	1	no	—
28	C₃×C₁₂₇	C₃²⋊C₉	1	1	yes	characteristic
29	C₃×C₁₂₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃×C₁₂₇	#731	1	1	yes	characteristic
31	C₃×C₁₂₇	#732	1	1	yes	characteristic
32	C₁₂₇⋊C₃	—	9	3	no	—
33	H(3,𝔽₃)⋊C₉	—	127	1	no	maximal, Sylow
34	C₁₂₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, complemented, Sylow
35	C₃×H(3,𝔽₃)	—	127	1	no	—
36	C₃²×C₉	C₁₂₇⋊C₃	1	1	yes	characteristic, complemented
37	C₃²⋊C₉	—	127	2	no	—
38	C₃³	—	381	1	no	—
39	C₃³	C₃×C₁₂₇⋊C₃	1	1	yes	characteristic
40	H(3,𝔽₃)	—	381	1	no	—
41	H(3,𝔽₃)	—	381	1	no	—
42	H(3,𝔽₃)	—	127	3	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	381	2	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	1	no	—
49	C₃²	—	1143	1	no	—
50	C₃²	—	1143	1	no	—
51	C₃²	—	1143	1	no	—
52	C₃²	—	381	3	no	—
53	C₃²	C₉×C₁₂₇⋊C₃	1	1	yes	characteristic
54	C₃²	C₃²⋊(C₁₂₇⋊C₃)	1	1	yes	characteristic
55	C₃²	#16812182	1	1	yes	characteristic
56	C₃²	#16812183	1	1	yes	characteristic
57	C₉	—	3	3	no	—
58	C₉	—	1143	2	no	—
59	C₃	—	3	1	no	—
60	C₃	—	3	1	no	—
61	C₃	—	3	1	no	—
62	C₃	—	1143	3	no	—
63	C₃	#23779831	1	1	yes	characteristic
64	C₃	#23779842	1	1	yes	characteristic
65	C₃	#23779845	1	1	yes	characteristic
66	C₃	#23779848	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃²⋊(C₁₂₇⋊C₃)
commutator	C₃²×C₁₂₇	C₃×C₉
Frattini	C₃³	C₃×C₁₂₇⋊C₃
Fitting	C₃²×C₉×C₁₂₇	C₃
socle	C₃²×C₁₂₇	H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₁₂₇⋊C₃)	C₃
cosocle	C₃³×C₁₂₇	C₃²
hypercenter	C₃²×C₉	C₁₂₇⋊C₃
hypocenter	C₁₂₇	H(3,𝔽₃)⋊C₉
LCS	G ▷ C₃²×C₁₂₇ ▷ C₃×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. C₃²⋊#11765917	C₃	semidirect
3. C₃²×C₉×C₁₂₇	C₃	semidirect
7. C₃²⋊(C₁₂₇⋊C₃)	C₉	semidirect
34. C₁₂₇	H(3,𝔽₃)⋊C₉	semidirect
36. C₃²×C₉	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
C₃²⋊#11765917	3
C₃²×C₉×C₁₂₇	3
H(3,𝔽₃)⋊C₉	127