Subgroups — #46798486

Order 33777.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014629	C₃	1	1	yes	maximal, characteristic
2	#26014609	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂₇⋊C₃×C₁₃₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#16836730	C₉	1	2	yes	—
6	#16836729	—	3	1	no	—
7	#16836728	—	3	1	no	—
8	C₃²⋊(C₁₃₉⋊C₃)	C₉	1	1	yes	characteristic
9	C₉×C₁₃₉⋊C₃	—	3	1	no	—
10	C₁₃₉⋊C₂₇	—	3	1	no	—
11	C₁₃₉⋊C₂₇	—	3	1	no	—
12	C₃×C₉×C₁₃₉	C₃²	1	1	yes	characteristic
13	C₂₇×C₁₃₉	—	3	1	no	—
14	C₃²×C₁₃₉	C₃×C₉	1	1	yes	characteristic
15	C₃×C₁₃₉⋊C₃	—	3	1	no	—
16	C₉×C₁₃₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₉×C₁₃₉	3₋¹⁺²	1	1	yes	characteristic
18	C₉×C₁₃₉	3₋¹⁺²	1	1	yes	characteristic
19	C₁₃₉⋊C₉	—	3	2	no	—
20	C₃×C₁₃₉	—	3	1	no	—
21	C₃×C₁₃₉	C₃²⋊C₉	1	1	yes	characteristic
22	C₁₃₉⋊C₃	—	9	1	no	—
23	#6872	—	139	1	no	maximal, Sylow
24	C₁₃₉	#6872	1	1	yes	characteristic, complemented, Sylow
25	#737	—	139	1	no	—
26	C₂₇⋊C₃	—	139	1	no	—
27	C₂₇⋊C₃	C₁₃₉⋊C₃	1	1	yes	characteristic, complemented
28	C₃×C₂₇	—	139	1	no	—
29	3₋¹⁺²	—	139	2	no	—
30	3₋¹⁺²	—	417	1	no	—
31	3₋¹⁺²	—	417	1	no	—
32	H(3,𝔽₃)	—	139	1	no	—
33	C₃×C₉	—	417	1	no	—
34	C₃×C₉	C₃×C₁₃₉⋊C₃	1	1	yes	characteristic
35	C₂₇	—	3	1	no	—
36	C₂₇	—	417	1	no	—
37	C₂₇	—	417	1	no	—
38	C₃²	—	417	1	no	—
39	C₃²	C₉×C₁₃₉⋊C₃	1	1	yes	characteristic
40	C₉	—	417	2	no	—
41	C₉	C₃²⋊(C₁₃₉⋊C₃)	1	1	yes	characteristic
42	C₉	#16836728	1	1	yes	characteristic
43	C₉	#16836729	1	1	yes	characteristic
44	C₃	—	3	1	no	—
45	C₃	—	1251	1	no	—
46	C₃	(C₃×C₁₃₉⋊C₃)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊(C₁₃₉⋊C₃)
commutator	C₃²×C₁₃₉	C₃×C₉
Frattini	C₃×C₉	C₃×C₁₃₉⋊C₃
Fitting	C₂₇⋊C₃×C₁₃₉	C₃
socle	C₃×C₁₃₉	C₃²⋊C₉
prime radical	C₃²⋊(C₁₃₉⋊C₃)	C₉
cosocle	C₃×C₉×C₁₃₉	C₃²
hypercenter	C₂₇⋊C₃	C₁₃₉⋊C₃
hypocenter	C₁₃₉	#6872
LCS	G ▷ C₃²×C₁₃₉ ▷ C₃×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. #26014609	C₃	semidirect
3. C₃×C₁₃₉⋊C₂₇	C₃	semidirect
4. C₂₇⋊C₃×C₁₃₉	C₃	semidirect
24. C₁₃₉	#6872	semidirect
27. C₂₇⋊C₃	C₁₃₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#26014629	3
#26014609	3
C₃×C₁₃₉⋊C₂₇	3
C₂₇⋊C₃×C₁₃₉	3
#6872	139