Subgroups — C₁₃₁×#6868 #44541391

Order 31833.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₁₃₁	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊C₉×C₁₃₁	C₃	1	3	yes	maximal
3	C₃³×C₁₃₁	C₃²	1	1	yes	characteristic
4	C₃×C₉×C₁₃₁	—	3	1	no	—
5	C₃×C₉×C₁₃₁	—	3	2	no	—
6	C₃×C₉×C₁₃₁	—	3	3	no	—
7	C₃×C₉×C₁₃₁	C₉	1	3	yes	complemented
8	C₃²×C₁₃₁	—	3	1	no	—
9	C₃²×C₁₃₁	—	3	2	no	—
10	C₃²×C₁₃₁	C₃×C₉	1	1	yes	characteristic
11	C₃²×C₁₃₁	H(3,𝔽₃)	1	1	yes	characteristic
12	C₃²×C₁₃₁	3₋¹⁺²	1	2	yes	—
13	C₉×C₁₃₁	—	3	3	no	—
14	C₉×C₁₃₁	—	9	3	no	—
15	C₃×C₁₃₁	—	3	1	no	—
16	C₃×C₁₃₁	—	3	2	no	—
17	C₃×C₁₃₁	C₃²⋊C₉	1	1	yes	characteristic
18	C₃×C₁₃₁	#732	1	1	yes	characteristic
19	C₃×C₁₃₁	#733	1	2	yes	—
20	#6868	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₁₃₁	#6868	1	1	yes	characteristic, direct product factor, Sylow
22	C₃²×C₉	C₃×C₁₃₁	1	1	yes	characteristic
23	C₃²⋊C₉	C₃×C₁₃₁	1	3	yes	—
24	C₃³	C₃²×C₁₃₁	1	1	yes	characteristic
25	C₃×C₉	—	3	1	no	—
26	C₃×C₉	—	3	2	no	—
27	C₃×C₉	—	3	3	no	—
28	C₃×C₉	C₉×C₁₃₁	1	3	yes	complemented
29	C₃²	—	3	1	no	—
30	C₃²	—	3	2	no	—
31	C₃²	C₃×C₉×C₁₃₁	1	1	yes	characteristic
32	C₃²	H(3,𝔽₃)×C₁₃₁	1	1	yes	characteristic
33	C₃²	3₋¹⁺²×C₁₃₁	1	2	yes	—
34	C₉	—	3	3	no	—
35	C₉	—	9	3	no	—
36	C₃	—	3	1	no	—
37	C₃	—	3	2	no	—
38	C₃	C₃²⋊C₉×C₁₃₁	1	1	yes	characteristic
39	C₃	#732×C₁₃₁	1	1	yes	characteristic
40	C₃	#733×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₁₃₁	H(3,𝔽₃)
commutator	C₃²	C₃×C₉×C₁₃₁
Frattini, cosocle	C₃³	C₃²×C₁₃₁
prime radical	C₃³×C₁₃₁	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₁₃₁ ◁ C₃³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃×C₉×C₁₃₁	C₉	semidirect
20. #6868	C₁₃₁	direct (N × H)
21. C₁₃₁	#6868	direct (N × H)
28. C₃×C₉	C₉×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₁₃₁	3
C₃²⋊C₉×C₁₃₁	3
#6868	131

Subgroups — C131×#6868 #44541391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#6868 #44541391