Subgroups — C₃×#21419308 #36382282

Order 22599.

3824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₃₁⋊C₃)	C₃	1	12	yes	maximal, complemented
2	#21419308	C₃	1	27	yes	maximal, direct product factor
3	C₃⁵×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₁⋊C₃	C₃²	1	9	yes	complemented
5	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	108	no	—
6	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	108	yes	complemented
7	C₃⁴×C₃₁	—	3	36	no	—
8	C₃⁴×C₃₁	C₃²	1	4	yes	—
9	C₃⁴×C₃₁	C₃²	1	9	yes	complemented
10	C₃³×C₃₁	—	3	243	no	—
11	C₃³×C₃₁	—	3	36	no	—
12	C₃³×C₃₁	—	3	108	no	—
13	C₃³×C₃₁	H(3,𝔽₃)	1	36	yes	complemented
14	C₃³×C₃₁	C₃³	1	12	yes	—
15	C₃³×C₃₁	C₃³	1	1	yes	characteristic
16	C₃²×C₃₁⋊C₃	—	3	108	no	—
17	C₃²×C₃₁⋊C₃	C₃³	1	27	yes	complemented
18	C₃²⋊(C₃₁⋊C₃)	—	3	324	no	—
19	C₃×C₃²⋊H(3,𝔽₃)	—	31	1	no	maximal, Sylow
20	C₃²×C₃₁	—	3	243	no	—
21	C₃²×C₃₁	—	3	36	no	—
22	C₃²×C₃₁	—	3	108	no	—
23	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	12	yes	—
24	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	36	yes	complemented
25	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
26	C₃×C₃₁⋊C₃	—	3	108	no	—
27	C₃×C₃₁⋊C₃	—	9	81	no	—
28	C₃⁵	C₃₁⋊C₃	1	1	yes	characteristic, complemented
29	C₃²×H(3,𝔽₃)	—	31	12	no	—
30	C₃²⋊H(3,𝔽₃)	—	31	27	no	—
31	C₃×C₃₁	—	3	36	no	—
32	C₃×C₃₁	C₃²⋊H(3,𝔽₃)	1	9	yes	complemented
33	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	4	yes	—
34	C₃₁⋊C₃	—	9	27	no	—
35	C₃⁴	—	3	36	no	—
36	C₃⁴	—	31	9	no	—
37	C₃⁴	C₃×C₃₁⋊C₃	1	4	yes	—
38	C₃⁴	C₃×C₃₁⋊C₃	1	9	yes	complemented
39	C₃×H(3,𝔽₃)	—	31	108	no	—
40	C₃×H(3,𝔽₃)	—	93	108	no	—
41	C₃₁	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	3	243	no	—
43	C₃³	—	3	36	no	—
44	C₃³	—	3	108	no	—
45	C₃³	—	31	27	no	—
46	C₃³	—	93	108	no	—
47	C₃³	C₃²⋊(C₃₁⋊C₃)	1	36	yes	complemented
48	C₃³	C₃²×C₃₁⋊C₃	1	12	yes	—
49	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
50	H(3,𝔽₃)	—	93	324	no	—
51	C₃²	—	3	243	no	—
52	C₃²	—	3	108	no	—
53	C₃²	—	3	36	no	—
54	C₃²	—	279	81	no	—
55	C₃²	—	93	108	no	—
56	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	12	yes	—
57	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	36	yes	complemented
58	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃	—	3	36	no	—
60	C₃	—	279	27	no	—
61	C₃	#21419308	1	9	yes	direct product factor
62	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	4	yes	—

	subgroup	G/H
center	C₃³	C₃²×C₃₁⋊C₃
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃⁵×C₃₁	C₃
socle	C₃³×C₃₁	C₃³
hypercenter	C₃⁵	C₃₁⋊C₃
hypocenter	C₃₁	C₃×C₃²⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃⁵

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₃₁⋊C₃)	C₃	semidirect
2. #21419308	C₃	direct (N × H)
3. C₃⁵×C₃₁	C₃	semidirect
4. C₃³×C₃₁⋊C₃	C₃²	semidirect
6. C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
9. C₃⁴×C₃₁	C₃²	semidirect
13. C₃³×C₃₁	H(3,𝔽₃)	semidirect
17. C₃²×C₃₁⋊C₃	C₃³	semidirect
24. C₃²×C₃₁	C₃×H(3,𝔽₃)	semidirect
28. C₃⁵	C₃₁⋊C₃	semidirect
32. C₃×C₃₁	C₃²⋊H(3,𝔽₃)	semidirect
38. C₃⁴	C₃×C₃₁⋊C₃	semidirect
41. C₃₁	C₃×C₃²⋊H(3,𝔽₃)	semidirect
47. C₃³	C₃²⋊(C₃₁⋊C₃)	semidirect
57. C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	semidirect
61. C₃	#21419308	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊(C₃₁⋊C₃)	3
#21419308	3
C₃⁵×C₃₁	3
C₃×C₃²⋊H(3,𝔽₃)	31

Subgroups — C3×#21419308 #36382282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419308 #36382282