Subgroups — C₃²×C₉×C₃₁⋊C₉ #36382235

Order 22599.

2132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₃₁⋊C₉	C₃	1	3	yes	maximal
2	C₃×C₉×C₃₁⋊C₉	C₃	1	36	yes	maximal, direct product factor
3	C₃³×C₉×C₃₁	C₃	1	1	yes	maximal, characteristic
4	C₃²×C₃₁⋊C₉	C₉	1	81	yes	direct product factor
5	C₃²×C₃₁⋊C₉	C₃²	1	36	yes	—
6	C₉×C₃₁⋊C₉	C₃²	1	81	yes	direct product factor
7	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₃₁	C₉	1	27	yes	complemented
9	C₃²×C₉×C₃₁	C₃²	1	12	yes	—
10	C₃³×C₃₁	C₃×C₉	1	27	yes	—
11	C₃³×C₃₁	C₃×C₉	1	9	yes	—
12	C₃³×C₃₁	C₃³	1	4	yes	—
13	C₃×C₉×C₃₁	C₃×C₉	1	108	yes	complemented
14	C₃×C₉×C₃₁	C₃³	1	9	yes	—
15	C₃×C₃₁⋊C₉	C₃×C₉	1	324	yes	direct product factor
16	C₃×C₃₁⋊C₉	C₃³	1	27	yes	—
17	C₃²×C₉²	—	31	1	no	maximal, Sylow
18	C₃²×C₃₁	C₉²	1	81	yes	complemented
19	C₃²×C₃₁	C₃²×C₉	1	36	yes	—
20	C₃²×C₃₁	C₃²×C₉	1	12	yes	—
21	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
22	C₉×C₃₁	C₃²×C₉	1	27	yes	complemented
23	C₃₁⋊C₉	C₃²×C₉	1	81	yes	direct product factor
24	C₃³×C₉	—	31	3	no	—
25	C₃³×C₉	C₃₁⋊C₃	1	1	yes	characteristic
26	C₃×C₉²	—	31	36	no	—
27	C₃×C₃₁	C₃×C₉²	1	36	yes	complemented
28	C₃×C₃₁	C₃³×C₉	1	3	yes	—
29	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
30	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
31	C₃²×C₉	—	31	81	no	—
32	C₃²×C₉	—	31	36	no	—
33	C₃²×C₉	C₃₁⋊C₉	1	27	yes	direct product factor
34	C₃²×C₉	C₃×C₃₁⋊C₃	1	12	yes	—
35	C₉²	—	31	81	no	—
36	C₃₁	C₃²×C₉²	1	1	yes	characteristic, complemented, Sylow
37	C₃³	C₉×C₃₁⋊C₃	1	27	yes	—
38	C₃³	C₃×C₃₁⋊C₉	1	9	yes	—
39	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
40	C₃×C₉	—	31	324	no	—
41	C₃×C₉	—	31	27	no	—
42	C₃×C₉	C₃×C₃₁⋊C₉	1	108	yes	direct product factor
43	C₃×C₉	C₃²×C₃₁⋊C₃	1	9	yes	—
44	C₃²	C₉×C₃₁⋊C₉	1	81	yes	direct product factor
45	C₃²	C₃×C₉×C₃₁⋊C₃	1	36	yes	—
46	C₃²	C₃²×C₃₁⋊C₉	1	12	yes	—
47	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
48	C₉	—	31	81	no	—
49	C₉	C₃²×C₃₁⋊C₉	1	27	yes	direct product factor
50	C₃	C₃×C₉×C₃₁⋊C₉	1	36	yes	direct product factor
51	C₃	C₃²×C₉×C₃₁⋊C₃	1	3	yes	—
52	C₃	C₃³×C₃₁⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃³×C₉	C₃₁⋊C₃
commutator, hypocenter	C₃₁	C₃²×C₉²
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃³×C₉×C₃₁	C₃
socle, prime radical	C₃⁴×C₃₁	C₃²
cosocle	C₃²×C₃₁	C₃⁴
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×C₃₁⋊C₉	C₃	direct (N × H)
4. C₃²×C₃₁⋊C₉	C₉	direct (N × H)
6. C₉×C₃₁⋊C₉	C₃²	direct (N × H)
8. C₃²×C₉×C₃₁	C₉	semidirect
13. C₃×C₉×C₃₁	C₃×C₉	semidirect
15. C₃×C₃₁⋊C₉	C₃×C₉	direct (N × H)
18. C₃²×C₃₁	C₉²	semidirect
22. C₉×C₃₁	C₃²×C₉	semidirect
23. C₃₁⋊C₉	C₃²×C₉	direct (N × H)
27. C₃×C₃₁	C₃×C₉²	semidirect
33. C₃²×C₉	C₃₁⋊C₉	direct (N × H)
36. C₃₁	C₃²×C₉²	semidirect
42. C₃×C₉	C₃×C₃₁⋊C₉	direct (N × H)
44. C₃²	C₉×C₃₁⋊C₉	direct (N × H)
49. C₉	C₃²×C₃₁⋊C₉	direct (N × H)
50. C₃	C₃×C₉×C₃₁⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₃₁⋊C₉	3
C₃×C₉×C₃₁⋊C₉	3
C₃³×C₉×C₃₁	3
C₃²×C₉²	31

Subgroups — C32×C9×C31⋊C9 #36382235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉×C₃₁⋊C₉ #36382235