Subgroups — C₃²×#14578900 #36382887

Order 22599.

1376 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₃₁⋊C₃)	C₃	1	3	yes	maximal, complemented
2	C₃×#14578900	C₃	1	36	yes	maximal, direct product factor
3	C₃³×C₉×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₁⋊C₃	—	3	3	no	—
5	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	24	no	—
6	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	36	yes	complemented
7	#14578900	C₃²	1	81	yes	direct product factor
8	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₁	—	3	8	no	—
10	C₃²×C₉×C₃₁	—	3	1	no	—
11	C₃²×C₉×C₃₁	C₃²	1	12	yes	complemented
12	C₃³×C₃₁	—	3	9	no	—
13	C₃³×C₃₁	H(3,𝔽₃)	1	8	yes	—
14	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃³×C₃₁	C₃³	1	4	yes	—
16	C₃²×C₃₁⋊C₃	—	3	36	no	—
17	C₃²×C₃₁⋊C₃	—	9	27	no	—
18	C₃×C₉×C₃₁	—	3	24	no	—
19	C₃×C₉×C₃₁	—	3	12	no	—
20	C₃×C₉×C₃₁	C₃³	1	9	yes	complemented
21	C₃²⋊(C₃₁⋊C₃)	—	3	72	no	—
22	C₃²⋊(C₃₁⋊C₃)	C₃³	1	27	yes	complemented
23	C₃²×#732	—	31	1	no	maximal, Sylow
24	C₃²×C₃₁	—	3	24	no	—
25	C₃²×C₃₁	—	3	12	no	—
26	C₃²×C₃₁	#732	1	9	yes	complemented
27	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	8	yes	—
28	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	4	yes	—
29	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
30	C₃×C₃₁⋊C₃	—	3	27	no	—
31	C₃×C₃₁⋊C₃	—	9	108	no	—
32	C₉×C₃₁	—	3	9	no	—
33	C₃²×H(3,𝔽₃)	—	31	3	no	—
34	C₃³×C₉	C₃₁⋊C₃	1	1	yes	characteristic, complemented
35	C₃×#732	—	31	36	no	—
36	C₃×C₃₁	—	3	8	no	—
37	C₃×C₃₁	—	3	1	no	—
38	C₃×C₃₁	C₃×#732	1	12	yes	complemented
39	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
40	C₃₁⋊C₃	—	9	27	no	—
41	C₃⁴	—	93	3	no	—
42	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
43	C₃×H(3,𝔽₃)	—	31	36	no	—
44	C₃×H(3,𝔽₃)	—	93	24	no	—
45	C₃²×C₉	—	3	8	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	C₃×C₃₁⋊C₃	1	12	yes	complemented
48	#732	—	31	81	no	—
49	C₃₁	C₃²×#732	1	1	yes	characteristic, complemented, Sylow
50	C₃³	—	3	9	no	—
51	C₃³	—	279	27	no	—
52	C₃³	—	93	36	no	—
53	C₃³	C₃²⋊(C₃₁⋊C₃)	1	8	yes	—
54	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
55	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
56	H(3,𝔽₃)	—	31	27	no	—
57	H(3,𝔽₃)	—	93	72	no	—
58	C₃×C₉	—	3	24	no	—
59	C₃×C₉	—	3	12	no	—
60	C₃×C₉	C₃²×C₃₁⋊C₃	1	9	yes	complemented
61	C₃²	—	3	24	no	—
62	C₃²	—	3	12	no	—
63	C₃²	—	279	108	no	—
64	C₃²	—	93	27	no	—
65	C₃²	#14578900	1	9	yes	direct product factor
66	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	8	yes	—
67	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	4	yes	—
68	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
69	C₉	—	3	9	no	—
70	C₃	—	3	8	no	—
71	C₃	—	3	1	no	—
72	C₃	—	279	27	no	—
73	C₃	C₃×#14578900	1	12	yes	direct product factor
74	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₁⋊C₃)
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₃³×C₉×C₃₁	C₃
socle	C₃³×C₃₁	H(3,𝔽₃)
hypercenter	C₃³×C₉	C₃₁⋊C₃
hypocenter	C₃₁	C₃²×#732
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃²⋊(C₃₁⋊C₃)	C₃	semidirect
2. C₃×#14578900	C₃	direct (N × H)
3. C₃³×C₉×C₃₁	C₃	semidirect
6. C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	semidirect
7. #14578900	C₃²	direct (N × H)
11. C₃²×C₉×C₃₁	C₃²	semidirect
20. C₃×C₉×C₃₁	C₃³	semidirect
22. C₃²⋊(C₃₁⋊C₃)	C₃³	semidirect
26. C₃²×C₃₁	#732	semidirect
34. C₃³×C₉	C₃₁⋊C₃	semidirect
38. C₃×C₃₁	C₃×#732	semidirect
47. C₃²×C₉	C₃×C₃₁⋊C₃	semidirect
49. C₃₁	C₃²×#732	semidirect
60. C₃×C₉	C₃²×C₃₁⋊C₃	semidirect
65. C₃²	#14578900	direct (N × H)
73. C₃	C₃×#14578900	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
C₃×#14578900	3
C₃³×C₉×C₃₁	3
C₃²×#732	31

Subgroups — C32×#14578900 #36382887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#14578900 #36382887