Subgroups — C₃×#21419312 #36382287

Order 22599.

2636 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11719689	C₃	1	3	yes	maximal, complemented
2	#21419312	C₃	1	27	yes	maximal, direct product factor
3	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	9	yes	maximal, complemented
4	C₃⁵×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₃×#11719689	—	3	27	no	—
6	C₃×#11719689	C₃²	1	27	yes	complemented
7	C₃²×C₃₁⋊C₉	C₃²	1	9	yes	complemented
8	C₃²⋊(C₃₁⋊C₉)	C₃²	1	81	yes	complemented
9	C₃⁴×C₃₁	—	3	9	no	—
10	C₃⁴×C₃₁	—	3	27	no	—
11	C₃⁴×C₃₁	C₃²	1	9	yes	—
12	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
13	C₃⁴×C₃₁	C₃²	1	3	yes	—
14	C₃³×C₃₁	—	3	243	no	—
15	C₃³×C₃₁	—	3	27	no	—
16	C₃³×C₃₁	—	3	9	no	—
17	C₃³×C₃₁	—	3	9	no	—
18	C₃³×C₃₁	—	3	18	no	—
19	C₃³×C₃₁	—	3	81	no	—
20	C₃³×C₃₁	H(3,𝔽₃)	1	9	yes	—
21	C₃³×C₃₁	3₋¹⁺²	1	27	yes	complemented
22	C₃³×C₃₁	C₃³	1	3	yes	—
23	C₃³×C₃₁	C₃³	1	9	yes	—
24	C₃³×C₃₁	C₃³	1	1	yes	characteristic
25	#11719689	—	3	81	no	—
26	C₃×C₃₁⋊C₉	—	3	27	no	—
27	C₃×C₃₁⋊C₉	C₃³	1	27	yes	complemented
28	C₃×#6891	—	31	1	no	maximal, Sylow
29	C₃²×C₃₁	—	3	243	no	—
30	C₃²×C₃₁	—	3	9	no	—
31	C₃²×C₃₁	—	3	9	no	—
32	C₃²×C₃₁	—	3	18	no	—
33	C₃²×C₃₁	—	3	27	no	—
34	C₃²×C₃₁	—	3	81	no	—
35	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	9	yes	—
36	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	3	yes	—
37	C₃²×C₃₁	C₃×3₋¹⁺²	1	9	yes	—
38	C₃²×C₃₁	C₃×3₋¹⁺²	1	27	yes	complemented
39	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
40	C₃₁⋊C₉	—	3	27	no	—
41	C₃⁵	C₃₁⋊C₃	1	1	yes	characteristic
42	C₃²×3₋¹⁺²	—	31	3	no	—
43	#6891	—	31	27	no	—
44	C₃×C₃²⋊C₉	—	31	9	no	—
45	C₃×C₃₁	—	3	9	no	—
46	C₃×C₃₁	—	3	27	no	—
47	C₃×C₃₁	#6891	1	9	yes	complemented
48	C₃×C₃₁	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
49	C₃×C₃₁	C₃²×3₋¹⁺²	1	3	yes	—
50	C₃⁴	—	3	27	no	—
51	C₃⁴	—	3	9	no	—
52	C₃⁴	C₃×C₃₁⋊C₃	1	9	yes	—
53	C₃⁴	C₃×C₃₁⋊C₃	1	3	yes	—
54	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	31	27	no	—
56	C₃×3₋¹⁺²	—	93	27	no	—
57	C₃²×C₉	—	31	9	no	—
58	C₃²⋊C₉	—	31	81	no	—
59	C₃₁	C₃×#6891	1	1	yes	characteristic, complemented, Sylow
60	C₃³	—	3	243	no	—
61	C₃³	—	3	27	no	—
62	C₃³	—	3	9	no	—
63	C₃³	—	3	18	no	—
64	C₃³	—	3	9	no	—
65	C₃³	—	3	81	no	—
66	C₃³	#11719689	1	27	yes	complemented
67	C₃³	C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
68	C₃³	C₃²×C₃₁⋊C₃	1	3	yes	—
69	C₃³	C₃²×C₃₁⋊C₃	1	9	yes	—
70	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
71	3₋¹⁺²	—	93	81	no	—
72	C₃×C₉	—	31	27	no	—
73	C₃×C₉	—	93	27	no	—
74	C₃²	—	3	243	no	—
75	C₃²	—	3	81	no	—
76	C₃²	—	3	27	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	3	18	no	—
80	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	9	yes	—
81	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	3	yes	—
82	C₃²	C₃×#11719689	1	9	yes	—
83	C₃²	C₃×#11719689	1	27	yes	complemented
84	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
85	C₉	—	93	27	no	—
86	C₃	—	3	9	no	—
87	C₃	—	3	27	no	—
88	C₃	#21419312	1	9	yes	direct product factor
89	C₃	C₃²×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
90	C₃	C₃²×#11719689	1	3	yes	—

	subgroup	G/H
center	C₃³	C₃²×C₃₁⋊C₃
commutator, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting, prime radical	C₃⁵×C₃₁	C₃
socle	C₃³×C₃₁	C₃³
hypercenter	C₃⁵	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6891
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×#11719689	C₃	semidirect
2. #21419312	C₃	direct (N × H)
3. C₃×C₃²⋊(C₃₁⋊C₉)	C₃	semidirect
6. C₃×#11719689	C₃²	semidirect
7. C₃²×C₃₁⋊C₉	C₃²	semidirect
8. C₃²⋊(C₃₁⋊C₉)	C₃²	semidirect
21. C₃³×C₃₁	3₋¹⁺²	semidirect
27. C₃×C₃₁⋊C₉	C₃³	semidirect
38. C₃²×C₃₁	C₃×3₋¹⁺²	semidirect
47. C₃×C₃₁	#6891	semidirect
59. C₃₁	C₃×#6891	semidirect
66. C₃³	#11719689	semidirect
83. C₃²	C₃×#11719689	semidirect
88. C₃	#21419312	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×#11719689	3
#21419312	3
C₃×C₃²⋊(C₃₁⋊C₉)	3
C₃⁵×C₃₁	3
C₃×#6891	31

Subgroups — C3×#21419312 #36382287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419312 #36382287