Subgroups — C₃×#21419336 #36382322

Order 22599.

836 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419336	C₃	1	27	yes	maximal, direct product factor
2	C₃×(C₃₁⋊C₉)⋊C₉	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊(C₃₁⋊C₉)	C₃	1	3	yes	maximal
4	C₃³×C₉×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₃²×C₃₁⋊C₉	C₃²	1	9	yes	—
6	(C₃₁⋊C₉)⋊C₉	C₃²	1	81	yes	complemented
7	C₃²⋊(C₃₁⋊C₉)	C₃²	1	27	yes	—
8	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₃₁	—	3	9	no	—
10	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
11	C₃²×C₉×C₃₁	C₃²	1	9	yes	—
12	C₃³×C₃₁	—	3	9	no	—
13	C₃³×C₃₁	3₋¹⁺²	1	9	yes	—
14	C₃³×C₃₁	C₃³	1	3	yes	—
15	C₃³×C₃₁	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₁	—	3	27	no	—
17	C₃×C₉×C₃₁	—	3	9	no	—
18	C₃×C₉×C₃₁	C₃³	1	9	yes	—
19	C₃×C₃₁⋊C₉	—	3	27	no	—
20	C₃×C₃₁⋊C₉	C₃³	1	27	yes	—
21	C₃×#6895	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	3	9	no	—
23	C₃²×C₃₁	—	3	27	no	—
24	C₃²×C₃₁	C₃×3₋¹⁺²	1	9	yes	—
25	C₃²×C₃₁	C₃×3₋¹⁺²	1	3	yes	—
26	C₃²×C₃₁	#737	1	9	yes	—
27	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
28	C₉×C₃₁	—	3	9	no	—
29	C₃₁⋊C₉	—	3	27	no	—
30	C₃³×C₉	C₃₁⋊C₃	1	1	yes	characteristic
31	#6895	—	31	27	no	—
32	C₃×C₉⋊C₉	—	31	9	no	—
33	C₃×C₃²⋊C₉	—	31	3	no	—
34	C₃×C₃₁	—	3	9	no	—
35	C₃×C₃₁	#6895	1	9	yes	complemented
36	C₃×C₃₁	C₃²×3₋¹⁺²	1	1	yes	characteristic
37	C₃×C₃₁	C₃×#737	1	3	yes	—
38	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	C₃²×C₉	—	3	9	no	—
40	C₃²×C₉	—	31	9	no	—
41	C₃²×C₉	C₃×C₃₁⋊C₃	1	9	yes	—
42	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
43	C₉⋊C₉	—	31	81	no	—
44	C₃²⋊C₉	—	31	27	no	—
45	C₃₁	C₃×#6895	1	1	yes	characteristic, complemented, Sylow
46	C₃³	—	3	9	no	—
47	C₃³	#11719689	1	9	yes	—
48	C₃³	C₃²×C₃₁⋊C₃	1	3	yes	—
49	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
50	C₃×C₉	—	3	27	no	—
51	C₃×C₉	—	3	9	no	—
52	C₃×C₉	—	31	27	no	—
53	C₃×C₉	—	93	27	no	—
54	C₃×C₉	C₃²×C₃₁⋊C₃	1	9	yes	—
55	C₃²	—	3	27	no	—
56	C₃²	—	3	9	no	—
57	C₃²	C₃×#11719689	1	9	yes	—
58	C₃²	C₃×#11719689	1	3	yes	—
59	C₃²	#14578912	1	9	yes	—
60	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
61	C₉	—	3	9	no	—
62	C₉	—	93	27	no	—
63	C₃	—	3	9	no	—
64	C₃	#21419336	1	9	yes	direct product factor
65	C₃	C₃²×#11719689	1	1	yes	characteristic
66	C₃	C₃×#14578912	1	3	yes	—

	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	C₃³
prime radical	C₃⁴×C₃₁	C₃²
hypercenter	C₃³×C₉	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6895
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉

Complements

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

N	H ≅ G/N	type
1. #21419336	C₃	direct (N × H)
2. C₃×(C₃₁⋊C₉)⋊C₉	C₃	semidirect
6. (C₃₁⋊C₉)⋊C₉	C₃²	semidirect
35. C₃×C₃₁	#6895	semidirect
45. C₃₁	C₃×#6895	semidirect
64. C₃	#21419336	direct (N × H)

Maximal subgroups

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

subgroup	index
#21419336	3
C₃×(C₃₁⋊C₉)⋊C₉	3
C₃×C₃²⋊(C₃₁⋊C₉)	3
C₃³×C₉×C₃₁	3
C₃×#6895	31

Subgroups — C3×#21419336 #36382322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419336 #36382322