Subgroups — #28356052

Order 13851.

1124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#10587025	C₃	1	3	yes	maximal, complemented
2	#17987429	C₃	1	27	yes	maximal, complemented
3	#17987421	C₃	1	9	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	27	no	—
6	C₃×#10587025	C₃²	1	9	yes	—
7	C₃×#10587025	C₃²	1	18	yes	complemented
8	C₃²×#4294	C₃²	1	9	yes	complemented
9	C₃²⋊#4294	C₃²	1	81	yes	complemented
10	C₃⁴×C₁₉	C₃²	1	3	yes	—
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₃×C₁₉×H(3,𝔽₃)	—	3	9	no	—
13	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	—
14	#10624881	—	19	1	no	maximal, Sylow
15	C₃³×C₁₉	—	3	9	no	—
16	C₃³×C₁₉	—	9	3	no	—
17	C₃³×C₁₉	—	9	9	no	—
18	C₃³×C₁₉	H(3,𝔽₃)	1	9	yes	—
19	C₃³×C₁₉	C₃³	1	9	yes	—
20	C₃³×C₁₉	C₃³	1	3	yes	—
21	C₃³×C₁₉	C₃³	1	1	yes	characteristic
22	C₁₉×H(3,𝔽₃)	—	3	27	no	—
23	#10587025	—	3	27	no	—
24	#10587025	—	3	54	no	—
25	C₃×#4294	—	3	27	no	—
26	C₃×#4294	C₃³	1	27	yes	—
27	C₃²×3₋¹⁺²	—	19	3	no	—
28	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
29	#6893	—	19	27	no	—
30	#6891	—	19	9	no	—
31	C₃²×C₁₉	—	3	3	no	—
32	C₃²×C₁₉	—	3	9	no	—
33	C₃²×C₁₉	—	3	9	no	—
34	C₃²×C₁₉	—	3	27	no	—
35	C₃²×C₁₉	—	9	9	no	—
36	C₃²×C₁₉	—	9	27	no	—
37	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	—
38	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
40	#4294	—	3	27	no	—
41	C₃⁴	C₃×C₁₉⋊C₃	1	3	yes	—
42	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	19	9	no	—
44	C₃×3₋¹⁺²	—	19	18	no	—
45	C₃×3₋¹⁺²	—	57	27	no	—
46	C₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	—
48	C₃²×C₉	—	19	9	no	—
49	C₃²⋊C₉	—	19	81	no	—
50	C₃×C₁₉	—	3	3	no	—
51	C₃×C₁₉	—	3	9	no	—
52	C₃×C₁₉	—	9	9	no	—
53	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
55	C₃³	—	3	9	no	—
56	C₃³	—	9	3	no	—
57	C₃³	—	9	9	no	—
58	C₃³	C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
59	C₃³	C₃²×C₁₉⋊C₃	1	9	yes	—
60	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
61	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
62	3₋¹⁺²	—	57	54	no	—
63	3₋¹⁺²	—	57	27	no	—
64	H(3,𝔽₃)	—	3	27	no	—
65	C₃×C₉	—	19	27	no	—
66	C₃×C₉	—	57	27	no	—
67	C₁₉	#10624881	1	1	yes	characteristic, complemented, Sylow
68	C₃²	—	3	3	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	9	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	9	27	no	—
74	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
75	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
76	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
77	C₉	—	57	27	no	—
78	C₃	—	3	3	no	—
79	C₃	—	3	9	no	—
80	C₃	—	9	9	no	—
81	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
82	C₃	#17987547	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624881
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#10587025	C₃	semidirect
2. #17987429	C₃	semidirect
3. #17987421	C₃	semidirect
7. C₃×#10587025	C₃²	semidirect
8. C₃²×#4294	C₃²	semidirect
9. C₃²⋊#4294	C₃²	semidirect
67. C₁₉	#10624881	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#10587025	3
#17987429	3
#17987421	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624881	19