Subgroups — #28356065

Order 13851.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987423	C₃	1	27	yes	maximal, complemented
2	#17987421	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊#4294	C₃	1	3	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	C₃²	1	27	yes	complemented
6	C₃²×#4294	C₃²	1	9	yes	complemented
7	C₃²⋊#4294	C₃²	1	9	yes	—
8	C₃²⋊#4294	C₃²	1	18	yes	complemented
9	C₃²⋊#4294	C₃²	1	54	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	#10624883	—	19	1	no	maximal, Sylow
15	C₃³×C₁₉	—	3	9	no	—
16	C₃³×C₁₉	—	9	1	no	—
17	C₃³×C₁₉	—	9	3	no	—
18	C₃³×C₁₉	—	9	2	no	—
19	C₃³×C₁₉	—	9	6	no	—
20	C₃³×C₁₉	3₋¹⁺²	1	9	yes	complemented
21	C₃³×C₁₉	C₃³	1	9	yes	—
22	C₃³×C₁₉	C₃³	1	3	yes	—
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₁₉×H(3,𝔽₃)	—	3	27	no	—
25	#10587025	—	9	27	no	—
26	C₃×#4294	—	9	9	no	—
27	C₃×#4294	C₃³	1	27	yes	—
28	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
29	C₉⋊H(3,𝔽₃)	—	19	27	no	—
30	#6891	—	19	9	no	—
31	C₃×C₃²⋊C₉	—	19	3	no	—
32	C₃²×C₁₉	—	3	3	no	—
33	C₃²×C₁₉	—	3	1	no	—
34	C₃²×C₁₉	—	3	2	no	—
35	C₃²×C₁₉	—	3	6	no	—
36	C₃²×C₁₉	—	3	9	no	—
37	C₃²×C₁₉	—	3	9	no	—
38	C₃²×C₁₉	—	3	18	no	—
39	C₃²×C₁₉	—	9	9	no	—
40	C₃²×C₁₉	—	9	27	no	—
41	C₃²×C₁₉	C₃×3₋¹⁺²	1	9	yes	—
42	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
43	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
44	#4294	—	3	27	no	—
45	C₃⁴	C₃×C₁₉⋊C₃	1	3	yes	—
46	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
47	C₃×3₋¹⁺²	—	19	27	no	—
48	C₃×H(3,𝔽₃)	—	3	9	no	—
49	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	—
50	C₃²×C₉	—	19	9	no	—
51	C₃²⋊C₉	—	19	9	no	—
52	C₃²⋊C₉	—	19	18	no	—
53	C₃²⋊C₉	—	19	54	no	—
54	C₃×C₁₉	—	3	3	no	—
55	C₃×C₁₉	—	3	9	no	—
56	C₃×C₁₉	—	9	9	no	—
57	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
58	C₃×C₁₉	H(5,𝔽₃)	1	1	yes	characteristic
59	C₃×C₁₉	3₋¹⁺⁴	1	2	yes	—
60	C₃³	—	3	9	no	—
61	C₃³	—	9	1	no	—
62	C₃³	—	9	3	no	—
63	C₃³	—	9	2	no	—
64	C₃³	—	9	6	no	—
65	C₃³	#10587025	1	9	yes	complemented
66	C₃³	C₃²×C₁₉⋊C₃	1	9	yes	—
67	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
68	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
69	3₋¹⁺²	—	171	27	no	—
70	H(3,𝔽₃)	—	3	27	no	—
71	C₃×C₉	—	19	27	no	—
72	C₃×C₉	—	171	9	no	—
73	C₁₉	#10624883	1	1	yes	characteristic, complemented, Sylow
74	C₃²	—	3	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	6	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	3	9	no	—
80	C₃²	—	3	18	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	27	no	—
83	C₃²	C₃×#10587025	1	9	yes	—
84	C₃²	C₃×#10587025	1	3	yes	—
85	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
86	C₉	—	57	27	no	—
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	C₃²×#10587025	1	1	yes	characteristic
91	C₃	C₁₉⋊H(5,𝔽₃)	1	1	yes	characteristic
92	C₃	#17987547	1	2	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	#10624883
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. #17987423	C₃	semidirect
2. #17987421	C₃	semidirect
3. C₃×C₃²⋊#4294	C₃	semidirect
5. C₃×#10587025	C₃²	semidirect
6. C₃²×#4294	C₃²	semidirect
8. C₃²⋊#4294	C₃²	semidirect
9. C₃²⋊#4294	C₃²	semidirect
20. C₃³×C₁₉	3₋¹⁺²	semidirect
65. C₃³	#10587025	semidirect
73. C₁₉	#10624883	semidirect

Maximal subgroups

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

subgroup	index
#17987423	3
#17987421	3
C₃×C₃²⋊#4294	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624883	19