Subgroups — #28356047

Order 13851.

1718 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	3	yes	maximal, complemented
2	#17987417	C₃	1	9	yes	maximal, complemented
3	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
4	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃³×C₁₉⋊C₃	C₃²	1	9	yes	complemented
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	27	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	18	yes	complemented
8	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	9	yes	complemented
9	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	81	yes	complemented
10	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
11	C₃⁴×C₁₉	C₃²	1	3	yes	complemented
12	C₃×C₁₉×H(3,𝔽₃)	—	3	9	no	—
13	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	complemented
14	#10624880	—	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	complemented
19	C₃³×C₁₉	C₃³	1	9	yes	—
20	C₃³×C₁₉	C₃³	1	3	yes	—
21	C₃³×C₁₉	C₃³	1	1	yes	characteristic
22	C₃²×C₁₉⋊C₃	—	3	27	no	—
23	C₃²×C₁₉⋊C₃	—	9	27	no	—
24	C₃²×C₁₉⋊C₃	C₃³	1	27	yes	—
25	C₁₉×H(3,𝔽₃)	—	3	27	no	—
26	C₃²⋊(C₁₉⋊C₃)	—	3	27	no	—
27	C₃²⋊(C₁₉⋊C₃)	—	3	54	no	—
28	C₃²⋊(C₁₉⋊C₃)	—	9	81	no	—
29	C₃²×H(3,𝔽₃)	—	19	3	no	—
30	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
31	C₃²⋊H(3,𝔽₃)	—	19	9	no	—
32	C₃²⋊H(3,𝔽₃)	—	19	27	no	—
33	C₃²×C₁₉	—	3	9	no	—
34	C₃²×C₁₉	—	3	3	no	—
35	C₃²×C₁₉	—	3	9	no	—
36	C₃²×C₁₉	—	3	27	no	—
37	C₃²×C₁₉	—	9	9	no	—
38	C₃²×C₁₉	—	9	27	no	—
39	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
40	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	9	yes	—
41	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
42	C₃×C₁₉⋊C₃	—	3	27	no	—
43	C₃×C₁₉⋊C₃	—	3	81	no	—
44	C₃×C₁₉⋊C₃	—	9	81	no	—
45	C₃⁴	—	19	9	no	—
46	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
47	C₃⁴	C₃×C₁₉⋊C₃	1	3	yes	complemented
48	C₃×H(3,𝔽₃)	—	3	9	no	—
49	C₃×H(3,𝔽₃)	—	19	9	no	—
50	C₃×H(3,𝔽₃)	—	19	18	no	—
51	C₃×H(3,𝔽₃)	—	19	81	no	—
52	C₃×H(3,𝔽₃)	—	57	27	no	—
53	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	complemented
54	C₃×C₁₉	—	3	9	no	—
55	C₃×C₁₉	—	3	3	no	—
56	C₃×C₁₉	—	9	9	no	—
57	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×C₁₉	H(5,𝔽₃)	1	3	yes	—
59	C₁₉⋊C₃	—	9	27	no	—
60	C₃³	—	3	9	no	—
61	C₃³	—	9	3	no	—
62	C₃³	—	9	9	no	—
63	C₃³	—	19	27	no	—
64	C₃³	—	57	27	no	—
65	C₃³	—	171	27	no	—
66	C₃³	C₃²⋊(C₁₉⋊C₃)	1	9	yes	complemented
67	C₃³	C₃²×C₁₉⋊C₃	1	9	yes	—
68	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
69	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
70	H(3,𝔽₃)	—	3	27	no	—
71	H(3,𝔽₃)	—	57	54	no	—
72	H(3,𝔽₃)	—	57	27	no	—
73	H(3,𝔽₃)	—	171	81	no	—
74	C₁₉	#10624880	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	3	no	—
76	C₃²	—	3	9	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	3	27	no	—
79	C₃²	—	9	9	no	—
80	C₃²	—	9	27	no	—
81	C₃²	—	57	27	no	—
82	C₃²	—	57	81	no	—
83	C₃²	—	171	81	no	—
84	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	9	yes	—
85	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
86	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	—	171	27	no	—
91	C₃	C₃²×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
92	C₃	C₁₉⋊H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₃²×C₁₉×H(3,𝔽₃)	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624880
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×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
2. #17987417	C₃	semidirect
3. (C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
4. C₃²×C₁₉×H(3,𝔽₃)	C₃	semidirect
5. C₃³×C₁₉⋊C₃	C₃²	semidirect
7. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
8. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
9. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
11. C₃⁴×C₁₉	C₃²	semidirect
13. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
18. C₃³×C₁₉	H(3,𝔽₃)	semidirect
30. C₃²×H(3,𝔽₃)	C₁₉⋊C₃	semidirect
47. C₃⁴	C₃×C₁₉⋊C₃	semidirect
53. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
66. C₃³	C₃²⋊(C₁₉⋊C₃)	semidirect
74. C₁₉	#10624880	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃²⋊(C₁₉⋊C₃)	3
#17987417	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624880	19