Subgroups — C₁₉⋊C₃×C₃²⋊H(3,𝔽₃) #28355937

Order 13851.

3824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	1	12	yes	maximal, complemented
3	#17987417	C₃	1	2	yes	maximal, complemented
4	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	24	yes	maximal, complemented
5	C₁₉×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₁₉⋊C₃	—	3	36	no	—
7	C₃³×C₁₉⋊C₃	C₃²	1	4	yes	—
8	C₃³×C₁₉⋊C₃	C₃²	1	8	yes	complemented
9	C₃³×C₁₉⋊C₃	C₃²	1	9	yes	complemented
10	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	72	yes	complemented
11	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	24	yes	complemented
12	H(3,𝔽₃)×C₁₉⋊C₃	—	3	108	no	—
13	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic, complemented
14	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	12	yes	complemented
15	C₃×C₃²⋊H(3,𝔽₃)	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	3	12	no	—
17	C₃³×C₁₉	C₃³	1	4	yes	—
18	C₃³×C₁₉	C₃³	1	9	yes	complemented
19	C₃²×C₁₉⋊C₃	—	3	243	no	—
20	C₃²×C₁₉⋊C₃	—	3	24	no	—
21	C₃²×C₁₉⋊C₃	—	3	36	no	—
22	C₃²×C₁₉⋊C₃	—	3	72	no	—
23	C₃²×C₁₉⋊C₃	—	3	108	no	—
24	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	36	yes	complemented
25	C₃²×C₁₉⋊C₃	C₃³	1	8	yes	—
26	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
27	C₃²×C₁₉⋊C₃	C₃³	1	18	yes	complemented
28	C₁₉×H(3,𝔽₃)	—	3	36	no	—
29	C₃²⋊(C₁₉⋊C₃)	—	3	72	no	—
30	C₃²⋊(C₁₉⋊C₃)	—	3	216	no	—
31	C₃⁵	—	19	1	no	—
32	C₃²×H(3,𝔽₃)	—	19	12	no	—
33	C₃²⋊H(3,𝔽₃)	—	19	2	no	—
34	C₃²⋊H(3,𝔽₃)	—	19	24	no	—
35	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
36	C₃²×C₁₉	—	3	27	no	—
37	C₃²×C₁₉	—	3	12	no	—
38	C₃²×C₁₉	—	3	36	no	—
39	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	12	yes	complemented
40	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
41	C₃×C₁₉⋊C₃	—	3	216	no	—
42	C₃×C₁₉⋊C₃	—	3	24	no	—
43	C₃×C₁₉⋊C₃	—	3	108	no	—
44	C₃×C₁₉⋊C₃	—	3	72	no	—
45	C₃×C₁₉⋊C₃	—	9	81	no	—
46	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	12	yes	—
47	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	24	yes	complemented
48	C₃⁴	—	19	4	no	—
49	C₃⁴	—	19	8	no	—
50	C₃⁴	—	19	9	no	—
51	C₃⁴	—	57	36	no	—
52	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
53	C₃×H(3,𝔽₃)	—	19	24	no	—
54	C₃×H(3,𝔽₃)	—	19	72	no	—
55	C₃×H(3,𝔽₃)	—	57	108	no	—
56	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	12	yes	complemented
57	C₃×C₁₉	—	3	12	no	—
58	C₃×C₁₉	—	9	9	no	—
59	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	4	yes	—
60	C₁₉⋊C₃	—	3	24	no	—
61	C₁₉⋊C₃	—	9	18	no	—
62	C₁₉⋊C₃	C₃²⋊H(3,𝔽₃)	1	9	yes	direct product factor
63	C₃³	—	3	12	no	—
64	C₃³	—	19	1	no	—
65	C₃³	—	19	36	no	—
66	C₃³	—	19	8	no	—
67	C₃³	—	19	18	no	—
68	C₃³	—	57	36	no	—
69	C₃³	—	57	72	no	—
70	C₃³	—	57	108	no	—
71	C₃³	—	57	243	no	—
72	C₃³	—	57	24	no	—
73	C₃³	C₃²×C₁₉⋊C₃	1	4	yes	—
74	C₃³	C₃²×C₁₉⋊C₃	1	9	yes	complemented
75	H(3,𝔽₃)	—	3	36	no	—
76	H(3,𝔽₃)	—	57	216	no	—
77	H(3,𝔽₃)	—	57	72	no	—
78	C₁₉	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
79	C₃²	—	3	27	no	—
80	C₃²	—	3	12	no	—
81	C₃²	—	3	36	no	—
82	C₃²	—	19	12	no	—
83	C₃²	—	19	24	no	—
84	C₃²	—	57	108	no	—
85	C₃²	—	57	216	no	—
86	C₃²	—	57	24	no	—
87	C₃²	—	57	72	no	—
88	C₃²	—	171	81	no	—
89	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	12	yes	complemented
90	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
91	C₃	—	3	12	no	—
92	C₃	—	9	9	no	—
93	C₃	—	19	9	no	—
94	C₃	—	57	24	no	—
95	C₃	—	171	18	no	—
96	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	4	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×C₃²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	C₃×C₃²⋊H(3,𝔽₃)
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. C₃⁴×C₁₉⋊C₃	C₃	semidirect
2. C₃×H(3,𝔽₃)×C₁₉⋊C₃	C₃	semidirect
3. #17987417	C₃	semidirect
4. (C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	semidirect
5. C₁₉×C₃²⋊H(3,𝔽₃)	C₃	semidirect
8. C₃³×C₁₉⋊C₃	C₃²	semidirect
9. C₃³×C₁₉⋊C₃	C₃²	semidirect
10. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
11. C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	semidirect
13. C₃⁴×C₁₉	C₃²	semidirect
14. C₃×C₁₉×H(3,𝔽₃)	C₃²	semidirect
18. C₃³×C₁₉	C₃³	semidirect
24. C₃²×C₁₉⋊C₃	H(3,𝔽₃)	semidirect
27. C₃²×C₁₉⋊C₃	C₃³	semidirect
35. C₃²⋊H(3,𝔽₃)	C₁₉⋊C₃	direct (N × H)
39. C₃²×C₁₉	C₃×H(3,𝔽₃)	semidirect
47. C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	semidirect
52. C₃⁴	C₃×C₁₉⋊C₃	semidirect
56. C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	semidirect
62. C₁₉⋊C₃	C₃²⋊H(3,𝔽₃)	direct (N × H)
74. C₃³	C₃²×C₁₉⋊C₃	semidirect
78. C₁₉	C₃×C₃²⋊H(3,𝔽₃)	semidirect
89. C₃²	H(3,𝔽₃)×C₁₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₁₉⋊C₃	3
C₃×H(3,𝔽₃)×C₁₉⋊C₃	3
#17987417	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
C₁₉×C₃²⋊H(3,𝔽₃)	3
C₃×C₃²⋊H(3,𝔽₃)	19

Subgroups — C19⋊C3×C32⋊H(3,𝔽3) #28355937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉⋊C₃×C₃²⋊H(3,𝔽₃) #28355937