Subgroups — #28354828

Order 13851.

304 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#4294	C₃	1	3	yes	maximal, complemented
2	C₃×#13095306	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)⋊#4294	C₃	1	6	yes	maximal, complemented
4	#17987339	C₃	1	2	yes	maximal, complemented
5	C₃×C₁₉×#732	C₃	1	1	yes	maximal, characteristic
6	C₃²×#4294	—	3	3	no	—
7	C₃²×#4294	C₃²	1	2	yes	—
8	C₃²×#4294	C₃²	1	1	yes	characteristic
9	#13095306	—	3	3	no	—
10	C₃²⋊#4294	—	3	6	no	—
11	C₃²⋊#4294	C₃²	1	6	yes	—
12	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	3	yes	—
13	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
14	C₁₉×#732	C₉	1	9	yes	complemented
15	#10624722	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	3	3	no	—
17	C₃³×C₁₉	C₃³	1	1	yes	characteristic
18	C₁₉×H(3,𝔽₃)	—	3	6	no	—
19	C₁₉×H(3,𝔽₃)	C₃×C₉	1	9	yes	complemented
20	C₃×C₉×C₁₉	—	3	1	no	—
21	C₃×C₉×C₁₉	—	3	2	no	—
22	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	complemented
23	C₃×#4294	—	3	6	no	—
24	C₃×#4294	—	3	2	no	—
25	C₃×#4294	—	9	1	no	—
26	C₃×#4294	—	9	9	no	—
27	C₃×#4294	—	9	2	no	—
28	C₃×#4294	H(3,𝔽₃)	1	3	yes	—
29	C₃×#732	C₁₉⋊C₃	1	1	yes	characteristic
30	C₉×H(3,𝔽₃)	—	19	3	no	—
31	C₃×C₉⋊C₉	—	19	1	no	—
32	H(3,𝔽₃)⋊C₉	—	19	6	no	—
33	#6868	—	19	2	no	—
34	C₃²×C₁₉	—	3	2	no	—
35	C₃²×C₁₉	—	3	1	no	—
36	C₃²×C₁₉	—	3	9	no	—
37	C₃²×C₁₉	—	9	6	no	—
38	C₃²×C₁₉	—	9	3	no	—
39	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
40	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₃²×C₁₉	#737	1	2	yes	—
42	C₉×C₁₉	—	3	3	no	—
43	#4294	—	3	3	no	—
44	#4294	—	9	2	no	—
45	#4294	—	9	6	no	—
46	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	3	yes	—
47	C₃²×C₉	—	19	1	no	—
48	C₃²×C₉	—	19	2	no	—
49	C₃²×C₉	—	57	3	no	—
50	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
51	#732	#4294	1	9	yes	complemented
52	C₉⋊C₉	—	57	3	no	—
53	C₃²⋊C₉	—	19	6	no	—
54	C₃²⋊C₉	—	57	6	no	—
55	C₃×C₁₉	—	3	1	no	—
56	C₃×C₁₉	—	3	2	no	—
57	C₃×C₁₉	—	9	9	no	—
58	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×C₁₉	#6911	1	1	yes	characteristic
60	C₃×C₁₉	#6913	1	2	yes	—
61	C₃³	—	3	3	no	—
62	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
63	H(3,𝔽₃)	—	3	6	no	—
64	H(3,𝔽₃)	C₃×#4294	1	9	yes	complemented
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	19	3	no	—
68	C₃×C₉	—	57	6	no	—
69	C₃×C₉	—	57	2	no	—
70	C₃×C₉	—	171	1	no	—
71	C₃×C₉	—	171	9	no	—
72	C₃×C₉	—	171	2	no	—
73	C₃×C₉	C₃×#4294	1	3	yes	complemented
74	C₁₉	#10624722	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	9	3	no	—
79	C₃²	—	9	6	no	—
80	C₃²	C₃²×#4294	1	1	yes	characteristic
81	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
82	C₃²	#13095331	1	2	yes	—
83	C₉	—	3	3	no	—
84	C₉	—	57	3	no	—
85	C₉	—	171	6	no	—
86	C₉	—	171	2	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	2	no	—
89	C₃	—	9	9	no	—
90	C₃	H(3,𝔽₃)×#4294	1	1	yes	characteristic
91	C₃	#17987512	1	1	yes	characteristic
92	C₃	#17987524	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting, prime radical	C₃×C₁₉×#732	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
cosocle	C₃³×C₁₉	C₃³
hypercenter	C₃×#732	C₁₉⋊C₃
hypocenter	C₁₉	#10624722
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)×#4294	C₃	semidirect
2. C₃×#13095306	C₃	semidirect
3. H(3,𝔽₃)⋊#4294	C₃	semidirect
4. #17987339	C₃	semidirect
14. C₁₉×#732	C₉	semidirect
19. C₁₉×H(3,𝔽₃)	C₃×C₉	semidirect
22. C₃×C₉×C₁₉	C₃×C₉	semidirect
51. #732	#4294	semidirect
64. H(3,𝔽₃)	C₃×#4294	semidirect
73. C₃×C₉	C₃×#4294	semidirect
74. C₁₉	#10624722	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)×#4294	3
C₃×#13095306	3
H(3,𝔽₃)⋊#4294	3
#17987339	3
C₃×C₁₉×#732	3
#10624722	19