Subgroups — #28355747

Order 13851.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095322	C₃	1	2	yes	maximal, complemented
2	C₃×#13095320	C₃	1	1	yes	maximal, characteristic
3	#17987369	C₃	1	9	yes	maximal, complemented
4	C₁₉×#6887	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#10587025	—	3	2	no	—
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
7	#13095322	—	3	4	no	—
8	#13095322	C₃²	1	6	yes	complemented
9	#13095320	—	3	2	no	—
10	#13095320	C₃²	1	3	yes	—
11	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
12	C₁₉×C₉⋊C₉	—	3	2	no	—
13	C₁₉×C₃²⋊C₉	—	3	1	no	—
14	C₉²×C₁₉	C₃²	1	3	yes	complemented
15	#10624838	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²×C₁₉⋊C₃	—	9	1	no	—
18	C₃×C₉×C₁₉	—	3	2	no	—
19	C₃×C₉×C₁₉	—	3	1	no	—
20	C₃×C₉×C₁₉	—	3	3	no	—
21	C₃×C₉×C₁₉	H(3,𝔽₃)	1	2	yes	—
22	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
23	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
24	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
25	#10587025	—	3	6	no	—
26	#10587025	—	9	2	no	—
27	#10587025	—	9	2	no	—
28	C₃×#4294	—	9	2	no	—
29	C₃×#733	—	19	2	no	—
30	C₃×#732	—	19	1	no	—
31	#6887	C₁₉⋊C₃	1	1	yes	characteristic, complemented
32	#6878	—	19	9	no	—
33	C₃²×C₁₉	—	9	1	no	—
34	C₃²×C₁₉	#732	1	2	yes	—
35	C₃²×C₁₉	#732	1	1	yes	characteristic
36	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	—	9	3	no	—
38	C₃×C₁₉⋊C₃	—	27	3	no	—
39	C₉×C₁₉	—	3	2	no	—
40	C₉×C₁₉	—	3	1	no	—
41	C₉×C₁₉	—	9	3	no	—
42	#4294	—	9	6	no	—
43	C₃×3₋¹⁺²	—	57	2	no	—
44	C₃×H(3,𝔽₃)	—	57	1	no	—
45	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
46	#733	—	19	6	no	—
47	#733	—	57	4	no	—
48	#732	—	19	3	no	—
49	#732	—	57	2	no	—
50	C₉⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	1	no	—
52	C₉²	C₃×C₁₉⋊C₃	1	3	yes	complemented
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	—	3	1	no	—
55	C₃×C₁₉	—	3	2	no	—
56	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
57	C₁₉⋊C₃	—	27	3	no	—
58	C₃³	—	171	1	no	—
59	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
60	3₋¹⁺²	—	57	6	no	—
61	3₋¹⁺²	—	171	2	no	—
62	3₋¹⁺²	—	171	2	no	—
63	H(3,𝔽₃)	—	57	3	no	—
64	H(3,𝔽₃)	—	171	2	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	3	no	—
68	C₃×C₉	—	171	2	no	—
69	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
70	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
71	C₁₉	#10624838	1	1	yes	characteristic, complemented, Sylow
72	C₃²	—	513	3	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	171	3	no	—
75	C₃²	#13095320	1	2	yes	—
76	C₃²	#13095320	1	1	yes	characteristic
77	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
78	C₉	—	3	2	no	—
79	C₉	—	3	1	no	—
80	C₉	—	9	3	no	—
81	C₉	—	171	6	no	—
82	C₃	—	513	3	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	2	no	—
86	C₃	C₃×#13095320	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×#13095320
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6887	C₃
socle	C₃×C₁₉	C₃×#732
prime radical	C₃×#13095320	C₃
hypercenter	#6887	C₁₉⋊C₃
hypocenter	C₁₉	#10624838
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. C₃×#13095322	C₃	semidirect
3. #17987369	C₃	semidirect
4. C₁₉×#6887	C₃	semidirect
8. #13095322	C₃²	semidirect
14. C₉²×C₁₉	C₃²	semidirect
31. #6887	C₁₉⋊C₃	semidirect
52. C₉²	C₃×C₁₉⋊C₃	semidirect
71. C₁₉	#10624838	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095322	3
C₃×#13095320	3
#17987369	3
C₁₉×#6887	3
#10624838	19