Subgroups — #28354561

Order 13851.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987421	C₃	1	1	yes	maximal, characteristic
2	#17987421	C₃	1	2	yes	maximal
3	C₁₉⋊#6862	C₃	1	9	yes	maximal, complemented
4	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
5	C₃×#10587025	—	3	1	no	—
6	C₃×#10587025	—	3	2	no	—
7	C₃²⋊#4294	—	3	2	no	—
8	C₃²⋊#4294	—	3	4	no	—
9	C₃²⋊#4294	C₃²	1	3	yes	—
10	C₃²⋊#4294	C₃²	1	6	yes	—
11	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
12	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
13	C₁₉×C₃²⋊C₉	—	3	2	no	—
14	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
15	#10624675	—	19	1	no	maximal, Sylow
16	C₃³×C₁₉	—	9	1	no	—
17	C₃³×C₁₉	—	9	2	no	—
18	C₃³×C₁₉	—	9	1	no	—
19	C₃³×C₁₉	H(3,𝔽₃)	1	2	yes	—
20	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃³×C₁₉	C₃³	1	1	yes	characteristic
22	C₁₉×3₋¹⁺²	—	9	3	no	—
23	C₃×C₉×C₁₉	—	3	3	no	—
24	#10587025	—	9	3	no	—
25	#10587025	—	9	6	no	—
26	C₃×#4294	—	3	3	no	—
27	C₃×#4294	—	3	6	no	—
28	#6891	—	19	1	no	—
29	#6891	—	19	2	no	—
30	#6891	C₁₉⋊C₃	1	1	yes	characteristic
31	#6862	—	19	9	no	—
32	C₃²×C₁₉	—	3	1	no	—
33	C₃²×C₁₉	—	3	2	no	—
34	C₃²×C₁₉	—	3	1	no	—
35	C₃²×C₁₉	—	3	2	no	—
36	C₃²×C₁₉	—	3	2	no	—
37	C₃²×C₁₉	—	3	4	no	—
38	C₃²×C₁₉	—	3	1	no	—
39	C₃²×C₁₉	—	3	1	no	—
40	C₃²×C₁₉	—	3	2	no	—
41	C₃²×C₁₉	—	9	9	no	—
42	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
43	C₉×C₁₉	—	9	3	no	—
44	#4294	—	9	3	no	—
45	#4294	—	9	6	no	—
46	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
47	C₃×3₋¹⁺²	—	3	1	no	—
48	C₃×3₋¹⁺²	—	57	1	no	—
49	C₃×3₋¹⁺²	—	57	2	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	19	3	no	—
52	C₃²⋊C₉	—	19	6	no	—
53	C₃²⋊C₉	—	57	2	no	—
54	C₃²⋊C₉	—	57	4	no	—
55	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
56	C₃×C₁₉	—	9	1	no	—
57	C₃×C₁₉	—	9	2	no	—
58	C₃×C₁₉	—	9	1	no	—
59	C₃×C₁₉	#6910	1	1	yes	characteristic
60	C₃×C₁₉	#6910	1	1	yes	characteristic
61	C₃×C₁₉	#6910	1	2	yes	—
62	C₃³	—	9	1	no	—
63	C₃³	—	9	1	no	—
64	C₃³	—	9	2	no	—
65	C₃³	C₃²⋊(C₁₉⋊C₃)	1	2	yes	—
66	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
67	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
68	3₋¹⁺²	—	9	3	no	—
69	3₋¹⁺²	—	171	3	no	—
70	3₋¹⁺²	—	171	6	no	—
71	C₃×C₉	—	3	3	no	—
72	C₃×C₉	—	57	3	no	—
73	C₃×C₉	—	57	6	no	—
74	C₁₉	#10624675	1	1	yes	characteristic, complemented, Sylow
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	1	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	3	1	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	3	4	no	—
83	C₃²	—	3	2	no	—
84	C₃²	—	9	9	no	—
85	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
86	C₉	—	9	3	no	—
87	C₉	—	171	3	no	—
88	C₉	—	171	6	no	—
89	C₃	—	9	1	no	—
90	C₃	—	9	2	no	—
91	C₃	—	9	1	no	—
92	C₃	#17987509	1	1	yes	characteristic
93	C₃	#17987510	1	1	yes	characteristic
94	C₃	#17987511	1	2	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624675
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
3. C₁₉⋊#6862	C₃	semidirect
74. C₁₉	#10624675	semidirect

Maximal subgroups

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

subgroup	index
#17987421	3
C₁₉⋊#6862	3
C₁₉×#6891	3
#10624675	19