Subgroups — #28353236

Order 13851.

154 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	#17987335	C₃	1	2	yes	maximal
3	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
4	C₃×#10587025	—	3	1	no	—
5	C₃²⋊#4294	—	3	3	no	—
6	C₃²⋊#4294	—	3	2	no	—
7	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
8	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
9	C₁₉×C₃²⋊C₉	—	3	3	no	—
10	#10624476	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	—	3	1	no	—
12	C₃³×C₁₉	—	3	1	no	—
13	C₃³×C₁₉	—	3	2	no	—
14	C₃³×C₁₉	—	9	3	no	—
15	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
16	C₁₉×3₋¹⁺²	—	9	1	no	—
17	C₁₉×3₋¹⁺²	—	9	2	no	—
18	C₃×C₉×C₁₉	—	3	3	no	—
19	#10587025	—	9	1	no	—
20	#10587025	—	9	2	no	—
21	C₃×#4294	—	3	3	no	—
22	C₃×#4294	—	9	2	no	—
23	#6891	—	19	1	no	—
24	#6891	C₁₉⋊C₃	1	1	yes	characteristic
25	#6866	—	19	2	no	—
26	C₃²×C₁₉	—	3	3	no	—
27	C₃²×C₁₉	—	3	3	no	—
28	C₃²×C₁₉	—	9	1	no	—
29	C₃²×C₁₉	—	9	6	no	—
30	C₃²×C₁₉	—	9	2	no	—
31	C₃²×C₁₉	—	9	2	no	—
32	C₃²×C₁₉	—	9	1	no	—
33	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
34	C₃²×C₁₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₃²×C₁₉	#732	1	2	yes	—
36	C₉×C₁₉	—	9	3	no	—
37	#4294	—	9	3	no	—
38	#4294	—	27	2	no	—
39	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
40	C₃×3₋¹⁺²	—	3	1	no	—
41	C₃×3₋¹⁺²	—	57	1	no	—
42	C₃²⋊C₉	—	3	3	no	—
43	C₃²⋊C₉	—	57	3	no	—
44	C₃²⋊C₉	—	57	2	no	—
45	C₃×C₁₉	—	3	2	no	—
46	C₃×C₁₉	—	3	1	no	—
47	C₃×C₁₉	—	3	1	no	—
48	C₃×C₁₉	—	9	3	no	—
49	C₃×C₁₉	#6856	1	1	yes	characteristic
50	C₃³	—	3	1	no	—
51	C₃³	—	3	1	no	—
52	C₃³	—	3	2	no	—
53	C₃³	—	9	3	no	—
54	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
55	3₋¹⁺²	—	9	1	no	—
56	3₋¹⁺²	—	9	2	no	—
57	3₋¹⁺²	—	171	1	no	—
58	3₋¹⁺²	—	171	2	no	—
59	C₃×C₉	—	3	3	no	—
60	C₃×C₉	—	57	3	no	—
61	C₃×C₉	—	171	2	no	—
62	C₁₉	#10624476	1	1	yes	characteristic, complemented, Sylow
63	C₃²	—	3	3	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	9	1	no	—
66	C₃²	—	9	6	no	—
67	C₃²	—	9	2	no	—
68	C₃²	—	9	2	no	—
69	C₃²	—	9	1	no	—
70	C₃²	#13095313	1	1	yes	characteristic
71	C₃²	#13095314	1	1	yes	characteristic
72	C₃²	#13095319	1	2	yes	—
73	C₉	—	513	2	no	—
74	C₉	—	9	3	no	—
75	C₉	—	171	3	no	—
76	C₃	—	3	2	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	1	no	—
79	C₃	—	9	3	no	—
80	C₃	#17987309	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987309
commutator, prime radical, cosocle	C₃⁴×C₁₉	C₃²
Frattini	C₃⁴	C₃×C₁₉⋊C₃
Fitting	C₁₉×#6891	C₃
socle	C₃×C₁₉	#6856
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624476
LCS	G ▷ C₃⁴×C₁₉ ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
62. C₁₉	#10624476	semidirect

Maximal subgroups

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

subgroup	index
#17987421	3
#17987335	3
C₁₉×#6891	3
#10624476	19