Subgroups — #28355924

Order 13851.

1178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)×#4294	C₃	1	27	yes	maximal, complemented
2	C₃×C₃²⋊#4294	C₃	1	12	yes	maximal, complemented
3	C₃²×C₁₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4294	C₃²	1	36	yes	complemented
5	C₃²⋊#4294	C₃²	1	9	yes	—
6	C₃²⋊#4294	C₃²	1	72	yes	complemented
7	C₃⁴×C₁₉	C₃²	1	4	yes	—
8	C₃×C₁₉×H(3,𝔽₃)	C₉	1	27	yes	complemented
9	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	9	yes	—
10	#10624862	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	—	9	4	no	—
12	C₃³×C₁₉	—	9	8	no	—
13	C₃³×C₁₉	C₃×C₉	1	36	yes	complemented
14	C₃³×C₁₉	C₃³	1	12	yes	—
15	C₃³×C₁₉	C₃³	1	1	yes	characteristic
16	C₁₉×H(3,𝔽₃)	C₃×C₉	1	81	yes	complemented
17	C₃×#4294	—	9	36	no	—
18	C₃×#4294	C₃³	1	27	yes	—
19	C₃²×H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic
20	C₉×H(3,𝔽₃)	—	19	27	no	—
21	C₃×C₃²⋊C₉	—	19	12	no	—
22	C₃²×C₁₉	—	3	12	no	—
23	C₃²×C₁₉	—	3	1	no	—
24	C₃²×C₁₉	—	3	2	no	—
25	C₃²×C₁₉	—	3	24	no	—
26	C₃²×C₁₉	—	9	36	no	—
27	C₃²×C₁₉	C₃²×C₉	1	36	yes	—
28	C₃²×C₁₉	C₃²×C₉	1	3	yes	—
29	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
30	#4294	—	3	27	no	—
31	C₃⁴	C₃×C₁₉⋊C₃	1	4	yes	—
32	C₃×H(3,𝔽₃)	#4294	1	27	yes	complemented
33	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	9	yes	—
34	C₃²×C₉	—	19	36	no	—
35	C₃²⋊C₉	—	19	9	no	—
36	C₃²⋊C₉	—	19	72	no	—
37	C₃×C₁₉	—	3	3	no	—
38	C₃×C₁₉	—	3	36	no	—
39	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
40	C₃×C₁₉	H(5,𝔽₃)	1	1	yes	characteristic
41	C₃×C₁₉	3₋¹⁺⁴	1	2	yes	—
42	C₃³	—	9	4	no	—
43	C₃³	—	9	8	no	—
44	C₃³	C₃×#4294	1	36	yes	complemented
45	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
46	C₃³	C₃²×C₁₉⋊C₃	1	12	yes	—
47	H(3,𝔽₃)	C₃×#4294	1	81	yes	complemented
48	C₃×C₉	—	19	27	no	—
49	C₃×C₉	—	171	36	no	—
50	C₁₉	#10624862	1	1	yes	characteristic, complemented, Sylow
51	C₃²	—	3	12	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	2	no	—
54	C₃²	—	3	24	no	—
55	C₃²	—	9	36	no	—
56	C₃²	C₃²×#4294	1	36	yes	—
57	C₃²	C₃²×#4294	1	3	yes	—
58	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
59	C₉	—	57	27	no	—
60	C₃	—	3	3	no	—
61	C₃	—	3	36	no	—
62	C₃	C₃³×#4294	1	1	yes	characteristic
63	C₃	C₁₉⋊H(5,𝔽₃)	1	1	yes	characteristic
64	C₃	#17987547	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Fitting, prime radical	C₃²×C₁₉×H(3,𝔽₃)	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
hypercenter	C₃²×H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624862
LCS	G ▷ C₃×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. H(3,𝔽₃)×#4294	C₃	semidirect
2. C₃×C₃²⋊#4294	C₃	semidirect
4. C₃²×#4294	C₃²	semidirect
6. C₃²⋊#4294	C₃²	semidirect
8. C₃×C₁₉×H(3,𝔽₃)	C₉	semidirect
13. C₃³×C₁₉	C₃×C₉	semidirect
16. C₁₉×H(3,𝔽₃)	C₃×C₉	semidirect
32. C₃×H(3,𝔽₃)	#4294	semidirect
44. C₃³	C₃×#4294	semidirect
47. H(3,𝔽₃)	C₃×#4294	semidirect
50. C₁₉	#10624862	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×C₃²⋊#4294	3
C₃²×C₁₉×H(3,𝔽₃)	3
#10624862	19