Subgroups — #28356800

Order 13851.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987560	C₃	1	3	yes	maximal
2	C₉²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#4296	—	3	3	no	—
4	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
5	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
6	C₉×#4294	—	3	2	no	—
7	C₉×#4294	C₉	1	2	yes	complemented
8	C₉²×C₁₉	C₉	1	1	yes	characteristic, complemented
9	#10624519	—	19	1	no	maximal, Sylow
10	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
11	C₃×#4296	—	9	3	no	—
12	C₃×C₉×C₁₉	—	3	1	no	—
13	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
14	C₃×#4294	—	3	2	no	—
15	C₃×#4294	3₋¹⁺²	1	2	yes	—
16	C₉×C₁₉⋊C₃	—	3	3	no	—
17	C₉×C₁₉⋊C₃	—	3	6	no	—
18	C₉×C₁₉⋊C₃	—	3	1	no	—
19	C₉×C₁₉⋊C₃	—	3	2	no	—
20	C₃×C₉²	—	19	1	no	—
21	#6868	—	19	3	no	—
22	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
23	C₃×C₁₉⋊C₃	—	3	1	no	—
24	C₃×C₁₉⋊C₃	—	3	2	no	—
25	C₃×C₁₉⋊C₃	#732	1	1	yes	characteristic
26	C₃×C₁₉⋊C₃	#733	1	2	yes	—
27	#4296	—	27	3	no	—
28	C₉×C₁₉	—	3	3	no	—
29	C₉×C₁₉	—	3	1	no	—
30	#4294	—	3	6	no	—
31	#4294	—	3	2	no	—
32	C₃²×C₉	—	19	1	no	—
33	C₃²×C₉	—	57	1	no	—
34	C₃²⋊C₉	—	57	3	no	—
35	C₉²	—	19	2	no	—
36	C₉²	—	57	2	no	—
37	C₉²	#4296	1	1	yes	characteristic, complemented
38	C₃×C₁₉	—	3	1	no	—
39	C₃×C₁₉	#6868	1	1	yes	characteristic
40	C₁₉⋊C₃	—	3	2	no	—
41	C₁₉⋊C₃	#6879	1	1	yes	characteristic
42	C₁₉⋊C₃	#6880	1	2	yes	—
43	C₃³	—	19	1	no	—
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	19	2	no	—
46	C₃×C₉	—	57	1	no	—
47	C₃×C₉	—	57	2	no	—
48	C₃×C₉	—	57	3	no	—
49	C₃×C₉	—	57	6	no	—
50	C₃×C₉	—	57	2	no	—
51	C₃×C₉	—	171	3	no	—
52	C₃×C₉	C₃×#4296	1	1	yes	characteristic
53	C₁₉	#10624519	1	1	yes	characteristic, complemented, Sylow
54	C₃²	—	19	1	no	—
55	C₃²	—	19	2	no	—
56	C₃²	—	57	1	no	—
57	C₃²	—	57	2	no	—
58	C₃²	C₃²⋊#4296	1	1	yes	characteristic
59	C₉	—	513	3	no	—
60	C₉	—	3	3	no	—
61	C₉	—	3	1	no	—
62	C₉	—	57	6	no	—
63	C₉	—	57	2	no	—
64	C₃	—	3	1	no	—
65	C₃	—	19	1	no	—
66	C₃	—	19	2	no	—
67	C₃	—	57	2	no	—
68	C₃	#17987560	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987560
commutator	C₃×C₉×C₁₉	C₃×C₉
Frattini	C₃×C₉	C₃×#4296
Fitting	C₉²×C₁₉	C₉
socle	C₃×C₁₉	#6868
prime radical	C₃²×C₁₉⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₁₉⋊C₃	C₃²
hypercenter	C₉²	#4296
hypocenter	C₁₉	#10624519
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
7. C₉×#4294	C₉	semidirect
8. C₉²×C₁₉	C₉	semidirect
37. C₉²	#4296	semidirect
53. C₁₉	#10624519	semidirect

Maximal subgroups

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

subgroup	index
#17987560	3
C₉²×C₁₉⋊C₃	3
#10624519	19