Subgroups — #28356808

Order 13851.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#4296	C₃	1	1	yes	maximal, characteristic
2	#17987560	C₃	1	2	yes	maximal
3	C₁₉⋊C₃×#733	C₃	1	1	yes	maximal, characteristic
4	C₃²×#4296	—	3	1	no	—
5	C₃²⋊#4296	—	3	2	no	—
6	C₉×#4296	—	3	3	no	—
7	3₋¹⁺²×C₁₉⋊C₃	—	3	1	no	—
8	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
9	#13095322	C₉	1	2	yes	complemented
10	#13095321	—	3	2	no	—
11	C₁₉×#733	C₉	1	1	yes	characteristic, complemented
12	#10624525	—	19	1	no	maximal, Sylow
13	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃×#4296	—	3	6	no	—
15	C₃×#4296	—	3	3	no	—
16	C₃×#4296	—	3	3	no	—
17	C₃×#4296	—	9	2	no	—
18	C₁₉×3₋¹⁺²	—	3	1	no	—
19	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
20	#10587025	—	3	2	no	—
21	C₃×#4294	3₋¹⁺²	1	2	yes	—
22	#10587023	—	9	1	no	—
23	#10587022	—	9	1	no	—
24	C₉×C₁₉⋊C₃	—	3	1	no	—
25	C₉×C₁₉⋊C₃	—	3	1	no	—
26	C₉×C₁₉⋊C₃	—	3	1	no	—
27	C₉×C₁₉⋊C₃	—	9	1	no	—
28	C₃×#733	—	19	1	no	—
29	C₃×C₉²	—	19	1	no	—
30	#6868	—	19	2	no	—
31	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
32	C₃×C₁₉⋊C₃	—	3	2	no	—
33	C₃×C₁₉⋊C₃	—	3	1	no	—
34	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
35	C₃×C₁₉⋊C₃	#731	1	1	yes	characteristic
36	C₃×C₁₉⋊C₃	#732	1	1	yes	characteristic
37	#4296	—	3	9	no	—
38	#4296	—	27	2	no	—
39	C₉×C₁₉	—	3	1	no	—
40	C₉×C₁₉	—	9	1	no	—
41	#4294	—	3	2	no	—
42	#4294	—	9	2	no	—
43	C₃×3₋¹⁺²	—	57	1	no	—
44	C₃²×C₉	—	19	1	no	—
45	C₃²×C₉	—	57	1	no	—
46	#733	—	19	2	no	—
47	#733	—	57	2	no	—
48	#733	#4296	1	1	yes	characteristic, complemented
49	C₃²⋊C₉	—	57	2	no	—
50	C₉²	—	57	3	no	—
51	C₃×C₁₉	—	3	1	no	—
52	C₃×C₁₉	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
53	C₁₉⋊C₃	—	3	1	no	—
54	C₁₉⋊C₃	—	3	1	no	—
55	C₁₉⋊C₃	#6878	1	2	yes	—
56	C₁₉⋊C₃	#6880	1	1	yes	characteristic
57	C₃³	—	19	1	no	—
58	3₋¹⁺²	—	3	1	no	—
59	3₋¹⁺²	—	57	2	no	—
60	3₋¹⁺²	—	171	1	no	—
61	3₋¹⁺²	—	171	1	no	—
62	C₃×C₉	—	19	2	no	—
63	C₃×C₉	—	57	1	no	—
64	C₃×C₉	—	57	1	no	—
65	C₃×C₉	—	57	1	no	—
66	C₃×C₉	—	57	3	no	—
67	C₃×C₉	—	57	6	no	—
68	C₃×C₉	—	57	3	no	—
69	C₃×C₉	—	171	2	no	—
70	C₃×C₉	—	171	1	no	—
71	C₃×C₉	C₃×#4296	1	1	yes	characteristic
72	C₁₉	#10624525	1	1	yes	characteristic, complemented, Sylow
73	C₃²	—	19	1	no	—
74	C₃²	—	19	1	no	—
75	C₃²	—	19	1	no	—
76	C₃²	—	57	2	no	—
77	C₃²	—	57	1	no	—
78	C₃²	C₃²⋊#4296	1	1	yes	characteristic
79	C₉	—	513	2	no	—
80	C₉	—	3	1	no	—
81	C₉	—	9	1	no	—
82	C₉	—	57	9	no	—
83	C₉	—	57	2	no	—
84	C₉	—	171	2	no	—
85	C₃	—	3	1	no	—
86	C₃	—	19	2	no	—
87	C₃	—	19	1	no	—
88	C₃	—	57	1	no	—
89	C₃	—	57	1	no	—
90	C₃	H(3,𝔽₃)⋊#4296	1	1	yes	characteristic

	subgroup	G/H
center	C₃	H(3,𝔽₃)⋊#4296
commutator	C₃×C₉×C₁₉	C₃×C₉
Frattini	C₃×C₉	C₃×#4296
Fitting	C₁₉×#733	C₉
socle	C₃×C₁₉	H(3,𝔽₃)⋊C₉
prime radical	C₃²×C₁₉⋊C₃	H(3,𝔽₃)
cosocle	C₃×C₉×C₁₉⋊C₃	C₃²
hypercenter	#733	#4296
hypocenter	C₁₉	#10624525
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #733

Complements

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

N	H ≅ G/N	type
9. #13095322	C₉	semidirect
11. C₁₉×#733	C₉	semidirect
48. #733	#4296	semidirect
72. C₁₉	#10624525	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#4296	3
#17987560	3
C₁₉⋊C₃×#733	3
#10624525	19