Subgroups — #28356774

Order 13851.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987558	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#4296	—	3	3	no	—
4	C₃³×C₁₉⋊C₃	C₉	1	2	yes	complemented
5	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
6	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
7	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
8	C₃×C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
9	C₃⁴×C₁₉	C₉	1	1	yes	characteristic, complemented
10	#10624505	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	—	3	3	no	—
12	C₃³×C₁₉	—	3	1	no	—
13	C₃³×C₁₉	—	9	3	no	—
14	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₁₉⋊C₃	—	3	2	no	—
16	C₃²×C₁₉⋊C₃	—	3	3	no	—
17	C₃²×C₁₉⋊C₃	—	3	6	no	—
18	C₃²×C₁₉⋊C₃	—	9	6	no	—
19	C₃²×C₁₉⋊C₃	—	9	3	no	—
20	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	2	yes	—
22	C₃×#4296	—	9	3	no	—
23	C₃²⋊(C₁₉⋊C₃)	—	3	18	no	—
24	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
25	C₃²×H(3,𝔽₃)	—	19	1	no	—
26	#6866	—	19	3	no	—
27	C₃²×C₁₉	—	3	3	no	—
28	C₃²×C₁₉	—	3	1	no	—
29	C₃²×C₁₉	—	3	3	no	—
30	C₃²×C₁₉	—	9	9	no	—
31	C₃²×C₁₉	—	9	3	no	—
32	C₃²×C₁₉	C₃²⋊C₉	1	1	yes	characteristic
33	C₃×C₁₉⋊C₃	—	3	6	no	—
34	C₃×C₁₉⋊C₃	—	3	2	no	—
35	C₃×C₁₉⋊C₃	—	9	1	no	—
36	C₃×C₁₉⋊C₃	—	9	9	no	—
37	C₃×C₁₉⋊C₃	—	9	6	no	—
38	C₃×C₁₉⋊C₃	—	9	18	no	—
39	C₃×C₁₉⋊C₃	—	9	2	no	—
40	C₃×C₁₉⋊C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
41	#4296	—	27	3	no	—
42	C₃⁴	—	19	1	no	—
43	C₃⁴	—	19	2	no	—
44	C₃⁴	#4296	1	1	yes	characteristic, complemented
45	C₃×H(3,𝔽₃)	—	57	1	no	—
46	C₃×H(3,𝔽₃)	—	57	2	no	—
47	C₃×H(3,𝔽₃)	—	57	9	no	—
48	C₃²⋊C₉	—	57	3	no	—
49	C₃×C₁₉	—	3	3	no	—
50	C₃×C₁₉	—	3	1	no	—
51	C₃×C₁₉	—	9	3	no	—
52	C₃×C₁₉	#6866	1	1	yes	characteristic
53	C₁₉⋊C₃	—	3	3	no	—
54	C₁₉⋊C₃	—	9	6	no	—
55	C₁₉⋊C₃	—	9	2	no	—
56	C₃³	—	3	3	no	—
57	C₃³	—	3	1	no	—
58	C₃³	—	9	3	no	—
59	C₃³	—	19	1	no	—
60	C₃³	—	19	2	no	—
61	C₃³	—	57	3	no	—
62	C₃³	—	57	6	no	—
63	C₃³	—	57	2	no	—
64	C₃³	—	171	3	no	—
65	C₃³	—	171	6	no	—
66	C₃³	C₃×#4296	1	1	yes	characteristic
67	H(3,𝔽₃)	—	57	9	no	—
68	H(3,𝔽₃)	—	57	18	no	—
69	C₃×C₉	—	171	3	no	—
70	C₁₉	#10624505	1	1	yes	characteristic, complemented, Sylow
71	C₃²	—	3	3	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	9	9	no	—
75	C₃²	—	9	3	no	—
76	C₃²	—	19	3	no	—
77	C₃²	—	57	2	no	—
78	C₃²	—	57	6	no	—
79	C₃²	—	171	1	no	—
80	C₃²	—	171	9	no	—
81	C₃²	—	171	2	no	—
82	C₃²	—	171	18	no	—
83	C₃²	—	171	6	no	—
84	C₃²	C₃²⋊#4296	1	1	yes	characteristic
85	C₉	—	513	3	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	3	no	—
88	C₃	—	9	3	no	—
89	C₃	—	57	3	no	—
90	C₃	—	171	2	no	—
91	C₃	—	171	6	no	—
92	C₃	#17987558	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987558
commutator	C₃³×C₁₉	C₃×C₉
Frattini	C₃³	C₃×#4296
Fitting	C₃⁴×C₁₉	C₉
socle	C₃×C₁₉	#6866
prime radical	C₃²×C₃²⋊(C₁₉⋊C₃)	C₃
cosocle	C₃³×C₁₉⋊C₃	C₃²
hypercenter	C₃⁴	#4296
hypocenter	C₁₉	#10624505
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃³ ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
1. #17987558	C₃	semidirect
4. C₃³×C₁₉⋊C₃	C₉	semidirect
9. C₃⁴×C₁₉	C₉	semidirect
44. C₃⁴	#4296	semidirect
70. C₁₉	#10624505	semidirect

Maximal subgroups

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

subgroup	index
#17987558	3
C₃²×C₃²⋊(C₁₉⋊C₃)	3
#10624505	19