Subgroups — #28357038

Order 13851.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987584	C₃	1	3	yes	maximal, complemented
2	#17987583	C₃	1	3	yes	maximal, complemented
3	#17987582	C₃	1	2	yes	maximal, complemented
4	(C₁₉⋊C₃)⋊C₈₁	C₃	1	3	yes	maximal, complemented
5	C₃×#13095335	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987481	C₃	1	1	yes	maximal, characteristic
7	#13095335	C₃²	1	1	yes	characteristic
8	#13095335	C₃²	1	6	yes	—
9	#13095335	C₃²	1	1	yes	characteristic
10	#13095335	C₃²	1	1	yes	characteristic
11	#13095332	C₉	1	1	yes	characteristic
12	#13095312	C₉	1	1	yes	characteristic
13	#13095311	C₉	1	3	yes	—
14	#13095310	C₉	1	3	yes	—
15	C₃×#10587018	C₃²	1	1	yes	characteristic
16	C₂₇×C₁₉⋊C₃	C₃²	1	3	yes	—
17	C₁₉×C₂₇⋊C₃	C₉	1	1	yes	characteristic
18	#10624833	—	19	1	no	maximal, Sylow
19	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
20	C₃²⋊(C₁₉⋊C₃)	C₂₇	1	1	yes	characteristic
21	#10587025	C₂₇	1	2	yes	—
22	#10587023	C₂₇	1	3	yes	—
23	#10587022	C₂₇	1	3	yes	—
24	C₉×C₁₉⋊C₃	C₃×C₉	1	3	yes	—
25	C₁₉×C₂₇	C₃×C₉	1	3	yes	—
26	#10587018	C₃×C₉	1	3	yes	—
27	#10587018	C₃×C₉	1	1	yes	characteristic
28	#10587018	C₃×C₉	1	1	yes	characteristic
29	#10587018	C₃³	1	1	yes	characteristic
30	#6903	—	19	1	no	—
31	C₈₁⋊C₃	—	19	3	no	—
32	C₈₁⋊C₃	—	19	2	no	—
33	C₈₁⋊C₃	—	19	3	no	—
34	C₃×C₈₁	—	19	1	no	—
35	C₃×C₈₁	—	19	3	no	—
36	C₃²×C₁₉	C₃×C₂₇	1	1	yes	characteristic
37	C₃×C₁₉⋊C₃	C₃×C₂₇	1	3	yes	—
38	C₉×C₁₉	C₃×C₂₇	1	1	yes	characteristic
39	C₉×C₁₉	C₃×C₂₇	1	1	yes	characteristic
40	C₉×C₁₉	C₃²×C₉	1	1	yes	characteristic
41	#4294	C₃×C₂₇	1	6	yes	—
42	#737	—	19	1	no	—
43	C₂₇⋊C₃	—	19	1	no	—
44	C₂₇⋊C₃	—	19	3	no	—
45	C₂₇⋊C₃	—	19	3	no	—
46	C₂₇⋊C₃	#4296	1	1	yes	characteristic
47	C₃×C₂₇	—	19	1	no	—
48	C₃×C₂₇	—	19	3	no	—
49	C₈₁	—	19	1	no	—
50	C₈₁	—	19	1	no	—
51	C₈₁	—	19	6	no	—
52	C₈₁	—	19	1	no	—
53	C₃×C₁₉	—	3	1	no	—
54	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
55	C₁₉⋊C₃	—	3	3	no	—
56	3₋¹⁺²	—	19	2	no	—
57	3₋¹⁺²	—	19	3	no	—
58	3₋¹⁺²	—	19	3	no	—
59	H(3,𝔽₃)	—	19	1	no	—
60	C₃×C₉	—	19	3	no	—
61	C₃×C₉	C₃×#4296	1	1	yes	characteristic
62	C₂₇	—	19	3	no	—
63	C₂₇	—	19	1	no	—
64	C₂₇	—	19	1	no	—
65	C₂₇	—	19	1	no	—
66	C₂₇	C₃×#4296	1	3	yes	—
67	C₁₉	#10624833	1	1	yes	characteristic, complemented, Sylow
68	C₃²	—	19	3	no	—
69	C₃²	(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic
70	C₉	—	19	6	no	—
71	C₉	(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic
72	C₉	(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic
73	C₉	C₃²×#4296	1	1	yes	characteristic
74	C₃	—	3	1	no	—
75	C₃	—	57	3	no	—
76	C₃	C₃×(C₁₉⋊C₃)⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₉	C₃²×#4296
commutator, socle	C₃×C₁₉	C₃²×C₂₇
Fitting	C₁₉×C₂₇⋊C₃	C₉
prime radical	C₃²⋊(C₁₉⋊C₃)	C₂₇
cosocle	#10587018	C₃³
hypercenter	C₂₇⋊C₃	#4296
hypocenter	C₁₉	#10624833
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
1. #17987584	C₃	semidirect
2. #17987583	C₃	semidirect
3. #17987582	C₃	semidirect
4. (C₁₉⋊C₃)⋊C₈₁	C₃	semidirect
5. C₃×#13095335	C₃	semidirect
67. C₁₉	#10624833	semidirect

Maximal subgroups

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

subgroup	index
#17987584	3
#17987583	3
#17987582	3
(C₁₉⋊C₃)⋊C₈₁	3
C₃×#13095335	3
#17987481	3
#10624833	19