Subgroups — #28356875

Order 13851.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal
2	#17987555	C₃	1	9	yes	maximal, complemented
3	C₉²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
4	(C₁₉⋊C₃)⋊C₂₇	C₉	1	27	yes	complemented
5	C₃×#10587020	C₃²	1	9	yes	—
6	C₃×C₉×C₁₉⋊C₃	C₉	1	3	yes	—
7	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
8	C₉×#4294	C₉	1	2	yes	—
9	C₉×#4294	C₉	1	3	yes	—
10	C₉×#4294	C₃²	1	3	yes	—
11	C₉²×C₁₉	C₉	1	1	yes	characteristic
12	C₃×#6864	—	19	1	no	maximal, Sylow
13	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
16	C₃×#4294	C₃×C₉	1	6	yes	—
17	C₃×#4294	C₃×C₉	1	1	yes	characteristic
18	C₃×#4294	C₃³	1	1	yes	characteristic
19	C₉×C₁₉⋊C₃	—	3	9	no	—
20	C₉×C₁₉⋊C₃	C₃×C₉	1	9	yes	—
21	#10587020	C₃×C₉	1	27	yes	complemented
22	C₃²×C₂₇	—	19	3	no	—
23	C₃×C₉²	—	19	1	no	—
24	#6864	—	19	9	no	—
25	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
26	C₃×C₁₉⋊C₃	C₉²	1	3	yes	—
27	C₃×C₁₉⋊C₃	C₂₇⋊C₃	1	9	yes	—
28	C₉×C₁₉	—	3	3	no	—
29	C₉×C₁₉	C₉²	1	3	yes	—
30	#4294	—	3	6	no	—
31	#4294	C₉²	1	3	yes	—
32	#4294	C₃²×C₉	1	3	yes	—
33	C₃²×C₉	—	19	1	no	—
34	C₃²×C₉	—	19	3	no	—
35	C₃×C₂₇	—	19	9	no	—
36	C₃×C₂₇	—	19	27	no	—
37	C₉²	—	19	3	no	—
38	C₉²	—	19	3	no	—
39	C₉²	—	19	2	no	—
40	C₉²	#4296	1	1	yes	characteristic
41	C₃×C₁₉	C₃×C₉²	1	1	yes	characteristic
42	C₃×C₁₉	C₃×C₂₇⋊C₃	1	3	yes	—
43	C₁₉⋊C₃	#6864	1	9	yes	complemented
44	C₃³	—	19	1	no	—
45	C₃×C₉	—	19	1	no	—
46	C₃×C₉	—	19	1	no	—
47	C₃×C₉	—	19	9	no	—
48	C₃×C₉	—	19	6	no	—
49	C₃×C₉	—	57	9	no	—
50	C₃×C₉	C₃×#4296	1	3	yes	—
51	C₃×C₉	C₃×#4296	1	1	yes	characteristic
52	C₂₇	—	19	27	no	—
53	C₁₉	C₃×#6864	1	1	yes	characteristic, complemented, Sylow
54	C₃²	—	19	3	no	—
55	C₃²	—	19	9	no	—
56	C₃²	C₃²×#4296	1	1	yes	characteristic
57	C₉	—	3	3	no	—
58	C₉	—	19	3	no	—
59	C₉	—	19	3	no	—
60	C₉	—	57	6	no	—
61	C₉	C₉×#4296	1	3	yes	—
62	C₃	—	19	9	no	—
63	C₃	C₃×C₉×#4296	1	1	yes	characteristic
64	C₃	(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×#4296
commutator	C₃×C₁₉	C₃×C₉²
Frattini	C₃²	C₃²×#4296
Fitting	C₉²×C₁₉	C₉
socle	C₃²×C₁₉	C₃²×C₉
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₉²	#4296
hypocenter	C₁₉	C₃×#6864
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃×C₉ ◁ C₉²

Complements

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

N	H ≅ G/N	type
2. #17987555	C₃	semidirect
4. (C₁₉⋊C₃)⋊C₂₇	C₉	semidirect
21. #10587020	C₃×C₉	semidirect
43. C₁₉⋊C₃	#6864	semidirect
53. C₁₉	C₃×#6864	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₁₉⋊C₃)⋊C₂₇	3
#17987555	3
C₉²×C₁₉⋊C₃	3
C₃×#6864	19