Subgroups — H(3,𝔽₃)×#10587020 #28356980

Order 13851.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#10587020	C₃	1	4	yes	maximal, complemented
2	C₃²⋊#10587020	C₃	1	8	yes	maximal, complemented
3	H(3,𝔽₃)×#4294	C₃	1	1	yes	maximal, characteristic
4	C₃×#10587020	—	3	12	no	—
5	C₃×#10587020	C₃²	1	8	yes	—
6	C₃×#10587020	C₃²	1	1	yes	characteristic
7	C₃²×#4294	C₃²	1	4	yes	—
8	C₃²⋊#4294	C₉	1	8	yes	—
9	C₃×C₁₉×H(3,𝔽₃)	C₉	1	1	yes	characteristic
10	C₂₇×H(3,𝔽₃)	—	19	1	no	maximal, Sylow
11	C₃³×C₁₉	C₃×C₉	1	4	yes	—
12	C₁₉×H(3,𝔽₃)	C₂₇	1	9	yes	complemented
13	C₃×#4294	—	3	4	no	—
14	C₃×#4294	—	3	8	no	—
15	C₃×#4294	C₃×C₉	1	8	yes	—
16	C₃×#4294	C₃³	1	1	yes	characteristic
17	#10587020	—	3	8	no	—
18	#10587020	H(3,𝔽₃)	1	3	yes	direct product factor
19	C₃²×C₂₇	—	19	4	no	—
20	C₉×H(3,𝔽₃)	—	19	1	no	—
21	C₃²⋊C₂₇	—	19	8	no	—
22	C₃²×C₁₉	—	3	4	no	—
23	C₃²×C₁₉	—	3	8	no	—
24	C₃²×C₁₉	C₃×C₂₇	1	12	yes	complemented
25	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
26	#4294	—	3	8	no	—
27	#4294	C₃×H(3,𝔽₃)	1	1	yes	characteristic
28	#4294	#737	1	2	yes	—
29	C₃×H(3,𝔽₃)	#4296	1	1	yes	characteristic
30	C₃²×C₉	—	19	4	no	—
31	C₃×C₂₇	—	19	1	no	—
32	C₃×C₂₇	—	19	8	no	—
33	C₃×C₂₇	—	57	12	no	—
34	C₃²⋊C₉	—	19	8	no	—
35	C₃×C₁₉	—	3	12	no	—
36	C₃×C₁₉	C₉×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃×C₁₉	C₃²×C₂₇	1	1	yes	characteristic
38	C₃×C₁₉	#6903	1	2	yes	—
39	C₃³	C₃×#4296	1	4	yes	—
40	H(3,𝔽₃)	#10587020	1	9	yes	direct product factor
41	C₃×C₉	—	19	1	no	—
42	C₃×C₉	—	19	8	no	—
43	C₃×C₉	—	57	4	no	—
44	C₃×C₉	—	57	8	no	—
45	C₂₇	—	19	3	no	—
46	C₂₇	—	57	8	no	—
47	C₁₉	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
48	C₃²	—	3	4	no	—
49	C₃²	—	3	8	no	—
50	C₃²	C₃×#10587020	1	12	yes	complemented
51	C₃²	C₃²×#4296	1	1	yes	characteristic
52	C₉	—	19	1	no	—
53	C₉	—	19	2	no	—
54	C₉	—	57	8	no	—
55	C₃	—	3	12	no	—
56	C₃	H(3,𝔽₃)×#4296	1	1	yes	characteristic
57	C₃	C₃²×#10587020	1	1	yes	characteristic
58	C₃	#17987598	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator	C₃×C₁₉	C₃²×C₂₇
Fitting, prime radical	C₃×C₁₉×H(3,𝔽₃)	C₉
socle	C₃²×C₁₉	C₃²×C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₃×H(3,𝔽₃)	#4296
hypocenter	C₁₉	C₂₇×H(3,𝔽₃)
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#10587020	C₃	semidirect
2. C₃²⋊#10587020	C₃	semidirect
12. C₁₉×H(3,𝔽₃)	C₂₇	semidirect
18. #10587020	H(3,𝔽₃)	direct (N × H)
24. C₃²×C₁₉	C₃×C₂₇	semidirect
40. H(3,𝔽₃)	#10587020	direct (N × H)
47. C₁₉	C₂₇×H(3,𝔽₃)	semidirect
50. C₃²	C₃×#10587020	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#10587020	3
C₃²⋊#10587020	3
H(3,𝔽₃)×#4294	3
C₂₇×H(3,𝔽₃)	19

Subgroups — H(3,𝔽3)×#10587020 #28356980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#10587020 #28356980