Subgroups — C₅₃×H(3,𝔽₃)⋊C₉ #27166205

Order 12879.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₃)×C₅₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₃³×C₅₃	—	3	1	no	—
5	C₃³×C₅₃	C₃²	1	1	yes	characteristic
6	H(3,𝔽₃)×C₅₃	—	3	2	no	—
7	H(3,𝔽₃)×C₅₃	C₉	1	3	yes	complemented
8	C₃×C₉×C₅₃	—	3	1	no	—
9	C₃×C₉×C₅₃	—	3	3	no	—
10	C₃×C₉×C₅₃	—	3	2	no	—
11	C₃²×C₅₃	—	3	1	no	—
12	C₃²×C₅₃	—	3	1	no	—
13	C₃²×C₅₃	—	3	1	no	—
14	C₃²×C₅₃	—	3	3	no	—
15	C₃²×C₅₃	—	9	1	no	—
16	C₃²×C₅₃	—	9	1	no	—
17	C₃²×C₅₃	—	9	1	no	—
18	C₃²×C₅₃	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₅₃	3₋¹⁺²	1	2	yes	—
21	C₉×C₅₃	—	3	3	no	—
22	C₉×C₅₃	—	9	2	no	—
23	H(3,𝔽₃)⋊C₉	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₃×C₅₃	—	3	1	no	—
25	C₃×C₅₃	—	3	2	no	—
26	C₃×C₅₃	—	9	3	no	—
27	C₃×C₅₃	C₃²⋊C₉	1	1	yes	characteristic
28	C₃×C₅₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
29	C₃×C₅₃	#731	1	1	yes	characteristic
30	C₃×C₅₃	#732	1	1	yes	characteristic
31	C₃×H(3,𝔽₃)	C₃×C₅₃	1	1	yes	characteristic
32	C₃²×C₉	C₃×C₅₃	1	1	yes	characteristic, complemented
33	C₃²⋊C₉	C₃×C₅₃	1	2	yes	complemented
34	C₅₃	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic, direct product factor, Sylow
35	C₃³	—	3	1	no	—
36	C₃³	C₃²×C₅₃	1	1	yes	characteristic
37	H(3,𝔽₃)	—	3	2	no	—
38	H(3,𝔽₃)	C₉×C₅₃	1	3	yes	complemented
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	9	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	C₃×C₉×C₅₃	1	1	yes	characteristic
50	C₃²	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
51	C₃²	3₋¹⁺²×C₅₃	1	2	yes	—
52	C₉	—	3	3	no	—
53	C₉	—	9	2	no	—
54	C₃	—	3	1	no	—
55	C₃	—	3	2	no	—
56	C₃	—	9	3	no	—
57	C₃	C₅₃×C₃²⋊C₉	1	1	yes	characteristic
58	C₃	C₅₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
59	C₃	C₅₃×#731	1	1	yes	characteristic
60	C₃	C₅₃×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₅₃	H(3,𝔽₃)
commutator	C₃²	C₃×C₉×C₅₃
Frattini, cosocle	C₃³	C₃²×C₅₃
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₉×C₅₃	C₃	semidirect
3. C₅₃×C₃²⋊C₉	C₃	semidirect
7. H(3,𝔽₃)×C₅₃	C₉	semidirect
23. H(3,𝔽₃)⋊C₉	C₅₃	direct (N × H)
32. C₃²×C₉	C₃×C₅₃	semidirect
33. C₃²⋊C₉	C₃×C₅₃	semidirect
34. C₅₃	H(3,𝔽₃)⋊C₉	direct (N × H)
38. H(3,𝔽₃)	C₉×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(3,𝔽₃)×C₅₃	3
C₃²×C₉×C₅₃	3
C₅₃×C₃²⋊C₉	3
H(3,𝔽₃)⋊C₉	53

Subgroups — C53×H(3,𝔽3)⋊C9 #27166205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×H(3,𝔽₃)⋊C₉ #27166205