Subgroups — C₅₃×#6913 #27166248

Order 12879.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#737	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#733	C₃	1	2	yes	maximal, complemented
4	C₅₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#731	C₃	1	3	yes	maximal, complemented
6	C₅₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
7	C₃³×C₅₃	C₃²	1	1	yes	characteristic
8	3₋¹⁺²×C₅₃	—	3	3	no	—
9	3₋¹⁺²×C₅₃	—	3	6	no	—
10	3₋¹⁺²×C₅₃	C₃²	1	6	yes	—
11	H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
12	C₃×C₉×C₅₃	—	3	3	no	—
13	C₃×C₉×C₅₃	C₃²	1	1	yes	characteristic
14	C₃×C₉×C₅₃	C₃²	1	1	yes	characteristic
15	C₃×C₉×C₅₃	C₃²	1	1	yes	characteristic
16	C₃²×C₅₃	—	3	3	no	—
17	C₃²×C₅₃	—	3	1	no	—
18	C₃²×C₅₃	—	9	1	no	—
19	C₃²×C₅₃	C₃³	1	1	yes	characteristic
20	C₉×C₅₃	—	3	6	no	—
21	C₉×C₅₃	—	3	1	no	—
22	C₉×C₅₃	—	3	1	no	—
23	C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
24	#6913	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃×C₅₃	—	3	1	no	—
26	C₃×C₅₃	—	9	3	no	—
27	C₃×C₅₃	—	9	1	no	—
28	C₃×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	#737	C₃×C₅₃	1	3	yes	complemented
30	C₃×3₋¹⁺²	C₃×C₅₃	1	1	yes	characteristic, complemented
31	#733	C₃×C₅₃	1	2	yes	complemented
32	#732	C₃×C₅₃	1	1	yes	characteristic, complemented
33	#731	C₃×C₅₃	1	3	yes	complemented
34	AΣL(1,𝔽₂₇)	C₃×C₅₃	1	3	yes	complemented
35	C₅₃	#6913	1	1	yes	characteristic, direct product factor, Sylow
36	C₃³	C₃²×C₅₃	1	1	yes	characteristic
37	3₋¹⁺²	—	3	3	no	—
38	3₋¹⁺²	—	3	6	no	—
39	3₋¹⁺²	C₃²×C₅₃	1	6	yes	—
40	H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	C₃²×C₅₃	1	1	yes	characteristic
43	C₃×C₉	C₃²×C₅₃	1	1	yes	characteristic
44	C₃×C₉	C₃²×C₅₃	1	1	yes	characteristic
45	C₃²	—	3	3	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	9	1	no	—
48	C₃²	C₃³×C₅₃	1	1	yes	characteristic
49	C₉	—	3	6	no	—
50	C₉	—	3	1	no	—
51	C₉	—	3	1	no	—
52	C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
53	C₃	—	3	1	no	—
54	C₃	—	9	1	no	—
55	C₃	—	9	3	no	—
56	C₃	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#737	C₃	semidirect
2. C₃×3₋¹⁺²×C₅₃	C₃	semidirect
3. C₅₃×#733	C₃	semidirect
4. C₅₃×#732	C₃	semidirect
5. C₅₃×#731	C₃	semidirect
6. C₅₃×AΣL(1,𝔽₂₇)	C₃	semidirect
24. #6913	C₅₃	direct (N × H)
29. #737	C₃×C₅₃	semidirect
30. C₃×3₋¹⁺²	C₃×C₅₃	semidirect
31. #733	C₃×C₅₃	semidirect
32. #732	C₃×C₅₃	semidirect
33. #731	C₃×C₅₃	semidirect
34. AΣL(1,𝔽₂₇)	C₃×C₅₃	semidirect
35. C₅₃	#6913	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#737	3
C₃×3₋¹⁺²×C₅₃	3
C₅₃×#733	3
C₅₃×#732	3
C₅₃×#731	3
C₅₃×AΣL(1,𝔽₂₇)	3
#6913	53

Subgroups — C53×#6913 #27166248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#6913 #27166248