Subgroups — C₅₃×#10624839 #50311724

Order 38637.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#732	C₃	1	3	yes	maximal, complemented
2	C₅₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#6879	C₃	1	9	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₅₃	—	3	3	no	—
5	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
6	C₅₃×#732	—	3	6	no	—
7	C₅₃×#732	C₃²	1	9	yes	complemented
8	C₅₃×C₉⋊C₉	—	3	2	no	—
9	C₅₃×C₃²⋊C₉	—	3	1	no	—
10	C₉²×C₅₃	C₃²	1	3	yes	complemented
11	C₃³×C₅₃	—	9	3	no	—
12	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
13	H(3,𝔽₃)×C₅₃	—	3	9	no	—
14	H(3,𝔽₃)×C₅₃	—	9	6	no	—
15	C₃×C₉×C₅₃	—	3	3	no	—
16	C₃×C₉×C₅₃	—	3	3	no	—
17	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
18	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
19	#10624839	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₃²×C₅₃	—	9	1	no	—
21	C₃²×C₅₃	—	9	9	no	—
22	C₃²×C₅₃	—	27	9	no	—
23	C₃²×C₅₃	#732	1	3	yes	—
24	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₉×C₅₃	—	3	2	no	—
26	C₉×C₅₃	—	3	1	no	—
27	C₉×C₅₃	—	9	3	no	—
28	C₃×#732	C₃×C₅₃	1	3	yes	complemented
29	#6887	C₃×C₅₃	1	1	yes	characteristic, complemented
30	#6879	C₃×C₅₃	1	9	yes	complemented
31	C₃×C₅₃	—	3	1	no	—
32	C₃×C₅₃	—	3	3	no	—
33	C₃×C₅₃	—	27	9	no	—
34	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
35	C₃×H(3,𝔽₃)	—	3	3	no	—
36	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
37	#732	—	3	6	no	—
38	#732	C₃²×C₅₃	1	9	yes	complemented
39	C₉⋊C₉	—	3	2	no	—
40	C₃²⋊C₉	—	3	1	no	—
41	C₉²	C₃²×C₅₃	1	3	yes	complemented
42	C₅₃	#10624839	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	—	9	3	no	—
44	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
45	H(3,𝔽₃)	—	3	9	no	—
46	H(3,𝔽₃)	—	9	6	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
50	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
51	C₃²	—	9	1	no	—
52	C₃²	—	9	9	no	—
53	C₃²	—	27	9	no	—
54	C₃²	C₅₃×#732	1	3	yes	—
55	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
56	C₉	—	3	2	no	—
57	C₉	—	3	1	no	—
58	C₉	—	9	3	no	—
59	C₃	—	3	1	no	—
60	C₃	—	3	3	no	—
61	C₃	—	27	9	no	—
62	C₃	C₃×C₅₃×#732	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₅₃	C₃×#732
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₅₃
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅₃ ◁ C₃³×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅₃×#732	C₃	semidirect
2. C₅₃×#6887	C₃	semidirect
3. C₅₃×#6879	C₃	semidirect
7. C₅₃×#732	C₃²	semidirect
10. C₉²×C₅₃	C₃²	semidirect
19. #10624839	C₅₃	direct (N × H)
28. C₃×#732	C₃×C₅₃	semidirect
29. #6887	C₃×C₅₃	semidirect
30. #6879	C₃×C₅₃	semidirect
38. #732	C₃²×C₅₃	semidirect
41. C₉²	C₃²×C₅₃	semidirect
42. C₅₃	#10624839	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅₃×#732	3
C₅₃×#6887	3
C₅₃×#6879	3
#10624839	53

Subgroups — C53×#10624839 #50311724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624839 #50311724