Subgroups — C₅₃×#10624813 #50311698

Order 38637.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6895	C₃	1	2	yes	maximal, complemented
2	C₅₃×#6893	C₃	1	1	yes	maximal, characteristic
3	C₃×C₉²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₅₃×3₋¹⁺²⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₅₃×H(3,𝔽₃)⋊C₉	C₃	1	3	yes	maximal
6	C₃×3₋¹⁺²×C₅₃	C₃²	1	2	yes	—
7	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	1	yes	characteristic
8	C₃²×C₉×C₅₃	C₃²	1	3	yes	—
9	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
10	C₅₃×C₉⋊C₉	—	3	6	no	—
11	C₅₃×C₃²⋊C₉	—	3	3	no	—
12	C₅₃×C₃²⋊C₉	C₃²	1	6	yes	—
13	C₉²×C₅₃	—	3	3	no	—
14	C₃³×C₅₃	—	3	1	no	—
15	C₃³×C₅₃	C₃³	1	1	yes	characteristic
16	3₋¹⁺²×C₅₃	—	3	6	no	—
17	H(3,𝔽₃)×C₅₃	—	3	3	no	—
18	C₃×C₉×C₅₃	—	3	3	no	—
19	C₃×C₉×C₅₃	—	3	3	no	—
20	C₃×C₉×C₅₃	—	3	9	no	—
21	C₃×C₉×C₅₃	—	3	2	no	—
22	C₃×C₉×C₅₃	—	3	6	no	—
23	C₃×C₉×C₅₃	H(3,𝔽₃)	1	3	yes	—
24	#10624813	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₅₃	—	3	3	no	—
26	C₃²×C₅₃	—	9	1	no	—
27	C₃²×C₅₃	—	9	3	no	—
28	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃²×C₅₃	#737	1	3	yes	—
30	C₉×C₅₃	—	3	3	no	—
31	C₉×C₅₃	—	3	9	no	—
32	C₉×C₅₃	—	9	2	no	—
33	C₉×C₅₃	—	9	6	no	—
34	#6895	C₃×C₅₃	1	2	yes	complemented
35	#6893	C₃×C₅₃	1	1	yes	characteristic
36	C₃×C₉²	C₃×C₅₃	1	1	yes	characteristic, complemented
37	3₋¹⁺²⋊C₉	C₃×C₅₃	1	6	yes	complemented
38	H(3,𝔽₃)⋊C₉	C₃×C₅₃	1	3	yes	—
39	C₃×C₅₃	—	3	3	no	—
40	C₃×C₅₃	—	27	1	no	—
41	C₃×C₅₃	#6893	1	1	yes	characteristic
42	C₃×C₅₃	#6908	1	3	yes	—
43	C₃×3₋¹⁺²	C₃²×C₅₃	1	2	yes	—
44	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	1	yes	characteristic
45	C₃²×C₉	C₃²×C₅₃	1	3	yes	—
46	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
47	C₉⋊C₉	—	3	6	no	—
48	C₃²⋊C₉	—	3	3	no	—
49	C₃²⋊C₉	C₃²×C₅₃	1	6	yes	—
50	C₉²	—	3	3	no	—
51	C₅₃	#10624813	1	1	yes	characteristic, direct product factor, Sylow
52	C₃³	—	3	1	no	—
53	C₃³	C₃³×C₅₃	1	1	yes	characteristic
54	3₋¹⁺²	—	3	6	no	—
55	H(3,𝔽₃)	—	3	3	no	—
56	C₃×C₉	—	3	3	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	—	3	9	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	3	6	no	—
61	C₃×C₉	H(3,𝔽₃)×C₅₃	1	3	yes	—
62	C₃²	—	3	3	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	3	no	—
65	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
66	C₃²	C₅₃×#737	1	3	yes	—
67	C₉	—	3	3	no	—
68	C₉	—	3	9	no	—
69	C₉	—	9	2	no	—
70	C₉	—	9	6	no	—
71	C₃	—	3	3	no	—
72	C₃	—	27	1	no	—
73	C₃	C₅₃×#6893	1	1	yes	characteristic
74	C₃	C₅₃×#6908	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₅₃×#6895	C₃	semidirect
3. C₃×C₉²×C₅₃	C₃	semidirect
4. C₅₃×3₋¹⁺²⋊C₉	C₃	semidirect
24. #10624813	C₅₃	direct (N × H)
34. #6895	C₃×C₅₃	semidirect
36. C₃×C₉²	C₃×C₅₃	semidirect
37. 3₋¹⁺²⋊C₉	C₃×C₅₃	semidirect
51. C₅₃	#10624813	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×#6895	3
C₅₃×#6893	3
C₃×C₉²×C₅₃	3
C₅₃×3₋¹⁺²⋊C₉	3
C₅₃×H(3,𝔽₃)⋊C₉	3
#10624813	53

Subgroups — C53×#10624813 #50311698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624813 #50311698