Subgroups — C₅₃×#10624528 #50311413

Order 38637.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅₃×#733	C₃	1	1	yes	maximal, characteristic
2	C₃×C₅₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₅₃×#6868	C₃	1	2	yes	maximal
4	C₃×3₋¹⁺²×C₅₃	—	3	1	no	—
5	C₃²×C₉×C₅₃	—	3	1	no	—
6	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
7	C₅₃×#733	—	3	2	no	—
8	C₅₃×#733	C₉	1	3	yes	complemented
9	C₅₃×C₉⋊C₉	—	3	3	no	—
10	C₅₃×C₃²⋊C₉	—	3	2	no	—
11	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
12	3₋¹⁺²×C₅₃	—	3	3	no	—
13	3₋¹⁺²×C₅₃	—	9	1	no	—
14	3₋¹⁺²×C₅₃	—	9	1	no	—
15	C₃×C₉×C₅₃	—	3	1	no	—
16	C₃×C₉×C₅₃	—	3	1	no	—
17	C₃×C₉×C₅₃	—	3	1	no	—
18	C₃×C₉×C₅₃	—	3	3	no	—
19	C₃×C₉×C₅₃	—	9	1	no	—
20	C₃×C₉×C₅₃	—	9	2	no	—
21	C₃×C₉×C₅₃	—	9	1	no	—
22	C₃×C₉×C₅₃	—	9	2	no	—
23	C₃×C₉×C₅₃	C₃×C₉	1	1	yes	characteristic
24	C₃×C₉×C₅₃	3₋¹⁺²	1	2	yes	—
25	#10624528	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₅₃	—	3	1	no	—
27	C₃²×C₅₃	—	3	2	no	—
28	C₃²×C₅₃	C₃²⋊C₉	1	1	yes	characteristic
29	C₃²×C₅₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₅₃	#731	1	1	yes	characteristic
31	C₃²×C₅₃	#732	1	1	yes	characteristic
32	C₉×C₅₃	—	3	1	no	—
33	C₉×C₅₃	—	3	2	no	—
34	C₉×C₅₃	—	9	3	no	—
35	C₉×C₅₃	—	9	3	no	—
36	C₉×C₅₃	—	27	2	no	—
37	C₃×#733	C₃×C₅₃	1	1	yes	characteristic
38	C₃×C₉⋊C₉	C₃×C₅₃	1	1	yes	characteristic
39	#6868	C₃×C₅₃	1	2	yes	—
40	C₃×C₅₃	—	3	1	no	—
41	C₃×C₅₃	—	3	1	no	—
42	C₃×C₅₃	—	3	1	no	—
43	C₃×C₅₃	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
44	C₃×C₅₃	#6881	1	1	yes	characteristic
45	C₃×C₅₃	#6882	1	2	yes	—
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
49	#733	—	3	2	no	—
50	#733	C₉×C₅₃	1	3	yes	complemented
51	C₉⋊C₉	—	3	3	no	—
52	C₃²⋊C₉	—	3	2	no	—
53	C₅₃	#10624528	1	1	yes	characteristic, direct product factor, Sylow
54	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
55	3₋¹⁺²	—	3	3	no	—
56	3₋¹⁺²	—	9	1	no	—
57	3₋¹⁺²	—	9	1	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	1	no	—
60	C₃×C₉	—	3	1	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	9	1	no	—
63	C₃×C₉	—	9	2	no	—
64	C₃×C₉	—	9	1	no	—
65	C₃×C₉	—	9	2	no	—
66	C₃×C₉	C₃×C₉×C₅₃	1	1	yes	characteristic
67	C₃×C₉	3₋¹⁺²×C₅₃	1	2	yes	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	2	no	—
70	C₃²	C₅₃×C₃²⋊C₉	1	1	yes	characteristic
71	C₃²	C₅₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
72	C₃²	C₅₃×#731	1	1	yes	characteristic
73	C₃²	C₅₃×#732	1	1	yes	characteristic
74	C₉	—	3	1	no	—
75	C₉	—	3	2	no	—
76	C₉	—	9	3	no	—
77	C₉	—	9	3	no	—
78	C₉	—	27	2	no	—
79	C₃	—	3	1	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	1	no	—
82	C₃	C₅₃×H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
83	C₃	C₅₃×#6881	1	1	yes	characteristic
84	C₃	C₅₃×#6882	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	#732
commutator	C₃×C₉	C₃×C₉×C₅₃
Frattini, cosocle	C₃²×C₉	C₃²×C₅₃
prime radical	C₃³×C₅₃	H(3,𝔽₃)
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₅₃ ◁ C₃³×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅₃×#733	C₉	semidirect
25. #10624528	C₅₃	direct (N × H)
50. #733	C₉×C₅₃	semidirect
53. C₅₃	#10624528	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×#733	3
C₃×C₅₃×C₉⋊C₉	3
C₅₃×#6868	3
#10624528	53

Subgroups — C53×#10624528 #50311413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624528 #50311413