Subgroups — C₅₃×#10624638 #50311523

Order 38637.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×C₉⋊H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₅₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#6861	C₃	1	3	yes	maximal, complemented
4	C₅₃×#6859	C₃	1	3	yes	maximal, complemented
5	C₅₃×#6856	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²×C₅₃	—	3	3	no	—
7	C₃×H(3,𝔽₃)×C₅₃	C₃²	1	3	yes	—
8	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
9	C₅₃×C₉⋊C₉	—	3	2	no	—
10	C₅₃×C₃²⋊C₉	—	3	6	no	—
11	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
12	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
13	C₅₃×C₃²⋊C₉	C₃²	1	3	yes	—
14	C₉²×C₅₃	—	3	1	no	—
15	C₃³×C₅₃	—	3	3	no	—
16	C₃³×C₅₃	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₅₃	—	9	9	no	—
18	H(3,𝔽₃)×C₅₃	—	9	3	no	—
19	C₃×C₉×C₅₃	—	3	3	no	—
20	C₃×C₉×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	—	3	3	no	—
22	C₃×C₉×C₅₃	—	3	3	no	—
23	C₃×C₉×C₅₃	H(3,𝔽₃)	1	2	yes	—
24	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
25	#10624638	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₅₃	—	3	3	no	—
27	C₃²×C₅₃	—	3	1	no	—
28	C₃²×C₅₃	—	9	3	no	—
29	C₃²×C₅₃	—	9	6	no	—
30	C₃²×C₅₃	—	9	3	no	—
31	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
32	C₉×C₅₃	—	3	2	no	—
33	C₉×C₅₃	—	9	3	no	—
34	C₉×C₅₃	—	9	3	no	—
35	C₉×C₅₃	—	9	3	no	—
36	C₉×C₅₃	#732	1	3	yes	—
37	C₉⋊H(3,𝔽₃)	C₃×C₅₃	1	3	yes	complemented
38	#6887	C₃×C₅₃	1	1	yes	characteristic, complemented
39	#6861	C₃×C₅₃	1	3	yes	complemented
40	#6859	C₃×C₅₃	1	3	yes	complemented
41	#6856	C₃×C₅₃	1	3	yes	complemented
42	C₃×C₅₃	—	9	1	no	—
43	C₃×C₅₃	—	27	3	no	—
44	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
45	C₃×C₅₃	#6913	1	3	yes	—
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃×H(3,𝔽₃)	C₃²×C₅₃	1	3	yes	—
48	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
49	C₉⋊C₉	—	3	2	no	—
50	C₃²⋊C₉	—	3	6	no	—
51	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
52	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
53	C₃²⋊C₉	C₃²×C₅₃	1	3	yes	—
54	C₉²	—	3	1	no	—
55	C₅₃	#10624638	1	1	yes	characteristic, direct product factor, Sylow
56	C₃³	—	3	3	no	—
57	C₃³	C₃³×C₅₃	1	1	yes	characteristic
58	3₋¹⁺²	—	9	9	no	—
59	H(3,𝔽₃)	—	9	3	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	H(3,𝔽₃)×C₅₃	1	2	yes	—
65	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
66	C₃²	—	3	1	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	9	3	no	—
69	C₃²	—	9	3	no	—
70	C₃²	—	9	6	no	—
71	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
72	C₉	—	3	2	no	—
73	C₉	—	9	3	no	—
74	C₉	—	9	3	no	—
75	C₉	—	9	3	no	—
76	C₉	C₅₃×#732	1	3	yes	—
77	C₃	—	9	1	no	—
78	C₃	—	27	3	no	—
79	C₃	C₃×C₅₃×#732	1	1	yes	characteristic
80	C₃	C₅₃×#6913	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₅₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₅₃
LCS	G ▷ C₃³ ▷ C₃² ▷ 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×C₉⋊H(3,𝔽₃)	C₃	semidirect
2. C₅₃×#6887	C₃	semidirect
3. C₅₃×#6861	C₃	semidirect
4. C₅₃×#6859	C₃	semidirect
5. C₅₃×#6856	C₃	semidirect
25. #10624638	C₅₃	direct (N × H)
37. C₉⋊H(3,𝔽₃)	C₃×C₅₃	semidirect
38. #6887	C₃×C₅₃	semidirect
39. #6861	C₃×C₅₃	semidirect
40. #6859	C₃×C₅₃	semidirect
41. #6856	C₃×C₅₃	semidirect
55. C₅₃	#10624638	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×C₉⋊H(3,𝔽₃)	3
C₅₃×#6887	3
C₅₃×#6861	3
C₅₃×#6859	3
C₅₃×#6856	3
#10624638	53

Subgroups — C53×#10624638 #50311523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624638 #50311523