Subgroups — C₅₃×#10624841 #50311726

Order 38637.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#6908	C₃	1	3	yes	maximal, complemented
2	C₅₃×#6887	C₃	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#6880	C₃	1	2	yes	maximal, complemented
4	C₅₃×#6879	C₃	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#6878	C₃	1	6	yes	maximal, complemented
6	C₅₃×#737	—	3	3	no	—
7	C₃²×C₉×C₅₃	C₃²	1	1	yes	characteristic
8	C₅₃×#733	C₃²	1	6	yes	—
9	C₅₃×#732	C₃²	1	3	yes	—
10	C₅₃×#731	—	3	3	no	—
11	C₅₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
12	C₅₃×C₉⋊C₉	—	3	1	no	—
13	C₅₃×C₉⋊C₉	—	3	1	no	—
14	C₅₃×C₃²⋊C₉	—	3	1	no	—
15	C₉²×C₅₃	C₃²	1	2	yes	—
16	C₉²×C₅₃	C₃²	1	1	yes	characteristic
17	C₃³×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
18	3₋¹⁺²×C₅₃	—	3	6	no	—
19	3₋¹⁺²×C₅₃	—	9	6	no	—
20	H(3,𝔽₃)×C₅₃	—	3	3	no	—
21	C₃×C₉×C₅₃	—	3	3	no	—
22	C₃×C₉×C₅₃	—	3	2	no	—
23	C₃×C₉×C₅₃	—	3	1	no	—
24	C₃×C₉×C₅₃	—	9	3	no	—
25	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
26	C₃×C₉×C₅₃	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃×C₉×C₅₃	C₃³	1	1	yes	characteristic
28	#10624841	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₅₃	—	3	1	no	—
30	C₃²×C₅₃	—	9	1	no	—
31	C₃²×C₅₃	—	9	3	no	—
32	C₃²×C₅₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₉×C₅₃	—	3	1	no	—
34	C₉×C₅₃	—	3	1	no	—
35	C₉×C₅₃	—	9	2	no	—
36	C₉×C₅₃	—	9	1	no	—
37	C₉×C₅₃	—	9	6	no	—
38	C₉×C₅₃	#732	1	3	yes	—
39	#6908	C₃×C₅₃	1	3	yes	complemented
40	#6887	C₃×C₅₃	1	1	yes	characteristic, complemented
41	#6880	C₃×C₅₃	1	2	yes	complemented
42	#6879	C₃×C₅₃	1	1	yes	characteristic, complemented
43	#6878	C₃×C₅₃	1	6	yes	complemented
44	C₃×C₅₃	—	3	1	no	—
45	C₃×C₅₃	—	9	1	no	—
46	C₃×C₅₃	—	27	3	no	—
47	C₃×C₅₃	C₃×#732	1	1	yes	characteristic
48	#737	—	3	3	no	—
49	C₃²×C₉	C₃²×C₅₃	1	1	yes	characteristic
50	#733	C₃²×C₅₃	1	6	yes	—
51	#732	C₃²×C₅₃	1	3	yes	—
52	#731	—	3	3	no	—
53	AΣL(1,𝔽₂₇)	—	3	3	no	—
54	C₉⋊C₉	—	3	1	no	—
55	C₉⋊C₉	—	3	1	no	—
56	C₃²⋊C₉	—	3	1	no	—
57	C₉²	C₃²×C₅₃	1	2	yes	—
58	C₉²	C₃²×C₅₃	1	1	yes	characteristic
59	C₅₃	#10624841	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
61	3₋¹⁺²	—	3	6	no	—
62	3₋¹⁺²	—	9	6	no	—
63	H(3,𝔽₃)	—	3	3	no	—
64	C₃×C₉	—	3	3	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	9	3	no	—
68	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
69	C₃×C₉	H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
70	C₃×C₉	C₃³×C₅₃	1	1	yes	characteristic
71	C₃²	—	3	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	9	3	no	—
74	C₃²	C₃×H(3,𝔽₃)×C₅₃	1	1	yes	characteristic
75	C₉	—	3	1	no	—
76	C₉	—	3	1	no	—
77	C₉	—	9	2	no	—
78	C₉	—	9	1	no	—
79	C₉	—	9	6	no	—
80	C₉	C₅₃×#732	1	3	yes	—
81	C₃	—	3	1	no	—
82	C₃	—	9	1	no	—
83	C₃	—	27	3	no	—
84	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 ◁ C₃×C₉×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#6908	C₃	semidirect
2. C₅₃×#6887	C₃	semidirect
3. C₅₃×#6880	C₃	semidirect
4. C₅₃×#6879	C₃	semidirect
5. C₅₃×#6878	C₃	semidirect
28. #10624841	C₅₃	direct (N × H)
39. #6908	C₃×C₅₃	semidirect
40. #6887	C₃×C₅₃	semidirect
41. #6880	C₃×C₅₃	semidirect
42. #6879	C₃×C₅₃	semidirect
43. #6878	C₃×C₅₃	semidirect
59. C₅₃	#10624841	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×#6908	3
C₅₃×#6887	3
C₅₃×#6880	3
C₅₃×#6879	3
C₅₃×#6878	3
#10624841	53

Subgroups — C53×#10624841 #50311726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10624841 #50311726