Subgroups — C₃×D₅₃×D₅₇ #49136306

Order 36252.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₁₉×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅₃×D₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₃₀₂₁	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₃×D₅₇	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×D₁₉×D₅₃	—	3	1	no	maximal
6	C₃²×C₁₉×C₅₃	C₂²	1	1	yes	characteristic, complemented, Hall
7	D₃₀₂₁	C₂×C₃	1	1	yes	characteristic, complemented
8	C₅₃×D₅₇	C₂×C₃	1	1	yes	characteristic, complemented
9	C₃×D₁₀₀₇	—	3	1	no	—
10	C₃×D₁₉×C₅₃	—	3	1	no	—
11	C₃×C₁₉×D₅₃	—	2	1	no	—
12	C₃×C₁₉×D₅₃	D₃	1	1	yes	characteristic, complemented
13	C₃×C₁₉×D₅₃	C₂×C₃	1	1	yes	characteristic, complemented
14	D₁₉×D₅₃	—	3	1	no	Hall
15	C₃×C₁₉×C₅₃	—	2	1	no	—
16	C₃×C₁₉×C₅₃	C₂×D₃	1	1	yes	characteristic, complemented
17	C₃×C₁₉×C₅₃	C₂²×C₃	1	1	yes	characteristic, complemented
18	D₁₀₀₇	—	3	1	no	—
19	C₁₉×D₅₃	C₃×D₃	1	1	yes	characteristic, complemented
20	D₁₉×C₅₃	—	3	1	no	—
21	C₃×D₃×D₅₃	—	19	1	no	maximal, Hall
22	C₁₉×C₅₃	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
23	C₃×D₁₅₉	—	19	1	no	—
24	C₃×D₃×C₅₃	—	19	1	no	—
25	C₃²×D₅₃	D₁₉	1	1	yes	characteristic, complemented
26	C₂×C₃×D₅₇	—	53	1	no	maximal, Hall
27	C₂×C₃×D₅₃	—	57	1	no	—
28	D₃×D₅₃	—	19	1	no	—
29	C₃²×C₅₃	C₂×D₁₉	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃²×C₁₉	—	53	1	no	—
31	C₃×D₅₇	—	53	1	no	—
32	C₃×D₅₇	D₅₃	1	1	yes	characteristic, direct product factor
33	C₂×C₃×C₅₃	—	57	1	no	—
34	D₁₅₉	—	19	1	no	—
35	C₃×D₅₃	—	2	1	no	—
36	C₃×D₅₃	—	57	1	no	—
37	C₃×D₅₃	C₃×D₁₉	1	1	yes	characteristic, complemented
38	C₃×D₅₃	D₅₇	1	1	yes	characteristic, direct product factor
39	D₃×C₅₃	—	19	1	no	—
40	C₂×D₅₇	—	53	1	no	—
41	C₂×C₃×D₁₉	—	159	1	no	—
42	C₂×D₅₃	—	57	1	no	Hall
43	C₃²×C₁₉	C₂×D₅₃	1	1	yes	characteristic, complemented, Hall
44	C₃×C₅₃	—	2	1	no	—
45	C₃×C₅₃	C₂×C₃×D₁₉	1	1	yes	characteristic, complemented
46	C₃×C₅₃	C₂×D₅₇	1	1	yes	characteristic, complemented
47	C₂×C₃×C₁₉	—	53	1	no	—
48	C₂×C₃×C₁₉	—	53	1	no	—
49	C₂×C₃×C₁₉	—	106	1	no	—
50	D₅₇	—	53	1	no	—
51	D₅₇	C₃×D₅₃	1	1	yes	characteristic, direct product factor
52	C₃×D₁₉	—	3	1	no	—
53	C₃×D₁₉	—	159	1	no	—
54	C₂×C₅₃	—	57	1	no	—
55	D₅₃	—	57	1	no	—
56	D₅₃	C₃×D₅₇	1	1	yes	characteristic, direct product factor
57	C₂×D₁₉	—	159	1	no	Hall
58	C₃×C₁₉	—	2	1	no	—
59	C₃×C₁₉	D₃×D₅₃	1	1	yes	characteristic, complemented
60	C₃×C₁₉	C₂×C₃×D₅₃	1	1	yes	characteristic, complemented
61	C₅₃	C₂×C₃×D₅₇	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₁₉	—	53	1	no	—
63	D₁₉	—	3	1	no	—
64	D₁₉	—	159	1	no	—
65	C₂×C₃×D₃	—	1007	1	no	Hall
66	C₁₉	C₃×D₃×D₅₃	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃²	—	53	1	no	—
68	C₃×D₃	—	19	1	no	—
69	C₃×D₃	—	1007	1	no	—
70	C₂²×C₃	—	3021	1	no	—
71	C₂×D₃	—	1007	1	no	—
72	C₃²	D₁₉×D₅₃	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	53	1	no	—
74	C₂×C₃	—	53	1	no	—
75	C₂×C₃	—	57	1	no	—
76	C₂×C₃	—	106	1	no	—
77	C₂×C₃	—	3021	1	no	—
78	D₃	—	19	1	no	—
79	D₃	—	1007	1	no	—
80	C₂²	—	3021	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	C₃×D₁₉×D₅₃	1	1	yes	characteristic, complemented
83	C₃	D₅₃×D₅₇	1	1	yes	characteristic, direct product factor
84	C₂	—	53	1	no	—
85	C₂	—	57	1	no	—
86	C₂	—	3021	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₅₃×D₅₇
commutator, cosocle, hypocenter	C₃×C₁₉×C₅₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₃²×C₁₉×C₅₃	C₂²
LCS	G ▷ C₃×C₁₉×C₅₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×C₁₉×D₅₃	C₂	semidirect
2. C₃×C₅₃×D₅₇	C₂	semidirect
3. C₃×D₃₀₂₁	C₂	semidirect
4. D₅₃×D₅₇	C₃	direct (N × H)
6. C₃²×C₁₉×C₅₃	C₂²	semidirect
7. D₃₀₂₁	C₂×C₃	semidirect
8. C₅₃×D₅₇	C₂×C₃	semidirect
12. C₃×C₁₉×D₅₃	D₃	semidirect
13. C₃×C₁₉×D₅₃	C₂×C₃	semidirect
16. C₃×C₁₉×C₅₃	C₂×D₃	semidirect
17. C₃×C₁₉×C₅₃	C₂²×C₃	semidirect
19. C₁₉×D₅₃	C₃×D₃	semidirect
22. C₁₉×C₅₃	C₂×C₃×D₃	semidirect
25. C₃²×D₅₃	D₁₉	semidirect
29. C₃²×C₅₃	C₂×D₁₉	semidirect
32. C₃×D₅₇	D₅₃	direct (N × H)
37. C₃×D₅₃	C₃×D₁₉	semidirect
38. C₃×D₅₃	D₅₇	direct (N × H)
43. C₃²×C₁₉	C₂×D₅₃	semidirect
45. C₃×C₅₃	C₂×C₃×D₁₉	semidirect
46. C₃×C₅₃	C₂×D₅₇	semidirect
51. D₅₇	C₃×D₅₃	direct (N × H)
56. D₅₃	C₃×D₅₇	direct (N × H)
59. C₃×C₁₉	D₃×D₅₃	semidirect
60. C₃×C₁₉	C₂×C₃×D₅₃	semidirect
61. C₅₃	C₂×C₃×D₅₇	semidirect
66. C₁₉	C₃×D₃×D₅₃	semidirect
72. C₃²	D₁₉×D₅₃	semidirect
82. C₃	C₃×D₁₉×D₅₃	semidirect
83. C₃	D₅₃×D₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₁₉×D₅₃	2
C₃×C₅₃×D₅₇	2
C₃×D₃₀₂₁	2
D₅₃×D₅₇	3
C₃×D₁₉×D₅₃	3
C₃×D₃×D₅₃	19
C₂×C₃×D₅₇	53

Subgroups — C3×D53×D57 #49136306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₅₃×D₅₇ #49136306