Subgroups — C₂⁴×D₈×C₅₃ #27270589

Order 13568.

15454 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₂³×D₈×C₅₃	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁶×C₅₃	—	2	2	no	—
5	C₂³×D₄×C₅₃	—	2	30	no	—
6	C₂³×D₄×C₅₃	C₂²	1	60	yes	complemented
7	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×D₈×C₅₃	C₂²	1	560	yes	direct product factor
9	C₂³×C₈×C₅₃	C₂²	1	30	yes	complemented
10	C₂⁵×C₅₃	—	2	60	no	—
11	C₂⁵×C₅₃	—	4	32	no	—
12	C₂⁵×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	—	2	420	no	—
14	C₂²×D₄×C₅₃	C₂³	1	280	yes	complemented
15	C₂³×C₄×C₅₃	D₄	1	15	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	15	yes	—
17	C₂×D₈×C₅₃	C₂³	1	960	yes	direct product factor
18	C₂²×C₈×C₅₃	C₂³	1	140	yes	complemented
19	C₂⁴×C₅₃	—	2	280	no	—
20	C₂⁴×C₅₃	—	4	480	no	—
21	C₂⁴×C₅₃	D₈	1	16	yes	direct product factor
22	C₂⁴×C₅₃	C₂×D₄	1	15	yes	—
23	C₂×D₄×C₅₃	—	2	840	no	—
24	C₂×D₄×C₅₃	C₂⁴	1	240	yes	complemented
25	C₂²×C₄×C₅₃	C₂×D₄	1	105	yes	—
26	C₂²×C₄×C₅₃	C₂⁴	1	35	yes	—
27	D₈×C₅₃	C₂⁴	1	256	yes	direct product factor
28	C₂×C₈×C₅₃	C₂⁴	1	120	yes	complemented
29	C₂³×C₅₃	—	2	240	no	—
30	C₂³×C₅₃	—	4	1120	no	—
31	C₂³×C₅₃	C₂×D₈	1	120	yes	direct product factor
32	C₂³×C₅₃	C₂²×D₄	1	35	yes	—
33	D₄×C₅₃	—	2	240	no	—
34	D₄×C₅₃	C₂⁵	1	32	yes	complemented
35	C₂×C₄×C₅₃	C₂²×D₄	1	105	yes	—
36	C₂×C₄×C₅₃	C₂⁵	1	15	yes	—
37	C₈×C₅₃	C₂⁵	1	16	yes	complemented
38	C₂⁴×D₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	—	2	32	no	—
40	C₂²×C₅₃	—	4	480	no	—
41	C₂²×C₅₃	C₂²×D₈	1	140	yes	direct product factor
42	C₂²×C₅₃	C₂³×D₄	1	15	yes	—
43	C₄×C₅₃	C₂³×D₄	1	15	yes	—
44	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
45	C₂⁴×D₄	C₂×C₅₃	1	2	yes	complemented
46	C₂³×D₈	C₂×C₅₃	1	60	yes	direct product factor
47	C₂⁴×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	—	4	32	no	—
49	C₂×C₅₃	C₂³×D₈	1	30	yes	direct product factor
50	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
51	C₂⁶	—	2	2	no	—
52	C₂³×D₄	—	2	30	no	—
53	C₂³×D₄	C₂²×C₅₃	1	60	yes	complemented
54	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
55	C₂²×D₈	C₂²×C₅₃	1	560	yes	direct product factor
56	C₂³×C₈	C₂²×C₅₃	1	30	yes	complemented
57	C₅₃	C₂⁴×D₈	1	1	yes	characteristic, direct product factor, Sylow
58	C₂⁵	—	2	60	no	—
59	C₂⁵	—	4	32	no	—
60	C₂⁵	D₄×C₅₃	1	1	yes	characteristic
61	C₂²×D₄	—	2	420	no	—
62	C₂²×D₄	C₂³×C₅₃	1	280	yes	complemented
63	C₂³×C₄	D₄×C₅₃	1	15	yes	—
64	C₂³×C₄	C₂³×C₅₃	1	15	yes	—
65	C₂×D₈	C₂³×C₅₃	1	960	yes	direct product factor
66	C₂²×C₈	C₂³×C₅₃	1	140	yes	complemented
67	C₂⁴	—	2	280	no	—
68	C₂⁴	—	4	480	no	—
69	C₂⁴	D₈×C₅₃	1	16	yes	direct product factor
70	C₂⁴	C₂×D₄×C₅₃	1	15	yes	—
71	C₂×D₄	—	2	840	no	—
72	C₂×D₄	C₂⁴×C₅₃	1	240	yes	complemented
73	C₂²×C₄	C₂×D₄×C₅₃	1	105	yes	—
74	C₂²×C₄	C₂⁴×C₅₃	1	35	yes	—
75	D₈	C₂⁴×C₅₃	1	256	yes	direct product factor
76	C₂×C₈	C₂⁴×C₅₃	1	120	yes	complemented
77	C₂³	—	2	240	no	—
78	C₂³	—	4	1120	no	—
79	C₂³	C₂×D₈×C₅₃	1	120	yes	direct product factor
80	C₂³	C₂²×D₄×C₅₃	1	35	yes	—
81	D₄	—	2	240	no	—
82	D₄	C₂⁵×C₅₃	1	32	yes	complemented
83	C₂×C₄	C₂²×D₄×C₅₃	1	105	yes	—
84	C₂×C₄	C₂⁵×C₅₃	1	15	yes	—
85	C₈	C₂⁵×C₅₃	1	16	yes	complemented
86	C₂²	—	2	32	no	—
87	C₂²	—	4	480	no	—
88	C₂²	C₂²×D₈×C₅₃	1	140	yes	direct product factor
89	C₂²	C₂³×D₄×C₅₃	1	15	yes	—
90	C₄	C₂³×D₄×C₅₃	1	15	yes	—
91	C₄	C₂⁶×C₅₃	1	1	yes	characteristic
92	C₂	—	4	32	no	—
93	C₂	C₂³×D₈×C₅₃	1	30	yes	direct product factor
94	C₂	C₂⁴×D₄×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₅₃	D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₅₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ C₂⁴×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₄×C₅₃	C₂	semidirect
2. C₂³×D₈×C₅₃	C₂	direct (N × H)
3. C₂⁴×C₈×C₅₃	C₂	semidirect
6. C₂³×D₄×C₅₃	C₂²	semidirect
8. C₂²×D₈×C₅₃	C₂²	direct (N × H)
9. C₂³×C₈×C₅₃	C₂²	semidirect
14. C₂²×D₄×C₅₃	C₂³	semidirect
17. C₂×D₈×C₅₃	C₂³	direct (N × H)
18. C₂²×C₈×C₅₃	C₂³	semidirect
21. C₂⁴×C₅₃	D₈	direct (N × H)
24. C₂×D₄×C₅₃	C₂⁴	semidirect
27. D₈×C₅₃	C₂⁴	direct (N × H)
28. C₂×C₈×C₅₃	C₂⁴	semidirect
31. C₂³×C₅₃	C₂×D₈	direct (N × H)
34. D₄×C₅₃	C₂⁵	semidirect
37. C₈×C₅₃	C₂⁵	semidirect
38. C₂⁴×D₈	C₅₃	direct (N × H)
41. C₂²×C₅₃	C₂²×D₈	direct (N × H)
45. C₂⁴×D₄	C₂×C₅₃	semidirect
46. C₂³×D₈	C₂×C₅₃	direct (N × H)
47. C₂⁴×C₈	C₂×C₅₃	semidirect
49. C₂×C₅₃	C₂³×D₈	direct (N × H)
53. C₂³×D₄	C₂²×C₅₃	semidirect
55. C₂²×D₈	C₂²×C₅₃	direct (N × H)
56. C₂³×C₈	C₂²×C₅₃	semidirect
57. C₅₃	C₂⁴×D₈	direct (N × H)
62. C₂²×D₄	C₂³×C₅₃	semidirect
65. C₂×D₈	C₂³×C₅₃	direct (N × H)
66. C₂²×C₈	C₂³×C₅₃	semidirect
69. C₂⁴	D₈×C₅₃	direct (N × H)
72. C₂×D₄	C₂⁴×C₅₃	semidirect
75. D₈	C₂⁴×C₅₃	direct (N × H)
76. C₂×C₈	C₂⁴×C₅₃	semidirect
79. C₂³	C₂×D₈×C₅₃	direct (N × H)
82. D₄	C₂⁵×C₅₃	semidirect
85. C₈	C₂⁵×C₅₃	semidirect
88. C₂²	C₂²×D₈×C₅₃	direct (N × H)
93. C₂	C₂³×D₈×C₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₅₃	2
C₂³×D₈×C₅₃	2
C₂⁴×C₈×C₅₃	2
C₂⁴×D₈	53

Subgroups — C24×D8×C53 #27270589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×D₈×C₅₃ #27270589