Subgroups — C₂³×#127×C₅₃ #27270176

Order 13568.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₅₃	C₂	1	6	yes	maximal, complemented
2	C₂²×#127×C₅₃	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁶×C₅₃	C₂²	1	4	yes	—
5	C₂³×D₄×C₅₃	—	2	96	no	—
6	C₂³×D₄×C₅₃	C₂²	1	168	yes	complemented
7	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
8	C₂×#127×C₅₃	C₂²	1	448	yes	direct product factor
9	C₂²×C₄²×C₅₃	C₂²	1	28	yes	complemented
10	C₂⁵×C₅₃	—	2	96	no	—
11	C₂⁵×C₅₃	C₂³	1	56	yes	—
12	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	—	2	1344	no	—
14	C₂²×D₄×C₅₃	C₂³	1	672	yes	complemented
15	C₂³×C₄×C₅₃	D₄	1	48	yes	complemented
16	C₂³×C₄×C₅₃	C₂³	1	42	yes	—
17	#127×C₅₃	C₂³	1	512	yes	direct product factor
18	C₂×C₄²×C₅₃	C₂³	1	112	yes	complemented
19	C₂⁴×C₅₃	—	2	672	no	—
20	C₂⁴×C₅₃	—	4	256	no	—
21	C₂⁴×C₅₃	C₂×D₄	1	24	yes	—
22	C₂⁴×C₅₃	C₂⁴	1	112	yes	—
23	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
24	C₂×D₄×C₅₃	—	2	2688	no	—
25	C₂×D₄×C₅₃	C₂⁴	1	384	yes	complemented
26	C₂²×C₄×C₅₃	C₂×D₄	1	336	yes	complemented
27	C₂²×C₄×C₅₃	C₂⁴	1	84	yes	—
28	C₄²×C₅₃	C₂⁴	1	64	yes	complemented
29	C₂³×C₅₃	—	2	672	no	—
30	C₂³×C₅₃	—	4	896	no	—
31	C₂³×C₅₃	#127	1	64	yes	direct product factor
32	C₂³×C₅₃	C₂²×D₄	1	84	yes	—
33	C₂³×C₅₃	C₂⁵	1	32	yes	—
34	C₂³×C₅₃	C₂⁵	1	7	yes	—
35	D₄×C₅₃	—	2	768	no	—
36	C₂×C₄×C₅₃	C₂²×D₄	1	336	yes	complemented
37	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
38	C₂³×#127	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂²×C₅₃	—	2	96	no	—
40	C₂²×C₅₃	—	4	448	no	—
41	C₂²×C₅₃	C₂×#127	1	112	yes	direct product factor
42	C₂²×C₅₃	C₂³×D₄	1	42	yes	—
43	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
44	C₄×C₅₃	C₂³×D₄	1	48	yes	complemented
45	C₂⁴×D₄	C₂×C₅₃	1	6	yes	complemented
46	C₂²×#127	C₂×C₅₃	1	56	yes	direct product factor
47	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	—	4	32	no	—
49	C₂×C₅₃	C₂²×#127	1	28	yes	direct product factor
50	C₂×C₅₃	C₂⁴×D₄	1	3	yes	—
51	C₂⁶	C₂²×C₅₃	1	4	yes	—
52	C₂³×D₄	—	2	96	no	—
53	C₂³×D₄	C₂²×C₅₃	1	168	yes	complemented
54	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
55	C₂×#127	C₂²×C₅₃	1	448	yes	direct product factor
56	C₂²×C₄²	C₂²×C₅₃	1	28	yes	complemented
57	C₅₃	C₂³×#127	1	1	yes	characteristic, direct product factor, Sylow
58	C₂⁵	—	2	96	no	—
59	C₂⁵	C₂³×C₅₃	1	56	yes	—
60	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
61	C₂²×D₄	—	2	1344	no	—
62	C₂²×D₄	C₂³×C₅₃	1	672	yes	complemented
63	C₂³×C₄	D₄×C₅₃	1	48	yes	complemented
64	C₂³×C₄	C₂³×C₅₃	1	42	yes	—
65	#127	C₂³×C₅₃	1	512	yes	direct product factor
66	C₂×C₄²	C₂³×C₅₃	1	112	yes	complemented
67	C₂⁴	—	2	672	no	—
68	C₂⁴	—	4	256	no	—
69	C₂⁴	C₂×D₄×C₅₃	1	24	yes	—
70	C₂⁴	C₂⁴×C₅₃	1	112	yes	—
71	C₂⁴	C₂⁴×C₅₃	1	7	yes	—
72	C₂×D₄	—	2	2688	no	—
73	C₂×D₄	C₂⁴×C₅₃	1	384	yes	complemented
74	C₂²×C₄	C₂×D₄×C₅₃	1	336	yes	complemented
75	C₂²×C₄	C₂⁴×C₅₃	1	84	yes	—
76	C₄²	C₂⁴×C₅₃	1	64	yes	complemented
77	C₂³	—	2	672	no	—
78	C₂³	—	4	896	no	—
79	C₂³	#127×C₅₃	1	64	yes	direct product factor
80	C₂³	C₂²×D₄×C₅₃	1	84	yes	—
81	C₂³	C₂⁵×C₅₃	1	32	yes	—
82	C₂³	C₂⁵×C₅₃	1	7	yes	—
83	D₄	—	2	768	no	—
84	C₂×C₄	C₂²×D₄×C₅₃	1	336	yes	complemented
85	C₂×C₄	C₂⁵×C₅₃	1	24	yes	—
86	C₂²	—	2	96	no	—
87	C₂²	—	4	448	no	—
88	C₂²	C₂×#127×C₅₃	1	112	yes	direct product factor
89	C₂²	C₂³×D₄×C₅₃	1	42	yes	—
90	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
91	C₄	C₂³×D₄×C₅₃	1	48	yes	complemented
92	C₂	—	4	32	no	—
93	C₂	C₂²×#127×C₅₃	1	28	yes	direct product factor
94	C₂	C₂⁴×D₄×C₅₃	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₅₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×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. C₂²×#127×C₅₃	C₂	direct (N × H)
3. C₂³×C₄²×C₅₃	C₂	semidirect
6. C₂³×D₄×C₅₃	C₂²	semidirect
8. C₂×#127×C₅₃	C₂²	direct (N × H)
9. C₂²×C₄²×C₅₃	C₂²	semidirect
14. C₂²×D₄×C₅₃	C₂³	semidirect
15. C₂³×C₄×C₅₃	D₄	semidirect
17. #127×C₅₃	C₂³	direct (N × H)
18. C₂×C₄²×C₅₃	C₂³	semidirect
25. C₂×D₄×C₅₃	C₂⁴	semidirect
26. C₂²×C₄×C₅₃	C₂×D₄	semidirect
28. C₄²×C₅₃	C₂⁴	semidirect
31. C₂³×C₅₃	#127	direct (N × H)
36. C₂×C₄×C₅₃	C₂²×D₄	semidirect
38. C₂³×#127	C₅₃	direct (N × H)
41. C₂²×C₅₃	C₂×#127	direct (N × H)
44. C₄×C₅₃	C₂³×D₄	semidirect
45. C₂⁴×D₄	C₂×C₅₃	semidirect
46. C₂²×#127	C₂×C₅₃	direct (N × H)
47. C₂³×C₄²	C₂×C₅₃	semidirect
49. C₂×C₅₃	C₂²×#127	direct (N × H)
53. C₂³×D₄	C₂²×C₅₃	semidirect
55. C₂×#127	C₂²×C₅₃	direct (N × H)
56. C₂²×C₄²	C₂²×C₅₃	semidirect
57. C₅₃	C₂³×#127	direct (N × H)
62. C₂²×D₄	C₂³×C₅₃	semidirect
63. C₂³×C₄	D₄×C₅₃	semidirect
65. #127	C₂³×C₅₃	direct (N × H)
66. C₂×C₄²	C₂³×C₅₃	semidirect
73. C₂×D₄	C₂⁴×C₅₃	semidirect
74. C₂²×C₄	C₂×D₄×C₅₃	semidirect
76. C₄²	C₂⁴×C₅₃	semidirect
79. C₂³	#127×C₅₃	direct (N × H)
84. C₂×C₄	C₂²×D₄×C₅₃	semidirect
88. C₂²	C₂×#127×C₅₃	direct (N × H)
91. C₄	C₂³×D₄×C₅₃	semidirect
93. C₂	C₂²×#127×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
C₂²×#127×C₅₃	2
C₂³×C₄²×C₅₃	2
C₂³×#127	53

Subgroups — C23×#127×C53 #27270176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#127×C₅₃ #27270176