Subgroups — C₂³×#13133214 #28290779

Order 13568.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	6	yes	maximal, complemented
2	C₂²×#13133214	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₅₃	C₂²	1	4	yes	—
5	C₂³×D₂₁₂	—	2	96	no	—
6	C₂³×D₂₁₂	C₂²	1	168	yes	complemented
7	C₂×#13133214	C₂²	1	448	yes	direct product factor
8	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
9	C₂²×C₄²×C₅₃	C₂²	1	28	yes	complemented
10	C₂⁴×D₅₃	—	2	96	no	—
11	C₂⁴×D₅₃	C₂³	1	56	yes	—
12	C₂²×D₂₁₂	—	2	1344	no	—
13	C₂²×D₂₁₂	C₂³	1	672	yes	complemented
14	#13133214	C₂³	1	512	yes	direct product factor
15	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	D₄	1	48	yes	complemented
17	C₂³×C₄×C₅₃	C₂³	1	42	yes	—
18	C₂×C₄²×C₅₃	C₂³	1	112	yes	complemented
19	C₂⁴×C₅₃	C₂×D₄	1	24	yes	—
20	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
21	C₂³×D₅₃	—	2	672	no	—
22	C₂³×D₅₃	—	4	256	no	—
23	C₂³×D₅₃	C₂⁴	1	112	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	336	yes	complemented
25	C₂²×C₄×C₅₃	C₂⁴	1	84	yes	—
26	C₂×D₂₁₂	—	2	2688	no	—
27	C₂×D₂₁₂	C₂⁴	1	384	yes	complemented
28	C₄²×C₅₃	C₂⁴	1	64	yes	complemented
29	C₂³×C₅₃	#127	1	64	yes	complemented
30	C₂³×C₅₃	C₂²×D₄	1	84	yes	—
31	C₂³×C₅₃	C₂⁵	1	7	yes	—
32	C₂²×D₅₃	—	2	672	no	—
33	C₂²×D₅₃	—	4	896	no	—
34	C₂²×D₅₃	C₂⁵	1	32	yes	—
35	C₂×C₄×C₅₃	C₂²×D₄	1	336	yes	complemented
36	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
37	D₂₁₂	—	2	768	no	—
38	C₂³×#127	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	C₂×#127	1	112	yes	complemented
40	C₂²×C₅₃	C₂³×D₄	1	42	yes	—
41	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
42	C₂×D₅₃	—	2	96	no	—
43	C₂×D₅₃	—	4	448	no	—
44	C₄×C₅₃	C₂³×D₄	1	48	yes	complemented
45	C₂⁴×D₄	—	53	6	no	—
46	C₂²×#127	—	53	56	no	—
47	C₂³×C₄²	D₅₃	1	1	yes	characteristic, complemented
48	C₂×C₅₃	C₂²×#127	1	28	yes	complemented
49	C₂×C₅₃	C₂⁴×D₄	1	3	yes	—
50	D₅₃	—	4	32	no	—
51	C₂⁶	—	53	4	no	—
52	C₂³×D₄	—	53	168	no	—
53	C₂³×D₄	—	106	96	no	—
54	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
55	C₂×#127	—	53	448	no	—
56	C₂²×C₄²	C₂×D₅₃	1	28	yes	complemented
57	C₅₃	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
58	C₂⁵	—	53	56	no	—
59	C₂⁵	—	106	96	no	—
60	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
61	C₂²×D₄	—	53	672	no	—
62	C₂²×D₄	—	106	1344	no	—
63	C₂³×C₄	D₂₁₂	1	48	yes	complemented
64	C₂³×C₄	C₂²×D₅₃	1	42	yes	—
65	#127	—	53	512	no	—
66	C₂×C₄²	C₂²×D₅₃	1	112	yes	complemented
67	C₂⁴	—	53	112	no	—
68	C₂⁴	—	106	672	no	—
69	C₂⁴	—	212	256	no	—
70	C₂⁴	C₂×D₂₁₂	1	24	yes	—
71	C₂⁴	C₂³×D₅₃	1	7	yes	—
72	C₂×D₄	—	53	384	no	—
73	C₂×D₄	—	106	2688	no	—
74	C₂²×C₄	C₂×D₂₁₂	1	336	yes	complemented
75	C₂²×C₄	C₂³×D₅₃	1	84	yes	—
76	C₄²	C₂³×D₅₃	1	64	yes	complemented
77	C₂³	—	53	32	no	—
78	C₂³	—	106	672	no	—
79	C₂³	—	212	896	no	—
80	C₂³	#13133214	1	64	yes	direct product factor
81	C₂³	C₂²×D₂₁₂	1	84	yes	—
82	C₂³	C₂⁴×D₅₃	1	7	yes	—
83	D₄	—	106	768	no	—
84	C₂×C₄	C₂²×D₂₁₂	1	336	yes	complemented
85	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
86	C₂²	—	106	96	no	—
87	C₂²	—	212	448	no	—
88	C₂²	C₂×#13133214	1	112	yes	direct product factor
89	C₂²	C₂³×D₂₁₂	1	42	yes	—
90	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
91	C₄	C₂³×D₂₁₂	1	48	yes	complemented
92	C₂	—	212	32	no	—
93	C₂	C₂²×#13133214	1	28	yes	direct product factor
94	C₂	C₂⁴×D₂₁₂	1	3	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂³×#127
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₁₂	C₂	semidirect
2. C₂²×#13133214	C₂	direct (N × H)
3. C₂³×C₄²×C₅₃	C₂	semidirect
6. C₂³×D₂₁₂	C₂²	semidirect
7. C₂×#13133214	C₂²	direct (N × H)
9. C₂²×C₄²×C₅₃	C₂²	semidirect
13. C₂²×D₂₁₂	C₂³	semidirect
14. #13133214	C₂³	direct (N × H)
16. C₂³×C₄×C₅₃	D₄	semidirect
18. C₂×C₄²×C₅₃	C₂³	semidirect
24. C₂²×C₄×C₅₃	C₂×D₄	semidirect
27. C₂×D₂₁₂	C₂⁴	semidirect
28. C₄²×C₅₃	C₂⁴	semidirect
29. C₂³×C₅₃	#127	semidirect
35. C₂×C₄×C₅₃	C₂²×D₄	semidirect
39. C₂²×C₅₃	C₂×#127	semidirect
44. C₄×C₅₃	C₂³×D₄	semidirect
47. C₂³×C₄²	D₅₃	semidirect
48. C₂×C₅₃	C₂²×#127	semidirect
56. C₂²×C₄²	C₂×D₅₃	semidirect
57. C₅₃	C₂³×#127	semidirect
63. C₂³×C₄	D₂₁₂	semidirect
66. C₂×C₄²	C₂²×D₅₃	semidirect
74. C₂²×C₄	C₂×D₂₁₂	semidirect
76. C₄²	C₂³×D₅₃	semidirect
80. C₂³	#13133214	direct (N × H)
84. C₂×C₄	C₂²×D₂₁₂	semidirect
88. C₂²	C₂×#13133214	direct (N × H)
91. C₄	C₂³×D₂₁₂	semidirect
93. C₂	C₂²×#13133214	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×D₂₁₂	2
C₂²×#13133214	2
C₂³×C₄²×C₅₃	2
C₂³×#127	53

Subgroups — C23×#13133214 #28290779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133214 #28290779