Subgroups — C₂³×#11719912 #20289416

Order 6784.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719912	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₅₃	C₄	1	1	yes	characteristic
6	C₂³×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₅₃	C₄	1	14	yes	—
8	C₂²×C₄×D₅₃	C₂²	1	14	yes	—
9	C₂×#11719912	C₂²	1	112	yes	direct product factor
10	C₂²×#88322	C₂²	1	14	yes	complemented
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂²	1	14	yes	complemented
13	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×D₅₃	—	2	8	no	—
15	C₂³×D₅₃	C₂×C₄	1	14	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	7	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
18	C₂²×Dic₅₃	C₂×C₄	1	14	yes	—
19	C₂×C₄×D₅₃	C₂×C₄	1	84	yes	—
20	C₂×C₄×D₅₃	C₂³	1	28	yes	—
21	C₂×C₈×C₅₃	C₂³	1	28	yes	complemented
22	C₂×#88322	C₂³	1	28	yes	complemented
23	#11719912	C₂³	1	64	yes	direct product factor
24	C₂³×C₅₃	M₁₆	1	8	yes	complemented
25	C₂³×C₅₃	C₂²×C₄	1	7	yes	—
26	C₂²×D₅₃	—	2	56	no	—
27	C₂²×D₅₃	C₂²×C₄	1	28	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	21	yes	—
29	C₂×C₄×C₅₃	C₂⁴	1	7	yes	—
30	C₂×Dic₅₃	C₂²×C₄	1	28	yes	—
31	C₄×D₅₃	C₂²×C₄	1	56	yes	—
32	C₄×D₅₃	C₂⁴	1	8	yes	—
33	C₈×C₅₃	C₂⁴	1	8	yes	complemented
34	#88322	C₂⁴	1	8	yes	complemented
35	C₂²×C₅₃	C₂×M₁₆	1	28	yes	complemented
36	C₂²×C₅₃	C₂³×C₄	1	7	yes	—
37	C₂×D₅₃	—	2	56	no	—
38	C₂×D₅₃	C₂³×C₄	1	8	yes	—
39	C₄×C₅₃	C₂³×C₄	1	7	yes	—
40	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
41	Dic₅₃	C₂³×C₄	1	8	yes	—
42	C₂³×M₁₆	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	C₂²×M₁₆	1	14	yes	complemented
44	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
45	D₅₃	—	2	8	no	—
46	C₂⁴×C₄	—	53	1	no	—
47	C₂²×M₁₆	—	53	28	no	—
48	C₂³×C₈	—	53	1	no	—
49	C₂³×C₈	D₅₃	1	1	yes	characteristic, complemented
50	C₅₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂⁵	—	53	1	no	—
52	C₂³×C₄	—	53	1	no	—
53	C₂³×C₄	—	53	14	no	—
54	C₂³×C₄	—	53	14	no	—
55	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
56	C₂×M₁₆	—	53	112	no	—
57	C₂²×C₈	—	53	14	no	—
58	C₂²×C₈	C₂×D₅₃	1	14	yes	complemented
59	C₂⁴	—	53	14	no	—
60	C₂⁴	—	106	8	no	—
61	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
62	C₂²×C₄	—	53	14	no	—
63	C₂²×C₄	—	53	84	no	—
64	C₂²×C₄	—	53	28	no	—
65	C₂²×C₄	C₄×D₅₃	1	7	yes	—
66	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
67	M₁₆	—	53	64	no	—
68	C₂×C₈	—	53	28	no	—
69	C₂×C₈	C₂²×D₅₃	1	28	yes	complemented
70	C₂³	—	53	28	no	—
71	C₂³	—	106	56	no	—
72	C₂³	#11719912	1	8	yes	direct product factor
73	C₂³	C₂×C₄×D₅₃	1	7	yes	—
74	C₂×C₄	—	53	28	no	—
75	C₂×C₄	—	53	56	no	—
76	C₂×C₄	—	53	8	no	—
77	C₂×C₄	C₂×C₄×D₅₃	1	21	yes	—
78	C₂×C₄	C₂³×D₅₃	1	7	yes	—
79	C₈	—	53	8	no	—
80	C₈	C₂³×D₅₃	1	8	yes	complemented
81	C₂²	—	53	8	no	—
82	C₂²	—	106	56	no	—
83	C₂²	C₂×#11719912	1	28	yes	direct product factor
84	C₂²	C₂²×C₄×D₅₃	1	7	yes	—
85	C₄	—	53	8	no	—
86	C₄	C₂²×C₄×D₅₃	1	7	yes	—
87	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
88	C₂	—	106	8	no	—
89	C₂	C₂²×#11719912	1	14	yes	direct product factor
90	C₂	C₂³×C₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×C₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂³×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#11719912	C₂	direct (N × H)
3. C₂³×#88322	C₂	semidirect
4. C₂³×C₈×C₅₃	C₂	semidirect
9. C₂×#11719912	C₂²	direct (N × H)
10. C₂²×#88322	C₂²	semidirect
12. C₂²×C₈×C₅₃	C₂²	semidirect
21. C₂×C₈×C₅₃	C₂³	semidirect
22. C₂×#88322	C₂³	semidirect
23. #11719912	C₂³	direct (N × H)
24. C₂³×C₅₃	M₁₆	semidirect
33. C₈×C₅₃	C₂⁴	semidirect
34. #88322	C₂⁴	semidirect
35. C₂²×C₅₃	C₂×M₁₆	semidirect
43. C₂×C₅₃	C₂²×M₁₆	semidirect
49. C₂³×C₈	D₅₃	semidirect
50. C₅₃	C₂³×M₁₆	semidirect
58. C₂²×C₈	C₂×D₅₃	semidirect
69. C₂×C₈	C₂²×D₅₃	semidirect
72. C₂³	#11719912	direct (N × H)
80. C₈	C₂³×D₅₃	semidirect
83. C₂²	C₂×#11719912	direct (N × H)
89. C₂	C₂²×#11719912	direct (N × H)

Subgroups — C23×#11719912 #20289416

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719912 #20289416