Subgroups — D₅₃×D₆₄ #27305133

Order 13568.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
2	D₃₃₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272927	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×D₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₆×D₅₃	—	2	2	no	—
7	D₁₆₉₆	C₂²	1	2	yes	complemented
8	#16810458	—	2	2	no	—
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	#16810342	C₂²	1	1	yes	characteristic, complemented
11	C₅₃×D₃₂	C₂²	1	2	yes	complemented
12	C₅₃×C₆₄	C₂²	1	1	yes	characteristic, complemented
13	D₈×D₅₃	—	4	2	no	—
14	D₈₄₈	—	2	2	no	—
15	#13133158	—	4	2	no	—
16	C₁₆×D₅₃	D₄	1	1	yes	characteristic
17	#13133123	D₄	1	1	yes	characteristic
18	D₁₆×C₅₃	—	2	2	no	—
19	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
20	D₄×D₅₃	—	8	2	no	—
21	D₈×C₅₃	—	4	2	no	—
22	#11719922	—	8	2	no	—
23	D₄₂₄	—	4	2	no	—
24	C₈×D₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
26	#11719908	D₈	1	1	yes	characteristic
27	C₂²×D₅₃	—	16	2	no	—
28	D₄×C₅₃	—	8	2	no	—
29	C₂²⋊D₅₃	—	16	2	no	—
30	D₂₁₂	—	8	2	no	—
31	C₄×D₅₃	D₁₆	1	1	yes	characteristic
32	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
33	#88322	D₁₆	1	1	yes	characteristic
34	C₂×D₆₄	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	—	16	2	no	—
36	C₂×D₅₃	—	16	2	no	—
37	C₂×D₅₃	—	32	4	no	—
38	C₂×D₅₃	D₃₂	1	1	yes	characteristic
39	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
40	Dic₅₃	D₃₂	1	1	yes	characteristic
41	C₂×D₃₂	—	53	2	no	—
42	D₆₄	—	53	2	no	—
43	D₆₄	—	53	1	no	—
44	D₆₄	D₅₃	1	1	yes	characteristic, direct product factor
45	C₂×C₆₄	—	53	1	no	—
46	C₂×C₅₃	—	32	2	no	—
47	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
48	D₅₃	—	32	2	no	—
49	D₅₃	D₆₄	1	2	yes	direct product factor
50	C₂×D₁₆	—	106	2	no	—
51	D₃₂	—	53	2	no	—
52	D₃₂	—	106	2	no	—
53	D₃₂	C₂×D₅₃	1	2	yes	complemented
54	C₂×C₃₂	—	53	1	no	—
55	C₆₄	—	53	1	no	—
56	C₆₄	C₂×D₅₃	1	1	yes	characteristic, complemented
57	C₅₃	C₂×D₆₄	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₈	—	212	2	no	—
59	D₁₆	—	2	2	no	—
60	D₁₆	—	106	2	no	—
61	D₁₆	—	212	2	no	—
62	C₂×C₁₆	—	53	1	no	—
63	C₃₂	—	53	1	no	—
64	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
65	C₂×D₄	—	424	2	no	—
66	D₈	—	4	2	no	—
67	D₈	—	424	2	no	—
68	D₈	—	212	2	no	—
69	C₂×C₈	—	53	1	no	—
70	C₁₆	—	53	1	no	—
71	C₁₆	D₄×D₅₃	1	1	yes	characteristic
72	C₂³	—	848	2	no	—
73	D₄	—	8	2	no	—
74	D₄	—	848	2	no	—
75	D₄	—	424	2	no	—
76	C₂×C₄	—	53	1	no	—
77	C₈	—	53	1	no	—
78	C₈	D₈×D₅₃	1	1	yes	characteristic
79	C₂²	—	16	2	no	—
80	C₂²	—	53	1	no	—
81	C₂²	—	848	2	no	—
82	C₂²	—	1696	4	no	—
83	C₄	—	53	1	no	—
84	C₄	D₁₆×D₅₃	1	1	yes	characteristic
85	C₂	—	32	2	no	—
86	C₂	—	53	2	no	—
87	C₂	—	1696	2	no	—
88	C₂	D₃₂×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₅₃
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₅₃×D₆₄	C₂
socle	C₂×C₅₃	C₂×D₃₂
hypercenter	D₆₄	D₅₃
hypocenter	C₅₃	C₂×D₆₄
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ D₆₄

Complements

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

N	H ≅ G/N	type
1. D₃₂×D₅₃	C₂	semidirect
2. D₃₃₉₂	C₂	semidirect
3. #20272927	C₂	semidirect
4. C₆₄×D₅₃	C₂	semidirect
5. C₅₃×D₆₄	C₂	semidirect
7. D₁₆₉₆	C₂²	semidirect
10. #16810342	C₂²	semidirect
11. C₅₃×D₃₂	C₂²	semidirect
12. C₅₃×C₆₄	C₂²	semidirect
44. D₆₄	D₅₃	direct (N × H)
49. D₅₃	D₆₄	direct (N × H)
53. D₃₂	C₂×D₅₃	semidirect
56. C₆₄	C₂×D₅₃	semidirect
57. C₅₃	C₂×D₆₄	semidirect

Maximal subgroups

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

subgroup	index
D₃₂×D₅₃	2
D₃₃₉₂	2
#20272927	2
C₆₄×D₅₃	2
C₅₃×D₆₄	2
C₂×D₆₄	53

Subgroups — D53×D64 #27305133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×D₆₄ #27305133