Subgroups — #20277035

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811055	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₈₄₈	C₂	1	2	yes	maximal, complemented
4	#16810462	C₂	1	2	yes	maximal, complemented
5	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	—	2	1	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	Dic₄₂₄	—	2	1	no	—
9	Dic₄₂₄	C₂²	1	2	yes	—
10	Dic₄₂₄	C₂²	1	1	yes	characteristic
11	#13133162	—	2	1	no	—
12	D₈₄₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂²	1	2	yes	—
15	#11719945	—	4	1	no	—
16	C₂×Dic₁₀₆	—	4	1	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	Dic₂₁₂	—	2	1	no	—
19	Dic₂₁₂	—	2	2	no	—
20	Dic₂₁₂	—	4	1	no	—
21	D₄₂₄	—	2	1	no	—
22	#11719913	—	4	1	no	—
23	C₁₆×C₅₃	D₄	1	1	yes	characteristic
24	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
26	C₂²⋊D₅₃	—	8	1	no	—
27	C₂×Dic₅₃	—	8	1	no	—
28	D₂₁₂	—	4	1	no	—
29	C₄×D₅₃	—	8	1	no	—
30	Dic₁₀₆	—	4	2	no	—
31	Dic₁₀₆	—	4	1	no	—
32	Dic₁₀₆	—	8	1	no	—
33	C₈×C₅₃	D₈	1	1	yes	characteristic
34	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
36	C₂×D₅₃	—	8	1	no	—
37	C₄×C₅₃	D₁₆	1	1	yes	characteristic
38	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
39	Dic₅₃	—	8	2	no	—
40	Dic₅₃	—	8	1	no	—
41	#2264	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
44	D₅₃	—	16	1	no	—
45	#508	—	53	1	no	—
46	C₂×Q₃₂	—	53	1	no	—
47	Q₆₄	—	53	2	no	—
48	SD₆₄	—	53	2	no	—
49	M₆₄	D₅₃	1	1	yes	characteristic, complemented
50	C₅₃	#2264	1	1	yes	characteristic, complemented, Sylow
51	#135	—	106	1	no	—
52	C₂×Q₁₆	—	106	1	no	—
53	Q₃₂	—	53	2	no	—
54	Q₃₂	—	53	1	no	—
55	Q₃₂	—	106	1	no	—
56	SD₃₂	—	106	1	no	—
57	D₁₆	—	53	1	no	—
58	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
59	C₃₂	C₂×D₅₃	1	2	yes	—
60	#41	—	212	1	no	—
61	C₂×Q₈	—	212	1	no	—
62	Q₁₆	—	106	1	no	—
63	Q₁₆	—	106	2	no	—
64	Q₁₆	—	212	1	no	—
65	SD₁₆	—	212	1	no	—
66	D₈	—	106	1	no	—
67	C₂×C₈	D₂₁₂	1	1	yes	characteristic
68	C₁₆	D₂₁₂	1	1	yes	characteristic
69	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
70	Q₈	—	424	1	no	—
71	Q₈	—	212	1	no	—
72	Q₈	—	212	2	no	—
73	D₄	—	424	1	no	—
74	D₄	—	212	1	no	—
75	C₂×C₄	—	424	1	no	—
76	C₂×C₄	—	424	1	no	—
77	C₂×C₄	D₄₂₄	1	1	yes	characteristic
78	C₈	D₄₂₄	1	1	yes	characteristic
79	C₈	C₂×D₂₁₂	1	1	yes	characteristic
80	C₂²	—	424	1	no	—
81	C₂²	D₈₄₈	1	1	yes	characteristic
82	C₄	—	424	1	no	—
83	C₄	—	424	2	no	—
84	C₄	D₈₄₈	1	1	yes	characteristic
85	C₄	C₂×D₄₂₄	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	848	1	no	—
88	C₂	C₂×D₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈₄₈
commutator	C₁₆×C₅₃	C₂³
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂×D₁₆
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#2264
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₄₂₄	C₂	semidirect
3. Dic₈₄₈	C₂	semidirect
4. #16810462	C₂	semidirect
5. C₅₃×M₆₄	C₂	semidirect
49. M₆₄	D₅₃	semidirect
50. C₅₃	#2264	semidirect