Subgroups — #20272751

Order 6784.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	—	2	1	no	—
5	C₂×D₄₂₄	—	2	1	no	—
6	#13133214	C₄	1	2	yes	—
7	#13133157	—	2	1	no	—
8	M₃₂×C₅₃	—	2	1	no	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₂₁₂	—	2	2	no	—
11	C₂×D₂₁₂	—	4	2	no	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	D₄₂₄	—	4	1	no	—
16	D₄₂₄	—	4	1	no	—
17	D₄₂₄	—	4	1	no	—
18	D₄₂₄	—	4	1	no	—
19	C₁₆×C₅₃	—	4	1	no	—
20	#11719908	—	4	1	no	—
21	C₂²×D₅₃	—	4	2	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	D₂₁₂	—	4	2	no	—
25	D₂₁₂	—	4	2	no	—
26	D₂₁₂	—	8	4	no	—
27	C₈×C₅₃	D₈	1	1	yes	characteristic
28	C₈×C₅₃	D₈	1	1	yes	characteristic
29	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₅₃	#100	1	1	yes	characteristic
32	C₂×D₅₃	—	8	4	no	—
33	C₂×D₅₃	—	8	2	no	—
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	#102	1	1	yes	characteristic
36	C₄×C₅₃	#102	1	1	yes	characteristic
37	#1357	—	53	1	no	maximal, Sylow
38	C₂×C₅₃	—	2	1	no	—
39	C₂×C₅₃	#331	1	1	yes	characteristic
40	D₅₃	—	16	2	no	—
41	#493	—	53	1	no	—
42	#364	—	53	1	no	—
43	#364	D₅₃	1	1	yes	characteristic, complemented
44	C₅₃	#1357	1	1	yes	characteristic, complemented, Sylow
45	C₂×D₈	—	106	1	no	—
46	C₂×D₈	—	106	1	no	—
47	#127	—	53	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	106	1	no	—
50	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
51	C₂×D₄	—	106	2	no	—
52	C₂×D₄	—	212	2	no	—
53	D₈	—	212	1	no	—
54	D₈	—	212	1	no	—
55	D₈	—	212	1	no	—
56	D₈	—	212	1	no	—
57	C₂×C₈	D₂₁₂	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₄²	C₄×D₅₃	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	212	1	no	—
62	C₂³	—	212	2	no	—
63	D₄	—	424	4	no	—
64	D₄	—	212	2	no	—
65	D₄	—	212	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#11719921	1	1	yes	characteristic
68	C₈	#11719913	1	1	yes	characteristic
69	C₈	D₄₂₄	1	1	yes	characteristic
70	C₈	#11719922	1	1	yes	characteristic
71	C₈	#11719924	1	1	yes	characteristic
72	C₂²	—	424	4	no	—
73	C₂²	—	424	2	no	—
74	C₂²	#13133133	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#13133137	1	1	yes	characteristic
77	C₄	#13133139	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	848	2	no	—
80	C₂	#16810366	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810366
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#364	C₂
socle	C₂×C₅₃	#331
hypercenter	#364	D₅₃
hypocenter	C₅₃	#1357
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #16810444	C₂	semidirect
3. C₅₃×#364	C₂	semidirect
43. #364	D₅₃	semidirect
44. C₅₃	#1357	semidirect