Subgroups — #20272753

Order 6784.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811003	C₂	1	1	yes	maximal, characteristic
2	#16810444	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	—	2	1	no	—
5	C₂×Dic₂₁₂	—	2	1	no	—
6	#13133217	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×Dic₁₀₆	—	2	2	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	4	2	no	—
15	Dic₂₁₂	—	4	1	no	—
16	Dic₂₁₂	—	4	1	no	—
17	Dic₂₁₂	—	4	1	no	—
18	Dic₂₁₂	—	4	1	no	—
19	C₁₆×C₅₃	—	4	1	no	—
20	#11719908	—	4	1	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	2	no	—
24	Dic₁₀₆	—	4	2	no	—
25	Dic₁₀₆	—	4	2	no	—
26	C₈×C₅₃	D₈	1	1	yes	characteristic
27	C₈×C₅₃	D₈	1	1	yes	characteristic
28	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
29	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₂²×C₅₃	#100	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	#102	1	1	yes	characteristic
33	C₄×C₅₃	#102	1	1	yes	characteristic
34	Dic₅₃	—	8	2	no	—
35	#1358	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	#331	1	1	yes	characteristic
38	#494	—	53	1	no	—
39	#364	—	53	1	no	—
40	#364	D₅₃	1	1	yes	characteristic
41	C₅₃	#1358	1	1	yes	characteristic, complemented, Sylow
42	C₂×Q₁₆	—	106	1	no	—
43	C₂×Q₁₆	—	106	1	no	—
44	C₄⋊Q₈	—	53	2	no	—
45	M₃₂	—	2	1	no	—
46	M₃₂	—	106	1	no	—
47	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
48	C₂×Q₈	—	106	2	no	—
49	Q₁₆	—	212	1	no	—
50	Q₁₆	—	212	1	no	—
51	Q₁₆	—	212	1	no	—
52	Q₁₆	—	212	1	no	—
53	C₂×C₈	D₂₁₂	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
55	C₄⋊C₄	—	212	2	no	—
56	C₄²	C₄×D₅₃	1	1	yes	characteristic
57	C₁₆	—	4	1	no	—
58	C₁₆	—	212	1	no	—
59	Q₈	—	212	2	no	—
60	Q₈	—	212	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	212	2	no	—
63	C₂×C₄	#11719921	1	1	yes	characteristic
64	C₈	#11719913	1	1	yes	characteristic
65	C₈	D₄₂₄	1	1	yes	characteristic
66	C₈	#11719922	1	1	yes	characteristic
67	C₈	#11719924	1	1	yes	characteristic
68	C₂²	#13133133	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	424	2	no	—
71	C₄	#13133137	1	1	yes	characteristic
72	C₄	#13133139	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	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	#1358
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
41. C₅₃	#1358	semidirect