Subgroups — #20276942

Order 6784.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	1	yes	maximal, characteristic
2	#16810427	C₂	1	4	yes	maximal
3	C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	#13133217	—	2	1	no	—
6	C₂×#11719908	C₂²	1	2	yes	—
7	#13133151	—	2	1	no	—
8	#13133151	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₅₃	—	2	1	no	—
10	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₃	—	2	2	no	—
14	C₂×Dic₁₀₆	—	4	1	no	—
15	Q₁₆×C₅₃	—	2	4	no	—
16	Q₁₆×C₅₃	—	4	2	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₄⋊C₄×C₅₃	—	4	1	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊Dic₅₃	—	4	1	no	—
24	#11719908	—	2	2	no	—
25	Q₈×C₅₃	—	4	2	no	—
26	Q₈×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	2	no	—
32	Dic₁₀₆	—	8	2	no	—
33	C₈×C₅₃	SD₁₆	1	2	yes	—
34	C₈×C₅₃	#41	1	2	yes	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₃	—	8	2	no	—
37	C₄×C₅₃	SD₃₂	1	2	yes	—
38	C₄×C₅₃	Q₃₂	1	2	yes	—
39	C₄×C₅₃	#124	1	1	yes	characteristic
40	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₅₃	—	8	2	no	—
42	#2241	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	#486	1	1	yes	characteristic
44	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
45	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
46	#500	—	53	1	no	—
47	#494	D₅₃	1	1	yes	characteristic
48	#358	—	53	4	no	—
49	C₄×C₁₆	—	53	1	no	—
50	C₅₃	#2241	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	2	1	no	—
52	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	106	1	no	—
55	C₂×C₁₆	—	53	2	no	—
56	#107	—	53	2	no	—
57	#107	—	106	1	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	212	1	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	106	2	no	—
68	C₄⋊C₄	—	212	1	no	—
69	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
70	C₁₆	—	106	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	424	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	212	2	no	—
76	C₂×C₄	#11719922	1	1	yes	characteristic
77	C₂×C₄	#11719922	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₈	#11719923	1	2	yes	—
80	C₈	#11719947	1	2	yes	—
81	C₂²	C₂×#11719922	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	424	2	no	—
84	C₄	#13133160	1	2	yes	—
85	C₄	#13133163	1	2	yes	—
86	C₄	#13133204	1	1	yes	characteristic
87	C₄	C₂×#11719923	1	1	yes	characteristic
88	C₂	#16810969	1	1	yes	characteristic
89	C₂	C₂×#13133160	1	1	yes	characteristic
90	C₂	C₂×#13133163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#494	D₅₃
hypocenter	C₅₃	#2241
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
50. C₅₃	#2241	semidirect