Subgroups — #20276960

Order 6784.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811025	C₂	1	1	yes	maximal, characteristic
2	#16810427	C₂	1	4	yes	maximal
3	#16810399	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	#13133209	—	2	1	no	—
6	C₂×#11719908	C₂²	1	2	yes	—
7	#13133151	C₂²	1	2	yes	—
8	#13133149	—	2	1	no	—
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	Q₁₆×C₅₃	—	2	4	no	—
15	Q₁₆×C₅₃	—	4	2	no	—
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	—	4	2	no	—
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	2	2	no	—
21	#11719919	—	4	2	no	—
22	#11719908	—	2	2	no	—
23	Q₈×C₅₃	—	4	2	no	—
24	Q₈×C₅₃	—	4	2	no	—
25	C₂×C₄×C₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	D₈	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	2	no	—
29	C₈×C₅₃	SD₁₆	1	2	yes	—
30	C₈×C₅₃	#41	1	2	yes	—
31	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	—	8	2	no	—
34	C₄×C₅₃	#124	1	1	yes	characteristic
35	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₅₃	—	8	2	no	—
37	#2245	—	53	1	no	maximal, Sylow
38	C₂×C₅₃	#486	1	1	yes	characteristic
39	C₂×C₅₃	#510	1	2	yes	—
40	#499	—	53	1	no	—
41	#494	D₅₃	1	1	yes	characteristic
42	#358	—	53	4	no	—
43	#346	—	53	1	no	—
44	C₅₃	#2245	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	2	1	no	—
46	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
47	C₄⋊Q₈	—	2	1	no	—
48	#125	—	106	1	no	—
49	C₂×C₁₆	—	53	2	no	—
50	#107	—	53	2	no	—
51	#106	—	106	1	no	—
52	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
53	C₂×Q₈	—	2	2	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	2	no	—
56	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	106	2	no	—
60	C₄⋊C₄	—	212	2	no	—
61	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
62	C₁₆	—	106	2	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	212	2	no	—
67	C₂×C₄	#11719922	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
69	C₈	#11719923	1	2	yes	—
70	C₈	#11719947	1	2	yes	—
71	C₂²	C₂×#11719922	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	424	2	no	—
75	C₄	#13133204	1	1	yes	characteristic
76	C₄	C₂×#11719923	1	1	yes	characteristic
77	C₂	#16810969	1	1	yes	characteristic
78	C₂	#16811064	1	2	yes	—

	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	#2245
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
44. C₅₃	#2245	semidirect