Subgroups — #20276991

Order 6784.

116 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	#16810450	C₂	1	2	yes	maximal
3	#16810448	C₂	1	2	yes	maximal
4	C₄×#11719908	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#499	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	1	no	—
7	C₂×#11719908	C₂²	1	1	yes	characteristic
8	C₂×#11719908	C₂²	1	1	yes	characteristic
9	#13133151	C₂²	1	2	yes	—
10	#13133149	—	2	1	no	—
11	#125×C₅₃	—	2	1	no	—
12	#107×C₅₃	C₂²	1	2	yes	—
13	#106×C₅₃	—	2	1	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	2	no	—
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	Q₈	1	2	yes	—
23	#11719908	Q₈	1	2	yes	—
24	C₂×C₄×C₅₃	—	4	2	no	—
25	C₂×C₄×C₅₃	D₈	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	2	no	—
28	C₈×C₅₃	Q₁₆	1	2	yes	—
29	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	—	8	2	no	—
34	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₅₃	—	8	2	no	—
37	#2253	—	53	1	no	maximal, Sylow
38	C₂×C₅₃	#500	1	1	yes	characteristic
39	C₂×C₅₃	#508	1	2	yes	—
40	#499	—	53	1	no	—
41	#499	D₅₃	1	1	yes	characteristic
42	#367	—	53	2	no	—
43	#366	—	53	2	no	—
44	C₄×C₁₆	—	53	1	no	—
45	C₅₃	#2253	1	1	yes	characteristic, complemented, Sylow
46	#125	—	2	1	no	—
47	#125	—	106	1	no	—
48	C₂×C₁₆	—	53	1	no	—
49	C₂×C₁₆	—	53	1	no	—
50	#107	—	53	2	no	—
51	#107	C₂×D₅₃	1	2	yes	—
52	#106	—	2	1	no	—
53	#106	—	106	1	no	—
54	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
57	C₄⋊C₄	—	2	2	no	—
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	—	53	2	no	—
63	C₁₆	—	53	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	212	2	no	—
66	C₂×C₄	#11719922	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
68	C₈	#11719925	1	2	yes	—
69	C₈	Q₈×D₅₃	1	1	yes	characteristic
70	C₈	Q₈×D₅₃	1	1	yes	characteristic
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	#13133215	1	1	yes	characteristic
76	C₄	C₂×#11719925	1	1	yes	characteristic
77	C₂	#16811026	1	1	yes	characteristic
78	C₂	#16811053	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#499	D₅₃
hypocenter	C₅₃	#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
45. C₅₃	#2253	semidirect