Subgroups — #20276998

Order 6784.

120 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	#16810399	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	1	no	—
7	C₂×#11719908	C₂²	1	2	yes	—
8	#13133151	C₂²	1	1	yes	characteristic
9	#13133151	C₂²	1	1	yes	characteristic
10	#13133149	—	2	1	no	—
11	C₄⋊Q₈×C₅₃	—	2	1	no	—
12	#107×C₅₃	—	2	1	no	—
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	C₄⋊Dic₅₃	—	2	1	no	—
22	C₄⋊Dic₅₃	—	2	1	no	—
23	#11719919	—	4	1	no	—
24	#11719919	—	4	1	no	—
25	#11719908	Q₈	1	4	yes	complemented
26	Q₈×C₅₃	—	8	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	1	no	—
32	C₂×Dic₅₃	—	4	1	no	—
33	C₈×C₅₃	Q₁₆	1	2	yes	—
34	C₈×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	—	8	2	no	—
39	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₅₃	—	8	1	no	—
42	Dic₅₃	—	8	1	no	—
43	#2255	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	#500	1	1	yes	characteristic
45	C₂×C₅₃	#509	1	1	yes	characteristic
46	C₂×C₅₃	#510	1	1	yes	characteristic
47	#500	D₅₃	1	1	yes	characteristic
48	#499	—	53	1	no	—
49	#367	—	53	2	no	—
50	#366	—	53	2	no	—
51	#346	—	53	1	no	—
52	C₅₃	#2255	1	1	yes	characteristic, complemented, Sylow
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	106	1	no	—
55	C₂×C₁₆	—	53	2	no	—
56	#107	—	2	1	no	—
57	#107	—	53	1	no	—
58	#107	—	53	1	no	—
59	#107	C₂×D₅₃	1	2	yes	—
60	#106	—	106	1	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	106	1	no	—
68	C₄⋊C₄	—	106	1	no	—
69	C₄⋊C₄	—	212	1	no	—
70	C₄⋊C₄	—	212	1	no	—
71	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
72	C₁₆	—	53	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	212	1	no	—
76	C₂×C₄	—	212	1	no	—
77	C₂×C₄	#11719922	1	1	yes	characteristic
78	C₂×C₄	#11719922	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
80	C₈	#11719925	1	2	yes	—
81	C₈	Q₈×D₅₃	1	2	yes	—
82	C₂²	C₂×#11719922	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	424	1	no	—
87	C₄	—	424	1	no	—
88	C₄	#13133215	1	1	yes	characteristic
89	C₄	C₂×#11719925	1	1	yes	characteristic
90	C₂	#16811026	1	1	yes	characteristic
91	C₂	#16811059	1	1	yes	characteristic
92	C₂	#16811064	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#500	D₅₃
hypocenter	C₅₃	#2255
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
25. #11719908	Q₈	semidirect
52. C₅₃	#2255	semidirect