Subgroups — #20276992

Order 6784.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811024	C₂	1	2	yes	maximal
2	#16810450	C₂	1	2	yes	maximal
3	#16810449	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	#13133208	—	2	2	no	—
7	C₂×#11719908	C₂²	1	1	yes	characteristic
8	#13133151	C₂²	1	2	yes	—
9	#13133148	—	2	2	no	—
10	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	#107×C₅₃	C₂²	1	2	yes	—
13	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	2	2	no	—
15	C₄⋊Dic₅₃	—	2	2	no	—
16	C₄⋊Dic₅₃	—	4	2	no	—
17	#11719919	—	4	2	no	—
18	C₄×Dic₅₃	D₄	1	1	yes	characteristic
19	C₂×#88322	D₄	1	1	yes	characteristic
20	C₁₆×C₅₃	Q₈	1	2	yes	—
21	#11719908	Q₈	1	2	yes	—
22	C₂×C₄×C₅₃	—	4	2	no	—
23	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₅₃	—	4	2	no	—
25	C₂×Dic₅₃	D₈	1	1	yes	characteristic
26	C₂×Dic₅₃	D₈	1	1	yes	characteristic
27	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	#88322	Q₁₆	1	2	yes	—
30	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
31	C₄×C₅₃	—	8	2	no	—
32	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₅₃	—	2	1	no	—
35	Dic₅₃	—	2	1	no	—
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	1	yes	characteristic
40	C₂×C₅₃	#508	1	1	yes	characteristic
41	#499	—	53	2	no	—
42	#367	—	53	2	no	—
43	#366	—	53	1	no	—
44	#366	D₅₃	1	1	yes	characteristic
45	C₄×C₁₆	—	53	1	no	—
46	C₅₃	#2253	1	1	yes	characteristic, complemented, Sylow
47	#125	—	106	2	no	—
48	C₂×C₁₆	—	53	1	no	—
49	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
50	#107	—	53	2	no	—
51	#107	C₂×D₅₃	1	2	yes	—
52	#106	—	106	2	no	—
53	C₄×C₈	—	53	1	no	—
54	C₂×C₈	—	53	1	no	—
55	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	106	2	no	—
58	C₄⋊C₄	—	212	2	no	—
59	C₄⋊C₄	—	212	2	no	—
60	C₄²	—	53	1	no	—
61	C₁₆	—	53	2	no	—
62	C₁₆	Dic₁₀₆	1	2	yes	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	53	1	no	—
65	C₂×C₄	—	53	1	no	—
66	C₂×C₄	—	212	2	no	—
67	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
68	C₈	—	53	2	no	—
69	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
70	C₈	Q₈×D₅₃	1	1	yes	characteristic
71	C₂²	D₈×D₅₃	1	1	yes	characteristic
72	C₄	—	8	2	no	—
73	C₄	—	106	1	no	—
74	C₄	—	106	1	no	—
75	C₄	—	424	2	no	—
76	C₄	#13133216	1	1	yes	characteristic
77	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
78	C₂	#16811027	1	1	yes	characteristic
79	C₂	#16811052	1	1	yes	characteristic
80	C₂	#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#366	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#366	D₅₃
hypocenter	C₅₃	#2253
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

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