Subgroups — #20276989

Order 6784.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811027	C₂	1	2	yes	maximal
2	#16810449	C₂	1	1	yes	maximal, characteristic
3	#16810448	C₂	1	2	yes	maximal
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	#13133216	—	2	2	no	—
7	C₂×#11719908	C₂²	1	1	yes	characteristic
8	#13133151	C₂²	1	2	yes	—
9	#13133150	—	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×Dic₁₀₆	—	4	2	no	—
14	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₄⋊Dic₅₃	—	2	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	complemented
21	#11719908	Q₈	1	2	yes	complemented
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	Dic₁₀₆	—	8	4	no	—
28	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
30	#88322	Q₁₆	1	2	yes	—
31	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
32	C₄×C₅₃	—	8	2	no	—
33	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₅₃	—	8	2	no	—
36	Dic₅₃	D₁₆	1	2	yes	—
37	Dic₅₃	Q₃₂	1	2	yes	—
38	#2252	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#500	1	1	yes	characteristic
40	C₂×C₅₃	C₂×D₁₆	1	1	yes	characteristic
41	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
42	#500	—	53	2	no	—
43	#366	—	53	1	no	—
44	#366	—	53	2	no	—
45	#366	D₅₃	1	1	yes	characteristic
46	C₄×C₁₆	—	53	1	no	—
47	C₅₃	#2252	1	1	yes	characteristic, complemented, Sylow
48	C₄⋊Q₈	—	106	2	no	—
49	C₂×C₁₆	—	53	1	no	—
50	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
51	#107	—	53	2	no	—
52	#107	—	106	2	no	—
53	#107	C₂×D₅₃	1	2	yes	—
54	C₄×C₈	—	53	1	no	—
55	C₂×Q₈	—	212	2	no	—
56	C₂×C₈	—	53	1	no	—
57	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	106	2	no	—
60	C₄⋊C₄	—	212	2	no	—
61	C₄²	—	53	1	no	—
62	C₁₆	—	53	2	no	—
63	C₁₆	Dic₁₀₆	1	2	yes	complemented
64	Q₈	—	424	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	53	1	no	—
67	C₂×C₄	—	53	1	no	—
68	C₂×C₄	—	212	2	no	—
69	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
70	C₈	—	53	2	no	—
71	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
72	C₈	Q₈×D₅₃	1	1	yes	characteristic
73	C₂²	D₈×D₅₃	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	53	2	no	—
76	C₄	—	53	2	no	—
77	C₄	—	424	2	no	—
78	C₄	#13133216	1	1	yes	characteristic
79	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
80	C₂	#16811027	1	1	yes	characteristic
81	C₂	D₁₆×D₅₃	1	1	yes	characteristic
82	C₂	Q₃₂×D₅₃	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	#2252
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
20. C₁₆×C₅₃	Q₈	semidirect
21. #11719908	Q₈	semidirect
47. C₅₃	#2252	semidirect
63. C₁₆	Dic₁₀₆	semidirect