Subgroups — #20276996

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	#16810451	C₂	1	1	yes	maximal, characteristic
3	#16810450	C₂	1	2	yes	maximal
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#367	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	2	yes	—
26	Dic₁₀₆	—	8	4	no	—
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	—	8	2	no	—
35	Dic₅₃	SD₃₂	1	4	yes	—
36	#2254	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	#500	1	1	yes	characteristic
38	C₂×C₅₃	C₂×SD₃₂	1	2	yes	—
39	#500	—	53	2	no	—
40	#367	—	53	1	no	—
41	#367	—	53	2	no	—
42	#367	D₅₃	1	1	yes	characteristic
43	C₄×C₁₆	—	53	1	no	—
44	C₅₃	#2254	1	1	yes	characteristic, complemented, Sylow
45	C₄⋊Q₈	—	106	2	no	—
46	C₂×C₁₆	—	53	1	no	—
47	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
48	#107	—	53	2	no	—
49	#107	—	106	2	no	—
50	#107	C₂×D₅₃	1	2	yes	—
51	C₄×C₈	—	53	1	no	—
52	C₂×Q₈	—	212	2	no	—
53	C₂×C₈	—	53	1	no	—
54	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	106	2	no	—
57	C₄⋊C₄	—	212	2	no	—
58	C₄²	—	53	1	no	—
59	C₁₆	—	53	2	no	—
60	C₁₆	Dic₁₀₆	1	2	yes	complemented
61	Q₈	—	424	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	53	2	no	—
64	C₂×C₄	—	212	2	no	—
65	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
66	C₈	—	53	2	no	—
67	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
68	C₈	Q₈×D₅₃	1	1	yes	characteristic
69	C₂²	D₈×D₅₃	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	53	4	no	—
72	C₄	—	424	2	no	—
73	C₄	#13133216	1	1	yes	characteristic
74	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
75	C₂	#16811027	1	1	yes	characteristic
76	C₂	SD₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#367	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#367	D₅₃
hypocenter	C₅₃	#2254
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

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
44. C₅₃	#2254	semidirect
60. C₁₆	Dic₁₀₆	semidirect