Subgroups — #20274872

Order 6784.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133149	C₂	1	4	yes	maximal
2	#16810493	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	—	2	4	no	—
5	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
6	#13133149	—	2	8	no	—
7	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
10	C₂²×C₄×C₅₃	D₄	1	2	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×Dic₅₃	—	2	4	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₈×C₅₃	D₄	1	4	yes	—
15	C₂×C₈×C₅₃	Q₈	1	4	yes	—
16	C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₄²×C₅₃	Q₈	1	2	yes	—
18	C₄⋊Dic₅₃	—	2	8	no	—
19	C₄⋊Dic₅₃	—	4	8	no	—
20	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₅₃	SD₁₆	1	8	yes	—
23	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	4	no	—
28	C₂×Dic₅₃	—	4	8	no	—
29	C₈×C₅₃	C₄⋊C₄	1	8	yes	complemented
30	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₅₃	#127	1	1	yes	characteristic
32	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₂²×C₅₃	C₂×SD₁₆	1	4	yes	—
34	C₄×C₅₃	#106	1	8	yes	complemented
35	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
37	Dic₅₃	—	8	4	no	—
38	#1844	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#384	1	1	yes	characteristic
40	C₂×C₅₃	C₂×#106	1	2	yes	—
41	C₂×C₅₃	#492	1	2	yes	—
42	C₂×C₅₃	#498	1	2	yes	—
43	C₂×#106	—	53	4	no	—
44	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
45	#384	—	53	2	no	—
46	C₅₃	#1844	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₈	C₂×D₅₃	1	2	yes	—
48	C₂×C₄⋊C₄	—	53	4	no	—
49	C₂×C₄⋊C₄	—	106	4	no	—
50	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
51	#106	—	106	8	no	—
52	C₄×C₈	Dic₅₃	1	4	yes	complemented
53	C₂²×C₄	—	106	4	no	—
54	C₂²×C₄	D₂₁₂	1	2	yes	—
55	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
56	C₂×C₈	Dic₁₀₆	1	4	yes	—
57	C₂×C₈	D₂₁₂	1	4	yes	—
58	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
59	C₄⋊C₄	—	106	8	no	—
60	C₄⋊C₄	—	212	8	no	—
61	C₄²	Dic₁₀₆	1	2	yes	—
62	C₄²	C₂×Dic₅₃	1	2	yes	—
63	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
64	C₂×C₄	—	212	4	no	—
65	C₂×C₄	—	212	8	no	—
66	C₂×C₄	#11719913	1	8	yes	—
67	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
68	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
69	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
70	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
71	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
72	C₈	C₄⋊Dic₅₃	1	8	yes	complemented
73	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
74	C₂²	#13133214	1	1	yes	characteristic
75	C₂²	#13133217	1	1	yes	characteristic
76	C₂²	C₂×#11719913	1	4	yes	—
77	C₄	—	424	4	no	—
78	C₄	#13133149	1	8	yes	complemented
79	C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
80	C₄	#13133217	1	2	yes	—
81	C₂	#16810493	1	1	yes	characteristic
82	C₂	C₂×#13133149	1	2	yes	—
83	C₂	#16810996	1	2	yes	—
84	C₂	#16811021	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1844
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₅₃	C₄	semidirect
29. C₈×C₅₃	C₄⋊C₄	semidirect
34. C₄×C₅₃	#106	semidirect
46. C₅₃	#1844	semidirect
52. C₄×C₈	Dic₅₃	semidirect
72. C₈	C₄⋊Dic₅₃	semidirect
78. C₄	#13133149	semidirect