Subgroups — #20276499

Order 6784.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810661	C₂	1	2	yes	maximal
2	#16810451	C₂	1	2	yes	maximal, complemented
3	#16810449	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133176	—	2	2	no	—
6	#13133151	C₂²	1	4	yes	—
7	#13133149	—	2	2	no	—
8	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	Q₈	1	2	yes	—
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²⋊Dic₅₃	—	4	2	no	—
18	C₄⋊Dic₅₃	—	2	4	no	—
19	C₄×Dic₅₃	—	4	2	no	—
20	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
21	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
22	C₂³×C₅₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	Q₁₆	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	4	no	—
28	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₅₃	—	2	2	no	—
32	C₂²×C₅₃	#107	1	2	yes	—
33	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
34	C₄×C₅₃	#107	1	2	yes	—
35	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₅₃	—	8	4	no	—
38	#2158	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	2	2	no	—
40	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
41	C₂×C₅₃	#508	1	2	yes	—
42	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
43	#427	—	53	2	no	—
44	#367	—	53	2	no	—
45	#366	—	53	2	no	—
46	C₅₃	#2158	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
48	#117	—	106	2	no	—
49	C₂×C₁₆	Dic₅₃	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
52	#107	—	53	4	no	—
53	#106	—	106	2	no	—
54	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
55	C₂×C₈	Dic₁₀₆	1	2	yes	—
56	C₂×C₈	D₂₁₂	1	1	yes	characteristic
57	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
58	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
59	C₄⋊C₄	—	106	4	no	—
60	C₂²⋊C₄	—	212	2	no	—
61	C₄²	—	212	2	no	—
62	C₁₆	C₂×Dic₅₃	1	2	yes	—
63	C₁₆	C₂×Dic₅₃	1	2	yes	—
64	C₂³	D₄₂₄	1	1	yes	characteristic
65	C₂×C₄	—	212	4	no	—
66	C₂×C₄	D₄₂₄	1	1	yes	characteristic
67	C₂×C₄	Dic₂₁₂	1	2	yes	—
68	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
69	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
70	C₈	C₄⋊Dic₅₃	1	2	yes	—
71	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
72	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#13133151	1	2	yes	—
75	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
76	C₄	—	424	4	no	—
77	C₄	#13133151	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
79	C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#13133151	1	1	yes	characteristic
82	C₂	#16811055	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #16810451	C₂	semidirect
3. #16810449	C₂	semidirect
9. C₂×C₁₆×C₅₃	C₄	semidirect
46. C₅₃	#2158	semidirect
49. C₂×C₁₆	Dic₅₃	semidirect