Subgroups — #20276943

Order 6784.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	1	yes	maximal, characteristic
2	#16811003	C₂	1	1	yes	maximal, characteristic
3	#16810428	C₂	1	4	yes	maximal
4	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₂	—	2	1	no	—
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	#13133217	—	2	1	no	—
8	#13133217	—	2	1	no	—
9	#13133151	—	2	1	no	—
10	#13133151	C₂²	1	2	yes	—
11	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₀₆	—	2	2	no	—
14	C₂×Dic₁₀₆	—	4	1	no	—
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₄⋊Dic₅₃	—	2	2	no	—
19	C₄⋊Dic₅₃	—	4	1	no	—
20	C₄⋊Dic₅₃	—	4	1	no	—
21	C₄⋊Dic₅₃	—	4	1	no	—
22	Dic₂₁₂	—	2	4	no	—
23	Dic₂₁₂	—	4	2	no	—
24	C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	2	no	—
29	C₂×Dic₅₃	—	4	2	no	—
30	Dic₁₀₆	—	4	2	no	—
31	Dic₁₀₆	—	4	2	no	—
32	Dic₁₀₆	—	8	2	no	—
33	C₈×C₅₃	SD₁₆	1	2	yes	—
34	C₈×C₅₃	#41	1	2	yes	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₄×C₅₃	SD₃₂	1	2	yes	—
37	C₄×C₅₃	Q₃₂	1	2	yes	—
38	C₄×C₅₃	#124	1	1	yes	characteristic
39	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
40	Dic₅₃	—	8	2	no	—
41	Dic₅₃	—	8	2	no	—
42	#2241	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	#486	1	1	yes	characteristic
44	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
45	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
46	#500	—	53	1	no	—
47	#494	—	53	1	no	—
48	#358	—	53	4	no	—
49	C₄×C₁₆	D₅₃	1	1	yes	characteristic
50	C₅₃	#2241	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	53	2	no	—
52	C₂×Q₁₆	—	106	1	no	—
53	C₄⋊Q₈	—	106	1	no	—
54	C₄⋊Q₈	—	106	1	no	—
55	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
56	#107	—	53	2	no	—
57	#107	—	106	1	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×Q₈	—	106	2	no	—
60	C₂×Q₈	—	212	1	no	—
61	Q₁₆	—	106	4	no	—
62	Q₁₆	—	212	2	no	—
63	C₂×C₈	D₂₁₂	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
65	C₄⋊C₄	—	106	2	no	—
66	C₄⋊C₄	—	212	1	no	—
67	C₄⋊C₄	—	212	1	no	—
68	C₄⋊C₄	—	212	1	no	—
69	C₄²	D₂₁₂	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	Q₈	—	424	2	no	—
72	Q₈	—	212	2	no	—
73	Q₈	—	212	2	no	—
74	C₂×C₄	—	212	2	no	—
75	C₂×C₄	—	212	2	no	—
76	C₂×C₄	D₄₂₄	1	1	yes	characteristic
77	C₂×C₄	D₄₂₄	1	1	yes	characteristic
78	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
79	C₈	#11719913	1	2	yes	—
80	C₈	#11719945	1	2	yes	—
81	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
82	C₄	—	424	2	no	—
83	C₄	—	424	2	no	—
84	C₄	#13133162	1	2	yes	—
85	C₄	Dic₄₂₄	1	2	yes	—
86	C₄	#13133205	1	1	yes	characteristic
87	C₄	C₂×#11719913	1	1	yes	characteristic
88	C₂	#16810970	1	1	yes	characteristic
89	C₂	C₂×#13133162	1	1	yes	characteristic
90	C₂	C₂×Dic₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2241
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
50. C₅₃	#2241	semidirect