Subgroups — #20272756

Order 6784.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810444	C₂	1	2	yes	maximal, complemented
2	C₅₃×#492	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	2	2	no	—
4	C₂×SD₁₆×C₅₃	—	2	2	no	—
5	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	#127×C₅₃	C₄	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×Q₈×C₅₃	—	2	1	no	—
9	C₂×D₄×C₅₃	—	2	1	no	—
10	C₂×D₄×C₅₃	—	4	1	no	—
11	SD₁₆×C₅₃	—	4	2	no	—
12	SD₁₆×C₅₃	—	4	2	no	—
13	C₂×C₈×C₅₃	D₄	1	2	yes	—
14	C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#11719908	—	4	2	no	—
17	C₂³×C₅₃	—	4	1	no	—
18	Q₈×C₅₃	—	4	2	no	—
19	D₄×C₅₃	—	4	2	no	—
20	D₄×C₅₃	—	8	2	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₈×C₅₃	D₈	1	2	yes	—
25	C₈×C₅₃	SD₁₆	1	2	yes	—
26	C₂²×C₅₃	—	8	2	no	—
27	C₂²×C₅₃	—	8	1	no	—
28	C₂²×C₅₃	#100	1	1	yes	characteristic
29	C₄×C₅₃	—	2	2	no	—
30	C₄×C₅₃	—	8	1	no	—
31	C₄×C₅₃	#102	1	2	yes	—
32	#1359	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	—	16	1	no	—
35	C₂×C₅₃	#331	1	1	yes	characteristic
36	#492	D₅₃	1	1	yes	characteristic
37	#364	—	53	2	no	—
38	C₅₃	#1359	1	1	yes	characteristic, complemented, Sylow
39	C₂×SD₁₆	—	2	2	no	—
40	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
41	#127	Dic₅₃	1	1	yes	characteristic
42	M₃₂	—	106	2	no	—
43	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	2	1	no	—
46	C₂×D₄	—	4	1	no	—
47	SD₁₆	—	4	2	no	—
48	SD₁₆	—	4	2	no	—
49	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
50	C₄⋊C₄	—	4	1	no	—
51	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
52	C₁₆	—	212	2	no	—
53	C₂³	—	4	1	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	4	2	no	—
56	D₄	—	8	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
60	C₈	#11719922	1	2	yes	—
61	C₈	#11719923	1	2	yes	—
62	C₂²	—	8	2	no	—
63	C₂²	—	8	1	no	—
64	C₂²	#13133134	1	1	yes	characteristic
65	C₄	—	2	2	no	—
66	C₄	—	8	1	no	—
67	C₄	#13133138	1	2	yes	—
68	C₂	—	2	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#16810367	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810367
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂×C₅₃	#331
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1359
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #492

Complements

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

N	H ≅ G/N	type
1. #16810444	C₂	semidirect
38. C₅₃	#1359	semidirect