Subgroups — #20272880

Order 6784.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810416	C₂	1	2	yes	maximal
2	C₅₃×#492	C₂	1	1	yes	maximal, characteristic
3	#13133132	—	2	2	no	—
4	C₂×SD₁₆×C₅₃	—	2	2	no	—
5	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
6	#127×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈×C₅₃	—	2	1	no	—
9	C₂×D₄×C₅₃	—	4	1	no	—
10	C₂×D₄×C₅₃	D₄	1	2	yes	—
11	SD₁₆×C₅₃	—	4	4	no	—
12	C₂×C₈×C₅₃	—	2	1	no	—
13	C₄⋊C₄×C₅₃	—	4	1	no	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²⋊Dic₅₃	—	4	2	no	—
16	C₂³×C₅₃	—	2	2	no	—
17	Q₈×C₅₃	—	4	2	no	—
18	D₄×C₅₃	—	4	2	no	—
19	D₄×C₅₃	—	8	2	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	—	8	2	no	—
24	C₈×C₅₃	—	2	2	no	—
25	C₂²×C₅₃	—	4	2	no	—
26	C₂²×C₅₃	—	8	2	no	—
27	C₂²×C₅₃	#99	1	1	yes	characteristic
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	—	2	1	no	—
30	C₄×C₅₃	—	8	1	no	—
31	Dic₅₃	—	16	2	no	—
32	#1409	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	—	8	2	no	—
35	C₂×C₅₃	#353	1	1	yes	characteristic
36	#492	D₅₃	1	1	yes	characteristic
37	#353	—	53	2	no	—
38	C₅₃	#1409	1	1	yes	characteristic, complemented, Sylow
39	C₂×SD₁₆	—	2	2	no	—
40	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
41	#127	C₂×D₅₃	1	1	yes	characteristic
42	#99	—	106	2	no	—
43	C₄×C₈	Dic₅₃	1	1	yes	characteristic, complemented
44	C₂×Q₈	—	2	1	no	—
45	C₂×D₄	—	4	1	no	—
46	C₂×D₄	C₂²⋊D₅₃	1	2	yes	—
47	SD₁₆	—	4	4	no	—
48	C₂×C₈	—	2	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₂²⋊C₄	—	212	2	no	—
51	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
52	C₂³	—	2	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	424	2	no	—
59	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	8	2	no	—
63	C₂²	#13133132	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	848	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	8	2	no	—
70	C₂	#16810416	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810416
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#127	C₂×D₅₃
Fitting	C₅₃×#492	C₂
socle	C₂×C₅₃	#353
hypercenter	#492	D₅₃
hypocenter	C₅₃	#1409
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #492

Complements

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

N	H ≅ G/N	type
5. C₄⋊Q₈×C₅₃	C₄	semidirect
7. C₄×C₈×C₅₃	C₄	semidirect
38. C₅₃	#1409	semidirect
40. C₄⋊Q₈	Dic₅₃	semidirect
43. C₄×C₈	Dic₅₃	semidirect