Subgroups — #20276995

Order 6784.

128 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	#16810450	C₂	1	4	yes	maximal
3	C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	#13133217	—	2	1	no	—
6	C₂×#11719908	C₂²	1	2	yes	—
7	#13133151	—	2	1	no	—
8	#13133151	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₅₃	—	2	1	no	—
10	#107×C₅₃	—	2	1	no	—
11	#107×C₅₃	C₂²	1	2	yes	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₅₃	—	4	1	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	C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄⋊C₄×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	2	2	no	—
21	C₄⋊Dic₅₃	—	4	1	no	—
22	#11719908	Q₈	1	4	yes	complemented
23	Q₈×C₅₃	—	8	2	no	—
24	C₂×C₄×C₅₃	—	4	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	Dic₁₀₆	—	8	2	no	—
30	C₈×C₅₃	Q₁₆	1	2	yes	—
31	C₈×C₅₃	C₂×Q₈	1	2	yes	—
32	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
33	C₄×C₅₃	—	8	2	no	—
34	C₄×C₅₃	SD₃₂	1	2	yes	—
35	C₄×C₅₃	SD₃₂	1	2	yes	—
36	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
37	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
38	Dic₅₃	—	8	2	no	—
39	#2254	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	#500	1	1	yes	characteristic
41	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
42	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
43	#500	—	53	1	no	—
44	#500	D₅₃	1	1	yes	characteristic
45	#367	—	53	4	no	—
46	C₄×C₁₆	—	53	1	no	—
47	C₅₃	#2254	1	1	yes	characteristic, complemented, Sylow
48	C₄⋊Q₈	—	2	1	no	—
49	C₄⋊Q₈	—	106	1	no	—
50	C₂×C₁₆	—	53	2	no	—
51	#107	—	2	1	no	—
52	#107	—	53	2	no	—
53	#107	—	106	1	no	—
54	#107	C₂×D₅₃	1	2	yes	—
55	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	212	1	no	—
58	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	106	2	no	—
63	C₄⋊C₄	—	212	1	no	—
64	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
65	C₁₆	—	53	4	no	—
66	Q₈	—	8	2	no	—
67	Q₈	—	424	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	212	2	no	—
70	C₂×C₄	#11719922	1	1	yes	characteristic
71	C₂×C₄	#11719922	1	1	yes	characteristic
72	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
73	C₈	#11719925	1	2	yes	—
74	C₈	Q₈×D₅₃	1	2	yes	—
75	C₂²	C₂×#11719922	1	1	yes	characteristic
76	C₄	—	8	2	no	—
77	C₄	—	424	2	no	—
78	C₄	#13133160	1	2	yes	—
79	C₄	#13133161	1	2	yes	—
80	C₄	#13133215	1	1	yes	characteristic
81	C₄	C₂×#11719925	1	1	yes	characteristic
82	C₂	#16811026	1	1	yes	characteristic
83	C₂	C₂×#13133160	1	1	yes	characteristic
84	C₂	C₂×#13133161	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#500	D₅₃
hypocenter	C₅₃	#2254
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
22. #11719908	Q₈	semidirect
47. C₅₃	#2254	semidirect