Subgroups — #20272770

Order 6784.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	#16810376	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₄⋊Q₈×C₅₃	—	2	1	no	—
7	#107×C₅₃	—	2	1	no	—
8	#107×C₅₃	C₄	1	2	yes	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Q₈×C₅₃	—	4	1	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×#88322	—	4	1	no	—
17	#11719908	—	4	1	no	—
18	Q₈×C₅₃	—	8	1	no	—
19	Q₈×C₅₃	—	8	1	no	—
20	C₂×C₄×C₅₃	—	4	2	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₈×C₅₃	—	2	1	no	—
25	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
26	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
27	#88322	—	8	1	no	—
28	C₂²×C₅₃	#102	1	1	yes	characteristic
29	C₄×C₅₃	—	2	1	no	—
30	C₄×C₅₃	—	8	2	no	—
31	C₄×C₅₃	#101	1	1	yes	characteristic
32	C₄×C₅₃	#103	1	1	yes	characteristic
33	C₄×C₅₃	D₁₆	1	1	yes	characteristic
34	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
35	#1364	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	#332	1	1	yes	characteristic
37	C₂×C₅₃	#357	1	1	yes	characteristic
38	C₂×C₅₃	#362	1	1	yes	characteristic
39	#500	D₅₃	1	1	yes	characteristic
40	C₄⋊C₁₆	—	53	1	no	—
41	#335	—	53	1	no	—
42	C₅₃	#1364	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊Q₈	—	2	1	no	—
44	C₂×C₁₆	—	106	1	no	—
45	#107	—	2	1	no	—
46	#107	Dic₅₃	1	2	yes	—
47	C₄⋊C₈	—	106	1	no	—
48	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
49	C₂×Q₈	—	4	1	no	—
50	C₂×C₈	—	212	1	no	—
51	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	1	no	—
55	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₁₆	—	212	1	no	—
57	Q₈	—	8	1	no	—
58	Q₈	—	8	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	#11719922	1	1	yes	characteristic
61	C₂×C₄	#11719923	1	1	yes	characteristic
62	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	424	1	no	—
65	C₈	#11719924	1	1	yes	characteristic
66	C₈	#11719925	1	1	yes	characteristic
67	C₂²	#13133138	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	8	2	no	—
70	C₄	#13133136	1	1	yes	characteristic
71	C₄	#13133141	1	1	yes	characteristic
72	C₄	#13133158	1	1	yes	characteristic
73	C₄	#13133161	1	1	yes	characteristic
74	C₂	#16810369	1	1	yes	characteristic
75	C₂	#16810424	1	1	yes	characteristic
76	C₂	#16810439	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133138
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	#102
hypercenter	#500	D₅₃
hypocenter	C₅₃	#1364
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×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
42. C₅₃	#1364	semidirect