Subgroups — #20289734

Order 6784.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811544	C₂	1	1	yes	maximal, characteristic
2	#16811487	C₂	1	1	yes	maximal, characteristic
3	#16811027	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	#13133216	—	2	1	no	—
7	#13133151	C₄	1	1	yes	characteristic
8	#13133150	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#107×C₅₃	C₄	1	1	yes	characteristic
11	C₂×Dic₁₀₆	—	4	1	no	—
12	C₂×#88324	—	4	1	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₄⋊Dic₅₃	—	2	1	no	—
16	#11719919	—	4	1	no	—
17	C₄×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×#88322	D₄	1	1	yes	characteristic
19	#11719910	—	4	1	no	—
20	C₂×C₄×C₅₃	—	4	1	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×Dic₅₃	—	4	1	no	—
23	C₂×Dic₅₃	D₈	1	1	yes	characteristic
24	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
25	Dic₁₀₆	—	8	2	no	—
26	#88324	—	8	1	no	—
27	C₈×C₅₃	—	2	1	no	—
28	#88322	SD₁₆	1	1	yes	characteristic
29	#88322	Q₁₆	1	1	yes	characteristic
30	C₂²×C₅₃	#102	1	1	yes	characteristic
31	C₄×C₅₃	—	8	1	no	—
32	C₄×C₅₃	#101	1	1	yes	characteristic
33	C₄×C₅₃	#103	1	1	yes	characteristic
34	Dic₅₃	—	2	1	no	—
35	Dic₅₃	—	8	1	no	—
36	Dic₅₃	SD₃₂	1	1	yes	characteristic
37	Dic₅₃	Q₃₂	1	1	yes	characteristic
38	#1365	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#332	1	1	yes	characteristic
40	C₂×C₅₃	#358	1	1	yes	characteristic
41	C₂×C₅₃	#361	1	1	yes	characteristic
42	#500	—	53	1	no	—
43	C₄⋊C₁₆	—	53	1	no	—
44	#335	—	53	1	no	—
45	C₅₃	#1365	1	1	yes	characteristic, complemented, Sylow
46	C₄⋊Q₈	—	106	1	no	—
47	C₂×C₁₆	—	106	1	no	—
48	#107	—	53	1	no	—
49	#107	—	106	1	no	—
50	#107	#6157	1	1	yes	characteristic
51	C₄⋊C₈	—	106	1	no	—
52	C₄×C₈	—	53	1	no	—
53	C₂×Q₈	—	212	1	no	—
54	C₂×C₈	—	53	1	no	—
55	C₂×C₈	—	212	1	no	—
56	C₂×C₈	C₂×#6157	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	106	1	no	—
59	C₄⋊C₄	—	212	1	no	—
60	C₄²	—	53	1	no	—
61	C₁₆	—	212	1	no	—
62	Q₈	—	424	2	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	—	53	1	no	—
65	C₂×C₄	—	53	1	no	—
66	C₂×C₄	—	212	1	no	—
67	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	53	1	no	—
70	C₈	—	53	1	no	—
71	C₈	—	424	1	no	—
72	C₂²	#13133279	1	1	yes	characteristic
73	C₄	—	8	1	no	—
74	C₄	—	53	1	no	—
75	C₄	—	53	1	no	—
76	C₄	—	106	1	no	—
77	C₄	—	424	1	no	—
78	C₄	#13133278	1	1	yes	characteristic
79	C₄	#13133280	1	1	yes	characteristic
80	C₂	#16811482	1	1	yes	characteristic
81	C₂	#16811536	1	1	yes	characteristic
82	C₂	#16811541	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133279
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×#6157
Fitting	#107×C₅₃	C₄
socle	C₂²×C₅₃	#102
hypercenter	#107	#6157
hypocenter	C₅₃	#1365
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
45. C₅₃	#1365	semidirect