Subgroups — #20272765

Order 6784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811003	C₂	1	1	yes	maximal, characteristic
2	#16810376	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	—	2	1	no	—
5	C₂×Dic₂₁₂	C₄	1	2	yes	—
6	#13133217	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₂×C₁₆×C₅₃	—	2	1	no	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀₆	—	2	2	no	—
11	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄²×C₅₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	4	1	no	—
15	C₄⋊Dic₅₃	—	4	1	no	—
16	C₂×#88322	—	4	1	no	—
17	Dic₂₁₂	—	4	2	no	—
18	Dic₂₁₂	—	4	1	no	—
19	Dic₂₁₂	—	4	1	no	—
20	C₁₆×C₅₃	—	4	1	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	C₂×Dic₅₃	—	4	2	no	—
25	Dic₁₀₆	—	4	2	no	—
26	Dic₁₀₆	—	4	2	no	—
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	D₈	1	1	yes	characteristic
29	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	#88322	—	8	1	no	—
31	C₂²×C₅₃	#102	1	1	yes	characteristic
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	#100	1	1	yes	characteristic
34	C₄×C₅₃	#102	1	1	yes	characteristic
35	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
36	C₄×C₅₃	Q₃₂	1	1	yes	characteristic
37	Dic₅₃	—	8	2	no	—
38	#1362	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	#331	1	1	yes	characteristic
40	C₂×C₅₃	#358	1	1	yes	characteristic
41	C₂×C₅₃	#362	1	1	yes	characteristic
42	#494	—	53	1	no	—
43	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
44	#335	—	53	1	no	—
45	C₅₃	#1362	1	1	yes	characteristic, complemented, Sylow
46	C₂×Q₁₆	—	53	2	no	—
47	C₂×Q₁₆	—	106	1	no	—
48	C₄⋊Q₈	—	106	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₄⋊C₈	—	106	1	no	—
51	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
52	C₂×Q₈	—	106	2	no	—
53	Q₁₆	—	212	2	no	—
54	Q₁₆	—	212	1	no	—
55	Q₁₆	—	212	1	no	—
56	C₂×C₈	—	212	1	no	—
57	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₄⋊C₄	—	212	1	no	—
60	C₄⋊C₄	—	212	1	no	—
61	C₄²	D₂₁₂	1	1	yes	characteristic
62	C₁₆	—	4	1	no	—
63	Q₈	—	212	2	no	—
64	Q₈	—	212	2	no	—
65	C₂×C₄	—	212	2	no	—
66	C₂×C₄	#11719913	1	1	yes	characteristic
67	C₂×C₄	D₄₂₄	1	1	yes	characteristic
68	C₂×C₄	#11719921	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	424	1	no	—
71	C₈	#11719922	1	1	yes	characteristic
72	C₈	#11719924	1	1	yes	characteristic
73	C₂²	#13133139	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	424	2	no	—
76	C₄	#13133133	1	1	yes	characteristic
77	C₄	#13133137	1	1	yes	characteristic
78	C₄	#13133162	1	1	yes	characteristic
79	C₄	Dic₄₂₄	1	1	yes	characteristic
80	C₂	#16810366	1	1	yes	characteristic
81	C₂	#16810428	1	1	yes	characteristic
82	C₂	#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#1362
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
45. C₅₃	#1362	semidirect