Subgroups — #20289871

Order 6784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811556	C₂	1	1	yes	maximal, characteristic
2	#16811552	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133288	—	2	1	no	—
5	#13133267	—	2	1	no	—
6	Q₁₆×D₅₃	—	2	1	no	—
7	Dic₄₂₄	C₄	1	1	yes	characteristic
8	#13133163	—	2	1	no	—
9	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	Q₃₂×C₅₃	C₄	1	1	yes	characteristic
11	Q₈×D₅₃	—	4	1	no	—
12	#11719936	—	4	1	no	—
13	Q₁₆×C₅₃	—	2	1	no	—
14	#11719925	—	4	1	no	—
15	Dic₂₁₂	—	2	1	no	—
16	C₈×D₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#11719908	D₄	1	1	yes	characteristic
19	Q₈×C₅₃	—	4	1	no	—
20	C₄×D₅₃	—	8	1	no	—
21	C₄×D₅₃	D₈	1	1	yes	characteristic
22	Dic₁₀₆	—	4	1	no	—
23	Dic₁₀₆	—	8	1	no	—
24	#88324	—	8	1	no	—
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#88322	SD₁₆	1	1	yes	characteristic
27	C₂×D₅₃	SD₃₂	1	1	yes	characteristic
28	C₄×C₅₃	—	8	1	no	—
29	C₄×C₅₃	#102	1	1	yes	characteristic
30	Dic₅₃	—	8	1	no	—
31	Dic₅₃	D₁₆	1	1	yes	characteristic
32	#1420	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	#357	1	1	yes	characteristic
34	D₅₃	—	2	1	no	—
35	C₂×Q₃₂	—	53	1	no	—
36	M₆₄	—	53	1	no	—
37	#368	—	53	1	no	—
38	C₅₃	#1420	1	1	yes	characteristic, complemented, Sylow
39	C₂×Q₁₆	—	106	1	no	—
40	Q₃₂	—	53	1	no	—
41	Q₃₂	—	106	1	no	—
42	Q₃₂	#6157	1	1	yes	characteristic
43	C₂×C₁₆	—	53	1	no	—
44	#108	—	106	1	no	—
45	C₃₂	—	106	1	no	—
46	C₂×Q₈	—	212	1	no	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	106	1	no	—
49	Q₁₆	—	212	1	no	—
50	M₁₆	—	212	1	no	—
51	C₂×C₈	—	53	1	no	—
52	C₁₆	—	53	1	no	—
53	C₁₆	C₂×#6157	1	1	yes	characteristic
54	Q₈	—	4	1	no	—
55	Q₈	—	424	1	no	—
56	Q₈	—	212	1	no	—
57	C₂×C₄	—	53	1	no	—
58	C₂×C₄	—	424	1	no	—
59	C₈	—	53	1	no	—
60	C₈	—	424	1	no	—
61	C₈	C₂²⋊#6157	1	1	yes	characteristic
62	C₂²	—	53	1	no	—
63	C₄	—	8	1	no	—
64	C₄	—	53	1	no	—
65	C₄	—	424	1	no	—
66	C₄	#13133279	1	1	yes	characteristic
67	C₂	—	106	1	no	—
68	C₂	#16811535	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811535
commutator	C₁₆×C₅₃	C₂×C₄
Frattini	C₁₆	C₂×#6157
Fitting	Q₃₂×C₅₃	C₄
socle	C₂×C₅₃	#357
hypercenter	Q₃₂	#6157
hypocenter	C₅₃	#1420
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
38. C₅₃	#1420	semidirect