Subgroups — #20289867

Order 6784.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811551	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811056	C₂	1	1	yes	maximal, characteristic
4	#13133288	—	2	1	no	—
5	#13133267	—	2	1	no	—
6	#13133240	—	2	1	no	—
7	#13133160	—	2	1	no	—
8	D₈₄₈	C₄	1	1	yes	characteristic
9	C₁₆×D₅₃	C₂²	1	1	yes	characteristic
10	Q₃₂×C₅₃	C₄	1	1	yes	characteristic
11	#11719949	—	4	1	no	—
12	#11719936	—	4	1	no	—
13	Q₁₆×C₅₃	—	2	1	no	—
14	#11719924	—	4	1	no	—
15	D₄₂₄	—	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	D₂₁₂	—	4	1	no	—
21	D₂₁₂	—	8	1	no	—
22	C₄×D₅₃	—	8	1	no	—
23	C₄×D₅₃	D₈	1	1	yes	characteristic
24	#88324	—	8	1	no	—
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#88322	SD₁₆	1	1	yes	characteristic
27	C₂×D₅₃	—	8	1	no	—
28	C₂×D₅₃	SD₃₂	1	1	yes	characteristic
29	C₄×C₅₃	—	8	1	no	—
30	C₄×C₅₃	#102	1	1	yes	characteristic
31	Dic₅₃	D₁₆	1	1	yes	characteristic
32	#1417	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	#357	1	1	yes	characteristic
34	D₅₃	—	2	1	no	—
35	D₅₃	—	16	1	no	—
36	#508	—	53	1	no	—
37	C₂×C₃₂	—	53	1	no	—
38	#368	—	53	1	no	—
39	C₅₃	#1417	1	1	yes	characteristic, complemented, Sylow
40	#135	—	106	1	no	—
41	Q₃₂	#6157	1	1	yes	characteristic
42	SD₃₂	—	106	1	no	—
43	D₁₆	—	53	1	no	—
44	C₂×C₁₆	—	53	1	no	—
45	#108	—	106	1	no	—
46	C₃₂	—	106	1	no	—
47	#41	—	212	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	212	1	no	—
50	D₈	—	106	1	no	—
51	M₁₆	—	212	1	no	—
52	C₂×C₈	—	53	1	no	—
53	C₁₆	—	53	1	no	—
54	C₁₆	C₂×#6157	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	424	1	no	—
57	D₄	—	212	1	no	—
58	C₂×C₄	—	53	1	no	—
59	C₂×C₄	—	424	1	no	—
60	C₈	—	53	1	no	—
61	C₈	—	424	1	no	—
62	C₈	C₂²⋊#6157	1	1	yes	characteristic
63	C₂²	—	53	1	no	—
64	C₂²	—	424	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	53	1	no	—
67	C₄	#13133279	1	1	yes	characteristic
68	C₂	—	848	1	no	—
69	C₂	—	106	1	no	—
70	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	#1417
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
1. D₅₃⋊C₃₂	C₂	semidirect
2. #16811551	C₂	semidirect
39. C₅₃	#1417	semidirect