Subgroups — #20272778

Order 6784.

76 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	C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₄×#88322	C₄	1	1	yes	characteristic
6	C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	—	2	1	no	—
8	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
10	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
11	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×#88322	—	2	1	no	—
13	C₂×#88322	C₈	1	1	yes	characteristic
14	C₂×#88322	C₈	1	1	yes	characteristic
15	C₁₆×C₅₃	—	4	1	no	—
16	#11719908	—	4	1	no	—
17	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
20	C₈×C₅₃	—	2	1	no	—
21	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
22	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
23	#88322	—	2	2	no	—
24	#88322	—	2	1	no	—
25	#88322	C₁₆	1	2	yes	complemented
26	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₅₃	—	2	1	no	—
28	C₄×C₅₃	#106	1	1	yes	characteristic
29	C₄×C₅₃	#108	1	1	yes	characteristic
30	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₄×C₅₃	M₃₂	1	1	yes	characteristic
32	#1367	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	#334	1	1	yes	characteristic
34	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	C₂×C₅₃	#364	1	1	yes	characteristic
36	C₄⋊C₁₆	—	53	1	no	—
37	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
38	C₈²	—	53	1	no	—
39	C₅₃	#1367	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	106	1	no	—
42	C₄×C₈	—	53	1	no	—
43	C₄×C₈	—	53	1	no	—
44	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
45	C₂×C₈	—	53	1	no	—
46	C₂×C₈	—	53	1	no	—
47	C₂×C₈	—	106	1	no	—
48	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
49	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
50	C₄²	C₄×D₅₃	1	1	yes	characteristic
51	C₁₆	—	4	1	no	—
52	C₁₆	—	212	1	no	—
53	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
54	C₂×C₄	#11719912	1	1	yes	characteristic
55	C₂×C₄	#11719919	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	53	2	no	—
58	C₈	—	106	2	no	—
59	C₈	—	106	1	no	—
60	C₈	#11719923	1	1	yes	characteristic
61	C₈	#11719924	1	1	yes	characteristic
62	C₂²	#13133146	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	#13133148	1	1	yes	characteristic
65	C₄	#13133152	1	1	yes	characteristic
66	C₄	C₁₆×D₅₃	1	1	yes	characteristic
67	C₄	#13133156	1	1	yes	characteristic
68	C₂	#16810373	1	1	yes	characteristic
69	C₂	#16810441	1	1	yes	characteristic
70	C₂	#16810443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#1367
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
25. #88322	C₁₆	semidirect
39. C₅₃	#1367	semidirect