Subgroups — #20289738

Order 6784.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719910	C₂	1	2	yes	maximal
2	C₈×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719910	—	2	2	no	—
4	C₄×#88322	C₄	1	1	yes	characteristic
5	C₄×#88322	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×C₅₃	—	2	1	no	—
8	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×#88322	C₈	1	1	yes	characteristic
10	C₂×#88322	C₈	1	1	yes	characteristic
11	C₂×#88322	D₄	1	1	yes	characteristic
12	C₂×#88322	Q₈	1	1	yes	characteristic
13	#11719910	—	4	2	no	—
14	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
17	C₈×C₅₃	—	2	2	no	—
18	#88322	—	2	1	no	—
19	#88322	—	2	1	no	—
20	#88322	C₁₆	1	2	yes	complemented
21	#88322	SD₁₆	1	2	yes	—
22	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₅₃	—	2	1	no	—
24	C₄×C₅₃	#106	1	1	yes	characteristic
25	C₄×C₅₃	#108	1	1	yes	characteristic
26	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	M₃₂	1	1	yes	characteristic
28	#1367	—	53	1	no	maximal, Sylow
29	C₂×C₅₃	#334	1	1	yes	characteristic
30	C₂×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₅₃	#364	1	1	yes	characteristic
32	C₄⋊C₁₆	—	53	2	no	—
33	C₈²	—	53	1	no	—
34	C₅₃	#1367	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₁₆	—	106	2	no	—
36	C₄×C₈	—	53	1	no	—
37	C₄×C₈	—	53	1	no	—
38	C₄×C₈	#6157	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	53	1	no	—
41	C₂×C₈	—	53	1	no	—
42	C₂×C₈	—	53	1	no	—
43	C₂×C₈	—	53	1	no	—
44	C₄²	C₂×#6157	1	1	yes	characteristic
45	C₁₆	—	212	2	no	—
46	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
47	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
48	C₂×C₄	#11719940	1	1	yes	characteristic
49	C₈	—	2	2	no	—
50	C₈	—	53	2	no	—
51	C₈	—	53	2	no	—
52	C₈	—	106	1	no	—
53	C₈	—	106	1	no	—
54	C₂²	#13133284	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	#13133285	1	1	yes	characteristic
57	C₄	#13133288	1	1	yes	characteristic
58	C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
59	C₄	#13133292	1	1	yes	characteristic
60	C₂	#16811485	1	1	yes	characteristic
61	C₂	#16811544	1	1	yes	characteristic
62	C₂	#16811546	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#6157
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄²	C₂×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1367
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
20. #88322	C₁₆	semidirect
34. C₅₃	#1367	semidirect