Subgroups — #20272884

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810422	C₂	1	2	yes	maximal
2	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
3	#13133136	—	2	2	no	—
4	C₂×Q₁₆×C₅₃	—	2	2	no	—
5	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
6	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×Q₈×C₅₃	—	2	1	no	—
9	C₂×Q₈×C₅₃	D₄	1	2	yes	—
10	Q₁₆×C₅₃	—	4	4	no	—
11	C₂×C₈×C₅₃	—	2	1	no	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₄⋊C₄×C₅₃	—	4	1	no	—
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	#11719917	—	4	2	no	—
16	Q₈×C₅₃	—	4	2	no	—
17	Q₈×C₅₃	—	4	2	no	—
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	—	2	2	no	—
20	C₂×C₄×C₅₃	—	4	1	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	—	2	2	no	—
23	#88322	—	8	2	no	—
24	C₂²×C₅₃	#99	1	1	yes	characteristic
25	C₄×C₅₃	—	2	2	no	—
26	C₄×C₅₃	—	2	1	no	—
27	C₄×C₅₃	—	4	2	no	—
28	C₄×C₅₃	—	8	1	no	—
29	#1411	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#353	1	1	yes	characteristic
32	#494	D₅₃	1	1	yes	characteristic
33	#356	—	53	2	no	—
34	C₅₃	#1411	1	1	yes	characteristic, complemented, Sylow
35	C₂×Q₁₆	—	2	2	no	—
36	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
37	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
38	#101	—	106	2	no	—
39	C₄×C₈	Dic₅₃	1	1	yes	characteristic
40	C₂×Q₈	—	2	1	no	—
41	C₂×Q₈	C₂²⋊D₅₃	1	2	yes	—
42	Q₁₆	—	4	4	no	—
43	M₁₆	—	212	2	no	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	4	1	no	—
53	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	424	2	no	—
56	C₂²	#13133132	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	1	no	—
59	C₄	—	4	2	no	—
60	C₄	—	8	1	no	—
61	C₂	—	2	1	no	—
62	C₂	#16810416	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810416
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂×C₅₃	#353
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1411
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #494

Complements

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

N	H ≅ G/N	type
34. C₅₃	#1411	semidirect