Subgroups — #20283113

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133217	C₂	1	6	yes	maximal
2	#16810493	C₂	1	8	yes	maximal
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
5	#13133217	—	2	24	no	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	24	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
9	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
11	C₂²×Dic₅₃	C₂³	1	8	yes	—
12	C₂×Dic₁₀₆	—	2	24	no	—
13	C₄²×C₅₃	D₄	1	12	yes	—
14	C₄²×C₅₃	Q₈	1	16	yes	complemented
15	C₄⋊Dic₅₃	—	2	48	no	—
16	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
17	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
18	C₂×C₄×C₅₃	C₂×D₄	1	12	yes	—
19	C₂×C₄×C₅₃	C₂×Q₈	1	24	yes	—
20	C₂×Dic₅₃	—	2	24	no	—
21	Dic₁₀₆	—	4	16	no	—
22	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
23	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
24	C₄×C₅₃	#127	1	4	yes	—
25	C₄×C₅₃	C₄⋊Q₈	1	24	yes	complemented
26	Dic₅₃	—	4	8	no	—
27	#2868	—	53	1	no	maximal, Sylow
28	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
29	C₂×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
30	C₂×C₄⋊Q₈	—	53	6	no	—
31	#384	—	53	8	no	—
32	C₄³	D₅₃	1	1	yes	characteristic
33	C₅₃	#2868	1	1	yes	characteristic, complemented, Sylow
34	C₂²×Q₈	—	53	4	no	—
35	C₄⋊Q₈	—	106	24	no	—
36	C₂×C₄⋊C₄	—	53	24	no	—
37	C₂×C₄²	C₂×D₅₃	1	4	yes	—
38	C₂×C₄²	C₂×D₅₃	1	3	yes	—
39	C₂×Q₈	—	106	24	no	—
40	C₂²×C₄	—	53	8	no	—
41	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
42	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
43	C₄⋊C₄	—	106	48	no	—
44	C₄²	Dic₁₀₆	1	16	yes	complemented
45	C₄²	D₂₁₂	1	12	yes	—
46	C₂³	C₂³×D₅₃	1	1	yes	characteristic
47	Q₈	—	212	16	no	—
48	C₂×C₄	—	106	24	no	—
49	C₂×C₄	C₂×Dic₁₀₆	1	24	yes	—
50	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
51	C₂×C₄	C₂×D₂₁₂	1	12	yes	—
52	C₂²	C₂²×D₂₁₂	1	3	yes	—
53	C₂²	C₂²×Dic₁₀₆	1	4	yes	—
54	C₄	—	212	8	no	—
55	C₄	#13133214	1	4	yes	—
56	C₄	#13133217	1	24	yes	complemented
57	C₂	C₂×#13133214	1	1	yes	characteristic
58	C₂	C₂×#13133217	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄³×C₅₃	C₂
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#2868
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
14. C₄²×C₅₃	Q₈	semidirect
25. C₄×C₅₃	C₄⋊Q₈	semidirect
33. C₅₃	#2868	semidirect
44. C₄²	Dic₁₀₆	semidirect
56. C₄	#13133217	semidirect