Subgroups — #20276968

Order 6784.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	4	yes	maximal
2	#16811003	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	—	2	2	no	—
5	C₂×Dic₂₁₂	C₂²	1	4	yes	—
6	#13133217	—	2	2	no	—
7	Dic₄₂₄	—	2	8	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₀₆	—	2	4	no	—
11	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
13	C₄²×C₅₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₅₃	—	4	4	no	—
15	Dic₂₁₂	—	2	8	no	—
16	Dic₂₁₂	—	4	4	no	—
17	C₁₆×C₅₃	D₄	1	4	yes	—
18	C₂×C₄×C₅₃	D₈	1	2	yes	—
19	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₅₃	—	4	4	no	—
21	Dic₁₀₆	—	4	4	no	—
22	Dic₁₀₆	—	4	4	no	—
23	C₈×C₅₃	D₈	1	2	yes	—
24	C₈×C₅₃	C₂×D₄	1	2	yes	—
25	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
26	C₄×C₅₃	Q₃₂	1	4	yes	complemented
27	C₄×C₅₃	#127	1	1	yes	characteristic
28	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
29	Dic₅₃	—	8	4	no	—
30	#2247	—	53	1	no	maximal, Sylow
31	C₂×C₅₃	#493	1	1	yes	characteristic
32	C₂×C₅₃	C₂×Q₃₂	1	2	yes	—
33	C₂×Q₃₂	—	53	4	no	—
34	#494	—	53	2	no	—
35	C₄×C₁₆	D₅₃	1	1	yes	characteristic
36	C₅₃	#2247	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	53	4	no	—
38	C₂×Q₁₆	—	106	2	no	—
39	C₄⋊Q₈	—	106	2	no	—
40	Q₃₂	—	106	8	no	—
41	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
42	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
43	C₂×Q₈	—	106	4	no	—
44	Q₁₆	—	106	8	no	—
45	Q₁₆	—	212	4	no	—
46	C₂×C₈	D₂₁₂	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
48	C₄⋊C₄	—	212	4	no	—
49	C₄²	D₂₁₂	1	1	yes	characteristic
50	C₁₆	D₂₁₂	1	4	yes	—
51	Q₈	—	212	4	no	—
52	Q₈	—	212	4	no	—
53	C₂×C₄	—	212	4	no	—
54	C₂×C₄	D₄₂₄	1	2	yes	—
55	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
56	C₈	D₄₂₄	1	2	yes	—
57	C₈	C₂×D₂₁₂	1	2	yes	—
58	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
59	C₄	—	424	4	no	—
60	C₄	Dic₄₂₄	1	4	yes	complemented
61	C₄	#13133214	1	1	yes	characteristic
62	C₄	C₂×D₄₂₄	1	1	yes	characteristic
63	C₂	#16811000	1	1	yes	characteristic
64	C₂	C₂×Dic₄₂₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2247
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₅₃	Q₃₂	semidirect
36. C₅₃	#2247	semidirect
60. C₄	Dic₄₂₄	semidirect