Subgroups — #20276965

Order 6784.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈₄₈	C₂	1	4	yes	maximal, complemented
2	#16811000	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	—	2	2	no	—
5	C₂×D₄₂₄	C₂²	1	4	yes	—
6	#13133214	—	2	2	no	—
7	D₈₄₈	—	2	8	no	—
8	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₂₁₂	—	2	4	no	—
11	C₂×D₂₁₂	—	4	4	no	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
14	C₄²×C₅₃	D₄	1	1	yes	characteristic
15	D₄₂₄	—	2	8	no	—
16	D₄₂₄	—	4	4	no	—
17	C₁₆×C₅₃	D₄	1	4	yes	complemented
18	C₂²×D₅₃	—	4	4	no	—
19	C₂×C₄×C₅₃	D₈	1	2	yes	—
20	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
21	D₂₁₂	—	4	4	no	—
22	D₂₁₂	—	4	4	no	—
23	D₂₁₂	—	8	8	no	—
24	C₈×C₅₃	D₈	1	2	yes	—
25	C₈×C₅₃	C₂×D₄	1	2	yes	—
26	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
27	C₂×D₅₃	—	8	4	no	—
28	C₂×D₅₃	—	8	8	no	—
29	C₄×C₅₃	D₁₆	1	4	yes	complemented
30	C₄×C₅₃	#127	1	1	yes	characteristic
31	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
32	#2246	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	#493	1	1	yes	characteristic
34	C₂×C₅₃	C₂×D₁₆	1	2	yes	—
35	D₅₃	—	16	4	no	—
36	C₂×D₁₆	—	53	4	no	—
37	#493	—	53	2	no	—
38	C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
39	C₅₃	#2246	1	1	yes	characteristic, complemented, Sylow
40	C₂×D₈	—	53	4	no	—
41	C₂×D₈	—	106	2	no	—
42	#127	—	106	2	no	—
43	D₁₆	—	106	8	no	—
44	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
45	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
46	C₂×D₄	—	106	4	no	—
47	C₂×D₄	—	212	4	no	—
48	D₈	—	106	8	no	—
49	D₈	—	212	4	no	—
50	C₂×C₈	D₂₁₂	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
52	C₄²	D₂₁₂	1	1	yes	characteristic
53	C₁₆	D₂₁₂	1	4	yes	complemented
54	C₂³	—	212	4	no	—
55	D₄	—	424	8	no	—
56	D₄	—	212	4	no	—
57	D₄	—	212	4	no	—
58	C₂×C₄	D₄₂₄	1	2	yes	—
59	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
60	C₈	D₄₂₄	1	2	yes	—
61	C₈	C₂×D₂₁₂	1	2	yes	—
62	C₂²	—	424	8	no	—
63	C₂²	—	424	4	no	—
64	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
65	C₄	D₈₄₈	1	4	yes	complemented
66	C₄	#13133214	1	1	yes	characteristic
67	C₄	C₂×D₄₂₄	1	1	yes	characteristic
68	C₂	—	848	4	no	—
69	C₂	#16811000	1	1	yes	characteristic
70	C₂	C₂×D₈₄₈	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	#2246
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
1. C₂×D₈₄₈	C₂	semidirect
2. #16811000	C₂	semidirect
3. C₄×C₁₆×C₅₃	C₂	semidirect
17. C₁₆×C₅₃	D₄	semidirect
29. C₄×C₅₃	D₁₆	semidirect
38. C₄×C₁₆	D₅₃	semidirect
39. C₅₃	#2246	semidirect
53. C₁₆	D₂₁₂	semidirect
65. C₄	D₈₄₈	semidirect