Subgroups — #20272852

Order 6784.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
5	C₂×#11719908	C₄	1	1	yes	characteristic
6	C₁₆×D₅₃	—	2	1	no	—
7	C₁₆×D₅₃	—	2	1	no	—
8	#13133123	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₅₃	—	2	1	no	—
11	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×#88322	C₈	1	1	yes	characteristic
14	C₈×D₅₃	—	2	1	no	—
15	C₈×D₅₃	—	2	1	no	—
16	C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₁₆×C₅₃	D₄	1	1	yes	characteristic
18	#11719908	—	2	1	no	—
19	C₂²×D₅₃	C₁₆	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
21	C₂×Dic₅₃	C₁₆	1	1	yes	characteristic
22	C₄×D₅₃	—	2	1	no	—
23	C₄×D₅₃	—	2	1	no	—
24	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₅₃	M₁₆	1	1	yes	characteristic
26	#88322	—	2	1	no	—
27	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₅₃	—	2	1	no	—
29	C₂×D₅₃	—	2	1	no	—
30	C₂×D₅₃	C₃₂	1	2	yes	complemented
31	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₅₃	M₃₂	1	1	yes	characteristic
33	Dic₅₃	—	2	1	no	—
34	C₂²⋊C₃₂	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₂×C₅₃	C₂×C₃₂	1	1	yes	characteristic
37	C₂×C₅₃	M₆₄	1	1	yes	characteristic
38	D₅₃	—	2	2	no	—
39	C₂²×C₁₆	—	53	1	no	—
40	C₂×C₃₂	—	53	1	no	—
41	C₂×C₃₂	D₅₃	1	1	yes	characteristic, complemented
42	C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₈	—	53	1	no	—
44	C₂×C₁₆	—	53	1	no	—
45	C₂×C₁₆	—	106	1	no	—
46	C₂×C₁₆	—	106	1	no	—
47	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
48	C₃₂	—	2	1	no	—
49	C₃₂	—	106	1	no	—
50	C₂²×C₄	—	53	1	no	—
51	C₂×C₈	—	53	1	no	—
52	C₂×C₈	—	106	1	no	—
53	C₂×C₈	—	106	1	no	—
54	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
55	C₁₆	—	106	1	no	—
56	C₁₆	D₂₁₂	1	1	yes	characteristic
57	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
58	C₂³	—	53	1	no	—
59	C₂×C₄	—	53	1	no	—
60	C₂×C₄	—	106	1	no	—
61	C₂×C₄	—	106	1	no	—
62	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
63	C₈	—	106	1	no	—
64	C₈	#11719912	1	1	yes	characteristic
65	C₈	#11719921	1	1	yes	characteristic
66	C₂²	—	53	2	no	—
67	C₂²	—	106	1	no	—
68	C₂²	—	106	1	no	—
69	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
70	C₄	—	106	1	no	—
71	C₄	#13133129	1	1	yes	characteristic
72	C₄	#13133156	1	1	yes	characteristic
73	C₂	—	106	2	no	—
74	C₂	#16810402	1	1	yes	characteristic
75	C₂	C₃₂×D₅₃	1	1	yes	characteristic
76	C₂	#16810456	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₃₂
Fitting	C₂×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₁₆
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	C₂²⋊C₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#13133123	C₂	semidirect
3. C₂×C₃₂×C₅₃	C₂	semidirect
30. C₂×D₅₃	C₃₂	semidirect
41. C₂×C₃₂	D₅₃	semidirect
42. C₅₃	C₂²⋊C₃₂	semidirect