Subgroups — #20272854

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
2	#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
5	#13133156	—	2	1	no	—
6	#13133156	—	2	1	no	—
7	C₂×#11719908	C₄	1	1	yes	characteristic
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	—	4	1	no	—
30	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₅₃	M₃₂	1	1	yes	characteristic
32	Dic₅₃	—	2	1	no	—
33	#1400	—	53	1	no	maximal, Sylow
34	C₂×C₅₃	—	2	1	no	—
35	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
36	D₅₃	—	4	1	no	—
37	C₂×M₃₂	—	53	1	no	—
38	M₆₄	—	53	1	no	—
39	M₆₄	D₅₃	1	1	yes	characteristic, complemented
40	C₅₃	#1400	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	—	53	1	no	—
42	M₃₂	—	106	1	no	—
43	M₃₂	—	106	1	no	—
44	C₂×C₁₆	—	53	1	no	—
45	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	106	1	no	—
48	C₂²×C₄	—	53	1	no	—
49	C₂×C₈	—	53	1	no	—
50	C₂×C₈	—	106	1	no	—
51	C₂×C₈	—	106	1	no	—
52	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
53	C₁₆	—	106	1	no	—
54	C₁₆	D₂₁₂	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₂³	—	53	1	no	—
57	C₂×C₄	—	53	1	no	—
58	C₂×C₄	—	106	1	no	—
59	C₂×C₄	—	106	1	no	—
60	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
61	C₈	—	106	1	no	—
62	C₈	#11719912	1	1	yes	characteristic
63	C₈	#11719921	1	1	yes	characteristic
64	C₂²	—	106	1	no	—
65	C₂²	—	212	1	no	—
66	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
67	C₄	—	106	1	no	—
68	C₄	#13133129	1	1	yes	characteristic
69	C₄	#13133156	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	212	1	no	—
72	C₂	#16810402	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#11719921
commutator	C₂²×C₅₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₅₃×M₆₄	C₂
socle	C₂×C₅₃	C₂²⋊C₁₆
hypercenter	M₆₄	D₅₃
hypocenter	C₅₃	#1400
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #16810457	C₂	semidirect
3. C₅₃×M₆₄	C₂	semidirect
39. M₆₄	D₅₃	semidirect
40. C₅₃	#1400	semidirect