Subgroups — #20272855

Order 6784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810457	C₂	1	2	yes	maximal, complemented
2	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133123	—	2	2	no	—
4	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
5	M₃₂×C₅₃	—	2	2	no	—
6	C₂×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₈×C₅₃	—	2	1	no	—
10	C₂×C₈×C₅₃	—	2	1	no	—
11	C₂×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₅₃	—	2	1	no	—
14	C₁₆×C₅₃	D₄	1	2	yes	—
15	C₂³×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₄×C₅₃	—	2	1	no	—
17	C₂×C₄×C₅₃	—	2	1	no	—
18	C₂×C₄×C₅₃	C₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₈×C₅₃	—	2	1	no	—
21	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂²×C₅₃	—	2	1	no	—
24	C₂²×C₅₃	—	4	1	no	—
25	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
26	C₄×C₅₃	—	2	1	no	—
27	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
28	C₄×C₅₃	M₃₂	1	1	yes	characteristic
29	#1400	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	—	4	1	no	—
32	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
33	C₂×M₃₂	D₅₃	1	1	yes	characteristic
34	M₆₄	—	53	2	no	—
35	C₅₃	#1400	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	C₂×C₁₆	Dic₅₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
40	C₃₂	—	106	2	no	—
41	C₂²×C₄	#88322	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	#88322	1	1	yes	characteristic
45	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
46	C₁₆	—	2	1	no	—
47	C₁₆	C₂²⋊D₅₃	1	2	yes	—
48	C₂³	#11719908	1	1	yes	characteristic
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	#11719908	1	1	yes	characteristic
52	C₂×C₄	C₂×#88322	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	#11719917	1	1	yes	characteristic
55	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
56	C₂²	—	2	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₂×#11719908	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	C₂²⋊#88322	1	1	yes	characteristic
61	C₄	#13133157	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	C₂²⋊#11719908	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂²⋊C₁₆
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#1400
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #16810457	C₂	semidirect
35. C₅₃	#1400	semidirect