Subgroups — #27271427

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272851	C₂	1	1	yes	maximal, characteristic
2	#20272635	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1398	C₂	1	1	yes	maximal, characteristic
4	#16810457	C₄	1	2	yes	—
5	#16810399	C₄	1	1	yes	characteristic
6	#16810344	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	C₄	1	2	yes	—
8	C₅₃×#346	C₂²	1	1	yes	characteristic
9	C₂×#11719908	—	2	1	no	—
10	C₄×#88322	—	2	1	no	—
11	#13133123	—	2	2	no	—
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₈	1	4	yes	complemented
16	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
17	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
18	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
19	C₂×#88322	—	4	1	no	—
20	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
21	C₁₆×C₅₃	M₁₆	1	2	yes	—
22	#11719908	—	4	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
25	C₈×C₅₃	—	2	1	no	—
26	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
27	C₈×C₅₃	#97	1	1	yes	characteristic
28	#88322	—	8	1	no	—
29	#7333	—	53	1	no	maximal, Sylow
30	C₂²×C₅₃	#347	1	1	yes	characteristic
31	C₄×C₅₃	—	4	1	no	—
32	C₄×C₅₃	#322	1	1	yes	characteristic
33	C₄×C₅₃	#347	1	1	yes	characteristic
34	#1398	—	53	1	no	—
35	#1398	D₅₃	1	1	yes	characteristic
36	#1313	—	53	1	no	—
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#1313	1	1	yes	characteristic
39	M₆₄	—	53	2	no	—
40	M₆₄	Dic₅₃	1	2	yes	—
41	#346	—	53	1	no	—
42	#346	C₂×D₅₃	1	1	yes	characteristic
43	#322	—	53	1	no	—
44	C₅₃	#7333	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	106	1	no	—
46	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
48	C₄×C₈	—	106	1	no	—
49	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
50	C₃₂	—	106	2	no	—
51	C₃₂	#88322	1	4	yes	complemented
52	C₂×C₈	—	212	1	no	—
53	C₂×C₈	#11719912	1	1	yes	characteristic
54	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
55	C₄²	#11719912	1	1	yes	characteristic
56	C₁₆	—	212	1	no	—
57	C₁₆	#11719912	1	2	yes	—
58	C₁₆	C₂×#88322	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#13133127	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	424	1	no	—
63	C₈	C₄×#88322	1	1	yes	characteristic
64	C₈	#13133127	1	1	yes	characteristic
65	C₂²	#16810400	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#16810344	1	1	yes	characteristic
68	C₄	#16810400	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#20272635	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810344
commutator	C₈×C₅₃	C₄×C₈
Frattini	#346	C₂×D₅₃
Fitting	C₅₃×#1398	C₂
socle	C₂×C₅₃	#1313
prime radical	C₂²×C₅₃	#347
cosocle	C₅₃×#346	C₂²
hypercenter	#1398	D₅₃
hypocenter	C₅₃	#7333
LCS	G ▷ C₈×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₅₃	C₈	semidirect
44. C₅₃	#7333	semidirect
51. C₃₂	#88322	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20272851	2
#20272635	2
C₅₃×#1398	2
#7333	53