Subgroups — #28298870

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289827	C₂	1	1	yes	maximal, characteristic
2	#20289632	C₂	1	1	yes	maximal, characteristic
3	#20272850	C₂	1	1	yes	maximal, characteristic
4	#16811555	C₄	1	2	yes	—
5	#16811512	C₄	1	1	yes	characteristic
6	#16811464	C₄	1	1	yes	characteristic
7	#16810457	C₄	1	1	yes	characteristic
8	#16810398	C₂²	1	1	yes	characteristic
9	C₅₃×M₆₄	C₄	1	1	yes	characteristic
10	C₂×#11719910	—	2	1	no	—
11	C₄×#88324	—	2	1	no	—
12	#13133267	—	2	2	no	—
13	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
14	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
15	#13133123	C₈	1	2	yes	complemented
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	C₈	1	2	yes	complemented
18	C₂×#88324	—	4	1	no	—
19	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
20	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
21	C₂×#88322	M₁₆	1	1	yes	characteristic
22	#11719910	—	4	1	no	—
23	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
25	#11719908	M₁₆	1	2	yes	—
26	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
27	C₂×Dic₅₃	—	2	1	no	—
28	#88324	—	8	1	no	—
29	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
30	C₈×C₅₃	#97	1	1	yes	characteristic
31	#88322	—	2	1	no	—
32	#7333	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	#347	1	1	yes	characteristic
34	C₄×C₅₃	#322	1	1	yes	characteristic
35	C₄×C₅₃	#347	1	1	yes	characteristic
36	Dic₅₃	—	4	1	no	—
37	#1398	—	53	1	no	—
38	#1398	—	53	1	no	—
39	#1313	—	53	1	no	—
40	C₂×C₅₃	—	2	1	no	—
41	C₂×C₅₃	#1313	1	1	yes	characteristic
42	M₆₄	—	53	1	no	—
43	M₆₄	—	53	2	no	—
44	M₆₄	#6157	1	1	yes	characteristic
45	#346	—	53	1	no	—
46	#346	—	53	1	no	—
47	#322	—	53	1	no	—
48	C₅₃	#7333	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	53	1	no	—
50	C₂×C₁₆	—	106	1	no	—
51	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
52	C₄×C₈	—	53	1	no	—
53	C₄×C₈	—	106	1	no	—
54	C₃₂	—	53	2	no	—
55	C₃₂	—	106	2	no	—
56	C₃₂	#88324	1	2	yes	complemented
57	C₂×C₈	—	53	1	no	—
58	C₂×C₈	—	212	1	no	—
59	C₂×C₈	C₄×#6157	1	1	yes	characteristic
60	C₄²	—	53	1	no	—
61	C₁₆	—	53	2	no	—
62	C₁₆	—	212	1	no	—
63	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
64	C₁₆	C₂×#88324	1	1	yes	characteristic
65	C₂×C₄	—	106	1	no	—
66	C₂×C₄	#13133271	1	1	yes	characteristic
67	C₈	—	106	1	no	—
68	C₈	—	424	1	no	—
69	C₈	C₄×#88324	1	1	yes	characteristic
70	C₈	#13133271	1	1	yes	characteristic
71	C₂²	#16811514	1	1	yes	characteristic
72	C₄	—	212	1	no	—
73	C₄	#16811464	1	1	yes	characteristic
74	C₄	#16811514	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	#20289632	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811464
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×M₆₄	C₄
socle	C₂×C₅₃	#1313
prime radical	C₂²×C₅₃	#347
cosocle	#16810398	C₂²
hypercenter	M₆₄	#6157
hypocenter	C₅₃	#7333
LCS	G ▷ C₈×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
15. #13133123	C₈	semidirect
17. C₃₂×C₅₃	C₈	semidirect
48. C₅₃	#7333	semidirect
56. C₃₂	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20289827	2
#20289632	2
#20272850	2
#7333	53