Subgroups — #28299377

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic
2	#20289876	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811549	—	2	1	no	—
5	#16811462	—	2	1	no	—
6	Q₃₂×D₅₃	—	2	1	no	—
7	Dic₈₄₈	C₄	1	1	yes	characteristic, complemented
8	#16810463	—	2	1	no	—
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
11	#13133286	—	4	1	no	—
12	Q₁₆×D₅₃	—	4	1	no	—
13	Dic₄₂₄	—	2	1	no	—
14	#13133163	—	4	1	no	—
15	C₁₆×D₅₃	D₄	1	1	yes	characteristic
16	#13133123	D₄	1	1	yes	characteristic
17	Q₃₂×C₅₃	—	2	1	no	—
18	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
19	Q₈×D₅₃	—	8	1	no	—
20	C₄⋊#6157	—	8	1	no	—
21	Q₁₆×C₅₃	—	4	1	no	—
22	#11719925	—	8	1	no	—
23	Dic₂₁₂	—	4	1	no	—
24	C₈×D₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#11719908	SD₁₆	1	1	yes	characteristic
27	C₂×#6157	—	16	1	no	—
28	Q₈×C₅₃	—	8	1	no	—
29	C₄×D₅₃	—	16	1	no	—
30	C₄×D₅₃	D₁₆	1	1	yes	characteristic
31	Dic₁₀₆	—	8	1	no	—
32	Dic₁₀₆	—	16	1	no	—
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	#88322	SD₃₂	1	1	yes	characteristic
35	#7538	—	53	1	no	maximal, Sylow
36	C₂×D₅₃	D₃₂	1	1	yes	characteristic
37	#6157	—	32	1	no	—
38	C₄×C₅₃	—	16	1	no	—
39	C₄×C₅₃	#357	1	1	yes	characteristic
40	Dic₅₃	—	16	1	no	—
41	Dic₅₃	SD₆₄	1	1	yes	characteristic
42	C₂×Q₆₄	—	53	1	no	—
43	C₂×C₆₄	—	53	1	no	—
44	#1423	—	53	1	no	—
45	C₂×C₅₃	#1415	1	1	yes	characteristic
46	D₅₃	SD₁₂₈	1	1	yes	characteristic
47	D₅₃	Q₁₂₈	1	1	yes	characteristic
48	C₂×Q₃₂	—	106	1	no	—
49	Q₆₄	—	53	1	no	—
50	Q₆₄	—	106	1	no	—
51	Q₆₄	#6157	1	1	yes	characteristic, complemented
52	C₂×C₃₂	—	53	1	no	—
53	#366	—	106	1	no	—
54	C₆₄	—	106	1	no	—
55	C₅₃	#7538	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₁₆	—	212	1	no	—
57	Q₃₂	—	2	1	no	—
58	Q₃₂	—	106	1	no	—
59	Q₃₂	—	212	1	no	—
60	C₂×C₁₆	—	53	1	no	—
61	#107	—	212	1	no	—
62	C₃₂	—	53	1	no	—
63	C₃₂	C₂×#6157	1	1	yes	characteristic
64	C₂×Q₈	—	424	1	no	—
65	Q₁₆	—	4	1	no	—
66	Q₁₆	—	424	1	no	—
67	Q₁₆	—	212	1	no	—
68	C₂×C₈	—	53	1	no	—
69	C₄⋊C₄	—	424	1	no	—
70	C₁₆	—	53	1	no	—
71	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	Q₈	—	848	1	no	—
74	Q₈	—	424	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	848	1	no	—
77	C₂×C₄	—	848	1	no	—
78	C₈	—	53	1	no	—
79	C₈	#13133279	1	1	yes	characteristic
80	C₂²	—	53	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	53	1	no	—
83	C₄	—	848	1	no	—
84	C₄	—	1696	1	no	—
85	C₄	#16811535	1	1	yes	characteristic
86	C₂	—	53	1	no	—
87	C₂	—	53	1	no	—
88	C₂	#20289864	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289864
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₃₂	C₂×#6157
Fitting	C₅₃×Q₆₄	C₄
socle	C₂×C₅₃	#1415
prime radical	C₂×D₅₃	D₃₂
cosocle	C₃₂×D₅₃	C₂²
hypercenter	Q₆₄	#6157
hypocenter	C₅₃	#7538
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
7. Dic₈₄₈	C₄	semidirect
10. C₅₃×Q₆₄	C₄	semidirect
51. Q₆₄	#6157	semidirect
55. C₅₃	#7538	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊C₆₄	2
#20289876	2
Q₆₄×D₅₃	2
#7538	53