Subgroups — #27271979

Order 13568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	1	yes	maximal, characteristic, complemented
2	#20272917	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2262	C₂	1	1	yes	maximal, characteristic
4	#16810449	—	2	1	no	—
5	#16810342	—	2	1	no	—
6	C₅₃×#508	—	2	1	no	—
7	C₅₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
8	C₅₃×SD₆₄	—	2	1	no	—
9	C₅₃×D₃₂	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	#13133151	—	4	1	no	—
12	#135×C₅₃	—	4	1	no	—
13	Q₃₂×C₅₃	—	2	1	no	—
14	SD₃₂×C₅₃	—	4	1	no	—
15	D₁₆×C₅₃	—	2	1	no	—
16	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	#41×C₅₃	—	8	1	no	—
20	Q₁₆×C₅₃	—	4	1	no	—
21	SD₁₆×C₅₃	—	8	1	no	—
22	D₈×C₅₃	—	4	1	no	—
23	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
24	C₄⋊Dic₅₃	—	8	1	no	—
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	1	no	—
28	D₄×C₅₃	—	8	1	no	—
29	D₄×C₅₃	—	16	1	no	—
30	C₂×C₄×C₅₃	—	16	1	no	—
31	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
32	C₂×Dic₅₃	—	16	1	no	—
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
35	#7540	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	—	16	1	no	—
37	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
38	C₄×C₅₃	—	16	1	no	—
39	C₄×C₅₃	#357	1	1	yes	characteristic
40	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
41	Dic₅₃	—	32	1	no	—
42	#2262	D₅₃	1	1	yes	characteristic
43	M₁₂₈	—	53	1	no	—
44	#1424	—	53	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	32	1	no	—
47	C₂×C₅₃	#1415	1	1	yes	characteristic
48	#508	—	2	1	no	—
49	Q₆₄	Dic₅₃	1	1	yes	characteristic, complemented
50	SD₆₄	—	2	1	no	—
51	D₃₂	Dic₅₃	1	1	yes	characteristic, complemented
52	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
53	#366	—	106	1	no	—
54	C₆₄	—	106	1	no	—
55	C₅₃	#7540	1	1	yes	characteristic, complemented, Sylow
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
61	#107	—	212	1	no	—
62	C₃₂	C₂×Dic₅₃	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	#11719922	1	1	yes	characteristic
69	C₄⋊C₄	—	424	1	no	—
70	C₁₆	#11719923	1	1	yes	characteristic
71	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	848	1	no	—
77	C₂×C₄	#13133158	1	1	yes	characteristic
78	C₈	#13133138	1	1	yes	characteristic
79	C₈	#13133161	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	#16810458	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	1696	1	no	—
84	C₄	#16810424	1	1	yes	characteristic
85	C₄	#16810461	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#20272893	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272893
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting, prime radical	C₅₃×#2262	C₂
socle	C₂×C₅₃	#1415
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#2262	D₅₃
hypocenter	C₅₃	#7540
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #20272926	C₂	semidirect
2. #20272917	C₂	semidirect
7. C₅₃×Q₆₄	C₄	semidirect
9. C₅₃×D₃₂	C₄	semidirect
49. Q₆₄	Dic₅₃	semidirect
51. D₃₂	Dic₅₃	semidirect
55. C₅₃	#7540	semidirect

Maximal subgroups

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

subgroup	index
#20272926	2
#20272917	2
C₅₃×#2262	2
#7540	53