Subgroups — #27271770

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133123	C₂	1	1	yes	maximal, characteristic
2	#20272787	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₂×#13133123	C₄	1	1	yes	characteristic
6	#16810376	—	2	1	no	—
7	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#335	—	2	1	no	—
9	C₄⋊#88322	—	4	1	no	—
10	#13133123	—	2	1	no	—
11	#13133123	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	4	1	no	—
15	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	—	8	1	no	—
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	D₈	1	1	yes	characteristic
20	C₂×#88322	—	8	1	no	—
21	C₁₆×C₅₃	—	2	1	no	—
22	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
25	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	—	16	1	no	—
29	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	#108	1	1	yes	characteristic
31	#88322	—	16	1	no	—
32	#7461	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	#366	1	1	yes	characteristic
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	#335	1	1	yes	characteristic
36	C₄×C₅₃	#368	1	1	yes	characteristic
37	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
38	C₄×C₅₃	SD₆₄	1	1	yes	characteristic
39	C₄×C₃₂	—	53	1	no	—
40	#1371	—	53	1	no	—
41	#1371	D₅₃	1	1	yes	characteristic
42	C₂×C₅₃	#1371	1	1	yes	characteristic
43	C₂×C₅₃	#1424	1	1	yes	characteristic
44	C₂×C₅₃	#1425	1	1	yes	characteristic
45	C₂×C₃₂	—	53	1	no	—
46	C₂×C₃₂	—	53	1	no	—
47	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
48	#335	—	2	1	no	—
49	#335	—	106	1	no	—
50	C₅₃	#7461	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
52	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
53	C₄⋊C₈	—	4	1	no	—
54	C₄⋊C₈	—	212	1	no	—
55	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₃₂	—	53	2	no	—
57	C₃₂	—	106	1	no	—
58	C₂×C₈	—	8	1	no	—
59	C₂×C₈	—	424	1	no	—
60	C₂×C₈	#11719919	1	1	yes	characteristic
61	C₂×C₈	#11719925	1	1	yes	characteristic
62	C₄²	#11719922	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₈×D₅₃	1	1	yes	characteristic
65	C₁₆	#11719912	1	1	yes	characteristic
66	C₂×C₄	#13133150	1	1	yes	characteristic
67	C₂×C₄	#13133158	1	1	yes	characteristic
68	C₂×C₄	#13133163	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	16	1	no	—
71	C₈	—	848	1	no	—
72	C₈	#13133146	1	1	yes	characteristic
73	C₈	#13133152	1	1	yes	characteristic
74	C₂²	#16810448	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	#16810375	1	1	yes	characteristic
77	C₄	#16810452	1	1	yes	characteristic
78	C₄	#16810460	1	1	yes	characteristic
79	C₄	#16810461	1	1	yes	characteristic
80	C₂	#20272786	1	1	yes	characteristic
81	C₂	#20272916	1	1	yes	characteristic
82	C₂	#20272918	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133158
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1371	C₂
socle, prime radical	C₂²×C₅₃	#366
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1371	D₅₃
hypocenter	C₅₃	#7461
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1371

Complements

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

N	H ≅ G/N	type
11. #13133123	C₈	semidirect
50. C₅₃	#7461	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133123	2
#20272787	2
C₅₃×#1371	2
#7461	53