Subgroups — #27271774

Order 13568.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272849	C₂	1	1	yes	maximal, characteristic
2	#20272789	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1372	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	SD₃₂	1	1	yes	characteristic
26	C₂×C₄×C₅₃	SD₃₂	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	#7463	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	#367	1	1	yes	characteristic
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	#335	1	1	yes	characteristic
37	C₄×C₅₃	#368	1	1	yes	characteristic
38	#1397	—	53	1	no	—
39	#1372	—	53	1	no	—
40	#1372	D₅₃	1	1	yes	characteristic
41	C₂×C₅₃	#1372	1	1	yes	characteristic
42	C₂×C₅₃	#1426	1	1	yes	characteristic
43	C₂×C₅₃	#1426	1	1	yes	characteristic
44	C₂×C₃₂	—	53	1	no	—
45	C₂×C₃₂	—	53	1	no	—
46	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
47	#335	—	2	1	no	—
48	#335	—	106	1	no	—
49	C₅₃	#7463	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
51	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	212	1	no	—
54	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
55	C₃₂	—	53	2	no	—
56	C₃₂	—	106	1	no	—
57	C₂×C₈	—	8	1	no	—
58	C₂×C₈	—	424	1	no	—
59	C₂×C₈	#11719919	1	1	yes	characteristic
60	C₂×C₈	#11719925	1	1	yes	characteristic
61	C₄²	#11719922	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	C₈×D₅₃	1	1	yes	characteristic
64	C₁₆	#11719912	1	1	yes	characteristic
65	C₂×C₄	#13133150	1	1	yes	characteristic
66	C₂×C₄	#13133160	1	1	yes	characteristic
67	C₂×C₄	#13133161	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	16	1	no	—
70	C₈	—	848	1	no	—
71	C₈	#13133146	1	1	yes	characteristic
72	C₈	#13133152	1	1	yes	characteristic
73	C₂²	#16810450	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#16810375	1	1	yes	characteristic
77	C₄	#16810452	1	1	yes	characteristic
78	C₂	#20272788	1	1	yes	characteristic
79	C₂	#20272920	1	1	yes	characteristic
80	C₂	#20272922	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810450
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1372	C₂
socle, prime radical	C₂²×C₅₃	#367
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1372	D₅₃
hypocenter	C₅₃	#7463
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

Complements

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

N	H ≅ G/N	type
11. #13133123	C₈	semidirect
49. C₅₃	#7463	semidirect

Maximal subgroups

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

subgroup	index
#20272849	2
#20272789	2
C₅₃×#1372	2
#7463	53