Subgroups — #28299155

Order 13568.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289748	C₂	1	1	yes	maximal, characteristic
2	#20289748	C₂	1	1	yes	maximal, characteristic
3	#20272848	C₂	1	1	yes	maximal, characteristic
4	#16811486	—	2	1	no	—
5	#16811486	—	2	1	no	—
6	C₂×#13133123	C₄	1	1	yes	characteristic
7	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
9	C₄⋊#88324	—	4	1	no	—
10	C₄⋊#88324	—	4	1	no	—
11	C₂×#11719908	Q₈	1	1	yes	characteristic
12	C₈×Dic₅₃	D₄	1	1	yes	characteristic
13	#13133123	C₈	1	2	yes	complemented
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	—	2	1	no	—
16	C₂×#88324	—	8	1	no	—
17	C₂×#88324	—	8	1	no	—
18	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	D₈	1	1	yes	characteristic
20	C₂×#88322	Q₁₆	1	1	yes	characteristic
21	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	#11719908	—	2	1	no	—
24	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
25	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
26	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
27	#88324	—	16	1	no	—
28	#88324	—	16	1	no	—
29	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₅₃	#108	1	1	yes	characteristic
31	#88322	—	2	1	no	—
32	#7463	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	#367	1	1	yes	characteristic
34	C₄×C₅₃	#335	1	1	yes	characteristic
35	C₄×C₅₃	#368	1	1	yes	characteristic
36	Dic₅₃	—	2	1	no	—
37	Dic₅₃	—	2	1	no	—
38	#1397	—	53	1	no	—
39	#1372	—	53	1	no	—
40	#1372	—	53	1	no	—
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	#6157	1	1	yes	characteristic
46	C₄×C₁₆	—	53	1	no	—
47	#335	—	106	1	no	—
48	#335	—	106	1	no	—
49	C₅₃	#7463	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	53	1	no	—
51	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
52	C₄⋊C₈	—	212	1	no	—
53	C₄⋊C₈	—	212	1	no	—
54	C₄×C₈	—	53	1	no	—
55	C₃₂	—	2	1	no	—
56	C₃₂	—	53	2	no	—
57	C₂×C₈	—	53	1	no	—
58	C₂×C₈	—	424	1	no	—
59	C₂×C₈	—	424	1	no	—
60	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
61	C₄²	—	53	1	no	—
62	C₁₆	—	106	1	no	—
63	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
64	C₁₆	#11719940	1	1	yes	characteristic
65	C₂×C₄	—	53	1	no	—
66	C₂×C₄	—	53	1	no	—
67	C₂×C₄	#13133286	1	1	yes	characteristic
68	C₈	—	848	1	no	—
69	C₈	—	848	1	no	—
70	C₈	—	106	1	no	—
71	C₈	#13133284	1	1	yes	characteristic
72	C₈	#13133287	1	1	yes	characteristic
73	C₂²	#16811550	1	1	yes	characteristic
74	C₄	—	106	1	no	—
75	C₄	—	106	1	no	—
76	C₄	#16811487	1	1	yes	characteristic
77	C₄	#16811551	1	1	yes	characteristic
78	C₂	#20289749	1	1	yes	characteristic
79	C₂	#20289880	1	1	yes	characteristic
80	C₂	#20289881	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16811550
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#367
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7463
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
13. #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
#20289748	2
#20272848	2
#7463	53