Subgroups — #28299150

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289746	C₂	1	2	yes	maximal
2	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16811486	—	2	2	no	—
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊#88324	—	4	2	no	—
8	C₂×#11719908	Q₈	1	1	yes	characteristic
9	C₈×Dic₅₃	D₄	1	1	yes	characteristic
10	#13133123	C₈	1	2	yes	complemented
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	—	2	1	no	—
13	C₂×#88324	—	8	2	no	—
14	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
15	C₄×Dic₅₃	D₈	1	1	yes	characteristic
16	C₂×#88322	Q₁₆	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
19	#11719908	—	2	1	no	—
20	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
21	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
22	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
23	#88324	—	16	2	no	—
24	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₅₃	#108	1	1	yes	characteristic
26	#88322	—	2	1	no	—
27	#7461	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	#366	1	1	yes	characteristic
29	C₄×C₅₃	#335	1	1	yes	characteristic
30	C₄×C₅₃	#368	1	1	yes	characteristic
31	Dic₅₃	—	2	1	no	—
32	Dic₅₃	SD₆₄	1	2	yes	—
33	C₄×C₃₂	—	53	1	no	—
34	#1371	—	53	2	no	—
35	C₂×C₅₃	#1371	1	1	yes	characteristic
36	C₂×C₅₃	#1424	1	1	yes	characteristic
37	C₂×C₅₃	#1425	1	1	yes	characteristic
38	C₂×C₃₂	—	53	1	no	—
39	C₂×C₃₂	#6157	1	1	yes	characteristic
40	C₄×C₁₆	—	53	1	no	—
41	#335	—	106	2	no	—
42	C₅₃	#7461	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	—	53	1	no	—
44	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
45	C₄⋊C₈	—	212	2	no	—
46	C₄×C₈	—	53	1	no	—
47	C₃₂	—	2	1	no	—
48	C₃₂	—	53	2	no	—
49	C₂×C₈	—	53	1	no	—
50	C₂×C₈	—	424	2	no	—
51	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
52	C₄²	—	53	1	no	—
53	C₁₆	—	106	1	no	—
54	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
55	C₁₆	#11719940	1	1	yes	characteristic
56	C₂×C₄	—	53	1	no	—
57	C₂×C₄	—	53	1	no	—
58	C₂×C₄	#13133286	1	1	yes	characteristic
59	C₈	—	848	2	no	—
60	C₈	—	106	1	no	—
61	C₈	#13133284	1	1	yes	characteristic
62	C₈	#13133287	1	1	yes	characteristic
63	C₂²	#16811549	1	1	yes	characteristic
64	C₄	—	53	2	no	—
65	C₄	—	106	1	no	—
66	C₄	#16811487	1	1	yes	characteristic
67	C₄	#16811552	1	1	yes	characteristic
68	C₂	#20289747	1	1	yes	characteristic
69	C₂	#20289877	1	1	yes	characteristic
70	C₂	#20289878	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16811549
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#366
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7461
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
10. #13133123	C₈	semidirect
42. C₅₃	#7461	semidirect

Maximal subgroups

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

subgroup	index
#20289746	2
C₃₂×Dic₅₃	2
#7461	53