Subgroups — #27271772

Order 13568.

78 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	#20272791	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	1	yes	characteristic
5	C₂×#13133123	C₄	1	1	yes	characteristic
6	#16810444	—	2	1	no	—
7	C₅₃×#364	—	2	1	no	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	#13133157	—	4	1	no	—
10	#13133123	—	2	1	no	—
11	#13133123	C₈	1	2	yes	—
12	M₃₂×C₅₃	—	4	1	no	—
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
18	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
19	C₁₆×C₅₃	—	2	1	no	—
20	C₁₆×C₅₃	—	8	1	no	—
21	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
22	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
23	#11719908	—	8	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
26	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₅₃	#108	1	1	yes	characteristic
28	C₈×C₅₃	D₁₆	1	1	yes	characteristic
29	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
30	#7462	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#368	1	1	yes	characteristic
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	#335	1	1	yes	characteristic
34	C₄×C₅₃	#366	1	1	yes	characteristic
35	C₄×C₃₂	—	53	1	no	—
36	#1373	—	53	1	no	—
37	#1373	D₅₃	1	1	yes	characteristic
38	C₂×C₅₃	—	2	1	no	—
39	C₂×C₅₃	#1371	1	1	yes	characteristic
40	C₂×C₃₂	—	53	1	no	—
41	C₂×C₃₂	—	53	1	no	—
42	#364	—	2	1	no	—
43	#364	—	106	1	no	—
44	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
45	C₅₃	#7462	1	1	yes	characteristic, complemented, Sylow
46	M₃₂	—	4	1	no	—
47	M₃₂	—	212	1	no	—
48	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
49	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
51	C₃₂	—	53	2	no	—
52	C₃₂	—	106	1	no	—
53	C₂×C₈	#11719919	1	1	yes	characteristic
54	C₂×C₈	#11719922	1	1	yes	characteristic
55	C₄²	#11719925	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	—	8	1	no	—
58	C₁₆	—	424	1	no	—
59	C₁₆	C₈×D₅₃	1	1	yes	characteristic
60	C₁₆	#11719912	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#13133150	1	1	yes	characteristic
63	C₈	#13133146	1	1	yes	characteristic
64	C₈	#13133152	1	1	yes	characteristic
65	C₈	#13133158	1	1	yes	characteristic
66	C₈	#13133163	1	1	yes	characteristic
67	C₂²	#16810452	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	#16810375	1	1	yes	characteristic
70	C₄	#16810448	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#20272786	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133158
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1373	C₂
socle	C₂×C₅₃	#1371
prime radical	C₂²×C₅₃	#368
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1373	D₅₃
hypocenter	C₅₃	#7462
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
45. C₅₃	#7462	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
#20272791	2
C₅₃×#1373	2
#7462	53