Subgroups — #27271773

Order 13568.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272791	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810444	—	2	2	no	—
4	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	#13133157	—	4	2	no	—
8	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
11	C₃₂×C₅₃	—	2	1	no	—
12	C₃₂×C₅₃	C₈	1	2	yes	—
13	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
15	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
16	C₁₆×C₅₃	—	2	1	no	—
17	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
19	#11719908	—	8	2	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
22	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₅₃	#108	1	1	yes	characteristic
24	C₈×C₅₃	D₁₆	1	1	yes	characteristic
25	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
26	#7462	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	#368	1	1	yes	characteristic
28	C₄×C₅₃	—	2	2	no	—
29	C₄×C₅₃	#335	1	1	yes	characteristic
30	C₄×C₅₃	#366	1	1	yes	characteristic
31	C₄×C₃₂	D₅₃	1	1	yes	characteristic
32	#1373	—	53	2	no	—
33	C₂×C₅₃	—	2	1	no	—
34	C₂×C₅₃	#1371	1	1	yes	characteristic
35	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
36	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
37	#364	—	106	2	no	—
38	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
39	C₅₃	#7462	1	1	yes	characteristic, complemented, Sylow
40	M₃₂	—	212	2	no	—
41	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
43	C₄×C₈	D₂₁₂	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#88322	1	2	yes	—
46	C₂×C₈	D₄₂₄	1	1	yes	characteristic
47	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
48	C₄²	Dic₂₁₂	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	424	2	no	—
51	C₁₆	C₂×#88322	1	1	yes	characteristic
52	C₁₆	#11719917	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	#13133151	1	1	yes	characteristic
55	C₈	C₄⋊#88322	1	1	yes	characteristic
56	C₈	#13133153	1	1	yes	characteristic
57	C₈	D₈₄₈	1	1	yes	characteristic
58	C₈	Dic₄₂₄	1	1	yes	characteristic
59	C₂²	#16810453	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#16810376	1	1	yes	characteristic
62	C₄	#16810449	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#20272787	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₈₄₈
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₄×C₃₂×C₅₃	C₂
socle	C₂×C₅₃	#1371
prime radical	C₂²×C₅₃	#368
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#7462
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
39. C₅₃	#7462	semidirect

Maximal subgroups

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

subgroup	index
#20272791	2
C₄×C₃₂×C₅₃	2
#7462	53