Subgroups — #27270708

Order 13568.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272633	C₂	1	2	yes	maximal
2	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719908	C₄	1	4	yes	—
4	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₈²×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719908	—	2	4	no	—
8	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
9	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
10	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
15	C₄²×C₅₃	C₄²	1	1	yes	characteristic
16	C₁₆×C₅₃	—	2	2	no	—
17	C₁₆×C₅₃	C₁₆	1	4	yes	complemented
18	#11719908	—	4	4	no	—
19	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
20	C₂×C₄×C₅₃	#97	1	2	yes	—
21	C₈×C₅₃	—	2	4	no	—
22	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
23	C₈×C₅₃	M₃₂	1	2	yes	—
24	#7053	—	53	1	no	maximal, Sylow
25	C₂²×C₅₃	#322	1	1	yes	characteristic
26	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	#346	1	1	yes	characteristic
28	C₄×C₅₃	#347	1	4	yes	—
29	#1312	—	53	2	no	—
30	#1311	D₅₃	1	1	yes	characteristic
31	C₂×C₅₃	#1312	1	1	yes	characteristic
32	C₂×C₅₃	#1313	1	2	yes	—
33	C₄×C₁₆	—	53	4	no	—
34	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
35	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
36	C₈²	C₂×D₅₃	1	1	yes	characteristic
37	C₅₃	#7053	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₁₆	—	106	4	no	—
39	C₂×C₁₆	#88322	1	2	yes	—
40	C₂×C₁₆	#88322	1	2	yes	—
41	C₄×C₈	C₄×D₅₃	1	2	yes	—
42	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
43	C₂×C₈	#11719912	1	4	yes	—
44	C₂×C₈	C₂×#88322	1	1	yes	characteristic
45	C₂×C₈	C₂×#88322	1	1	yes	characteristic
46	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
47	C₁₆	—	2	2	no	—
48	C₁₆	—	212	4	no	—
49	C₁₆	#11719908	1	4	yes	complemented
50	C₂×C₄	C₄×#88322	1	1	yes	characteristic
51	C₂×C₄	#13133127	1	2	yes	—
52	C₈	—	2	4	no	—
53	C₈	C₂×#11719908	1	2	yes	—
54	C₈	#13133157	1	2	yes	—
55	C₂²	#16810344	1	1	yes	characteristic
56	C₄	C₄×#11719908	1	1	yes	characteristic
57	C₄	#16810399	1	1	yes	characteristic
58	C₄	#16810400	1	4	yes	—
59	C₂	#20272633	1	1	yes	characteristic
60	C₂	#20272635	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1311	C₂
socle, prime radical	C₂²×C₅₃	#322
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#7053
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1311

Complements

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

N	H ≅ G/N	type
17. C₁₆×C₅₃	C₁₆	semidirect
37. C₅₃	#7053	semidirect
49. C₁₆	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20272633	2
C₅₃×#1311	2
#7053	53