Subgroups — #27271423

Order 13568.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133123	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133123	C₄	1	4	yes	—
4	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₈²×C₅₃	C₄	1	1	yes	characteristic
7	#13133123	—	2	4	no	—
8	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
9	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₁₆	1	2	yes	—
14	C₂×C₈×C₅₃	C₁₆	1	2	yes	—
15	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₅₃	—	2	2	no	—
19	C₁₆×C₅₃	M₁₆	1	4	yes	—
20	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
23	C₈×C₅₃	—	2	2	no	—
24	C₈×C₅₃	C₃₂	1	4	yes	complemented
25	C₈×C₅₃	#97	1	2	yes	—
26	C₈×C₅₃	M₃₂	1	2	yes	—
27	#7331	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₅₃	#322	1	1	yes	characteristic
30	C₄×C₅₃	#346	1	1	yes	characteristic
31	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
32	C₄×C₅₃	M₆₄	1	2	yes	—
33	C₄×C₃₂	—	53	2	no	—
34	C₈×C₁₆	D₅₃	1	1	yes	characteristic
35	C₂×C₅₃	#1312	1	1	yes	characteristic
36	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
37	C₂×C₅₃	#1397	1	1	yes	characteristic
38	C₂×C₃₂	—	53	4	no	—
39	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
40	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
41	C₈²	Dic₅₃	1	1	yes	characteristic
42	C₅₃	#7331	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	#88322	1	2	yes	—
44	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
45	C₄×C₈	#88322	1	1	yes	characteristic
46	C₄×C₈	#88322	1	1	yes	characteristic
47	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
48	C₃₂	—	106	4	no	—
49	C₂×C₈	#11719908	1	2	yes	—
50	C₂×C₈	#11719908	1	2	yes	—
51	C₂×C₈	C₂×#88322	1	1	yes	characteristic
52	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
53	C₄²	C₂×#88322	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₁₆	#11719912	1	4	yes	—
56	C₂×C₄	C₄×#88322	1	1	yes	characteristic
57	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
58	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	#13133123	1	4	yes	complemented
61	C₈	#13133127	1	2	yes	—
62	C₈	#13133157	1	2	yes	—
63	C₂²	C₄×#11719908	1	1	yes	characteristic
64	C₄	#16810344	1	1	yes	characteristic
65	C₄	#16810399	1	1	yes	characteristic
66	C₄	C₂×#13133123	1	2	yes	—
67	C₄	#16810457	1	2	yes	—
68	C₂	#20272633	1	1	yes	characteristic
69	C₂	C₄×#13133123	1	1	yes	characteristic
70	C₂	#20272849	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#7331
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₈×C₅₃	C₃₂	semidirect
42. C₅₃	#7331	semidirect
60. C₈	#13133123	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
C₈×C₁₆×C₅₃	2
#7331	53