Subgroups — #27271426

Order 13568.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272849	C₂	1	1	yes	maximal, characteristic
2	#20272631	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1397	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	2	yes	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₄	1	2	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	2	yes	—
10	C₄×#88322	C₈	1	2	yes	—
11	#13133123	—	2	2	no	—
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂×#88322	—	2	2	no	—
20	C₁₆×C₅₃	M₁₆	1	2	yes	—
21	C₁₆×C₅₃	M₁₆	1	2	yes	—
22	#11719908	C₁₆	1	4	yes	—
23	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
25	C₈×C₅₃	#97	1	1	yes	characteristic
26	C₈×C₅₃	#97	1	1	yes	characteristic
27	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
28	#88322	—	4	2	no	—
29	#7332	—	53	1	no	maximal, Sylow
30	C₂²×C₅₃	#346	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	#322	1	1	yes	characteristic
33	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
34	#1397	—	53	1	no	—
35	#1397	D₅₃	1	1	yes	characteristic
36	#1311	—	53	1	no	—
37	C₂×C₅₃	#1312	1	1	yes	characteristic
38	C₂×C₅₃	#1398	1	2	yes	—
39	C₂×C₃₂	—	53	2	no	—
40	C₂×C₃₂	Dic₅₃	1	2	yes	—
41	C₄×C₁₆	—	53	1	no	—
42	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
43	C₈²	—	53	1	no	—
44	C₅₃	#7332	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	53	2	no	—
46	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
48	C₄×C₈	—	53	2	no	—
49	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
50	C₃₂	—	2	2	no	—
51	C₃₂	—	106	2	no	—
52	C₂×C₈	—	106	2	no	—
53	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
54	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
55	C₄²	C₈×D₅₃	1	1	yes	characteristic
56	C₁₆	—	53	4	no	—
57	C₁₆	#11719912	1	2	yes	—
58	C₁₆	#11719917	1	2	yes	—
59	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
60	C₂×C₄	#13133156	1	2	yes	—
61	C₈	—	212	2	no	—
62	C₈	#13133127	1	1	yes	characteristic
63	C₈	#13133128	1	1	yes	characteristic
64	C₈	C₁₆×D₅₃	1	2	yes	—
65	C₂²	#16810398	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#16810345	1	1	yes	characteristic
68	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
69	C₂	#20272634	1	1	yes	characteristic
70	C₂	#20272850	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1397	C₂
socle, prime radical	C₂²×C₅₃	#346
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1397	D₅₃
hypocenter	C₅₃	#7332
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #1397

Complements

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

N	H ≅ G/N	type
44. C₅₃	#7332	semidirect

Maximal subgroups

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

subgroup	index
#20272849	2
#20272631	2
C₅₃×#1397	2
#7332	53