Subgroups — #27271760

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272913	C₂	1	2	yes	maximal
2	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic
3	#16810457	—	2	2	no	—
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	—	4	2	no	—
8	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	2	no	—
15	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
18	C₁₆×C₅₃	—	2	1	no	—
19	C₁₆×C₅₃	—	2	1	no	—
20	C₁₆×C₅₃	C₁₆	1	2	yes	—
21	C₁₆×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₅₃	—	4	2	no	—
25	C₈×C₅₃	#106	1	1	yes	characteristic
26	C₈×C₅₃	#108	1	1	yes	characteristic
27	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	M₃₂	1	1	yes	characteristic
29	#7456	—	53	1	no	maximal, Sylow
30	C₂²×C₅₃	#364	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	#334	1	1	yes	characteristic
33	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
34	#1422	—	53	2	no	—
35	#1311	D₅₃	1	1	yes	characteristic
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	#1367	1	1	yes	characteristic
38	M₆₄	—	106	2	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	#7456	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	#88322	1	1	yes	characteristic
44	C₂×C₁₆	#88322	1	1	yes	characteristic
45	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
46	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
49	C₃₂	—	212	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₂×#88322	1	1	yes	characteristic
52	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
53	C₄²	#11719917	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	#11719908	1	2	yes	—
57	C₁₆	#11719913	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#13133149	1	1	yes	characteristic
62	C₈	#13133153	1	1	yes	characteristic
63	C₈	C₂×#11719908	1	1	yes	characteristic
64	C₈	#13133157	1	1	yes	characteristic
65	C₂²	#16810444	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#16810374	1	1	yes	characteristic
68	C₄	C₄⋊#11719908	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#20272779	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133149
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1311	C₂
socle	C₂×C₅₃	#1367
prime radical	C₂²×C₅₃	#364
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#7456
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1311

Complements

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

N	H ≅ G/N	type
42. C₅₃	#7456	semidirect

Maximal subgroups

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

subgroup	index
#20272913	2
C₅₃×#1311	2
#7456	53