Subgroups — #27270834

Order 13568.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272785	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
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	—	4	2	no	—
8	C₂×C₁₆×C₅₃	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	1	no	—
15	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	2	no	—
21	C₁₆×C₅₃	—	2	1	no	—
22	C₁₆×C₅₃	C₁₆	1	2	yes	complemented
23	#11719908	—	8	2	no	—
24	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
26	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
27	C₈×C₅₃	—	2	2	no	—
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	—	2	1	no	—
30	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₅₃	M₃₂	1	1	yes	characteristic
32	C₈×C₅₃	SD₃₂	1	2	yes	—
33	#7101	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	#335	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₅₃	#364	1	1	yes	characteristic
38	C₄×C₅₃	#367	1	1	yes	characteristic
39	C₄×C₅₃	#368	1	1	yes	characteristic
40	#1370	—	53	2	no	—
41	C₈×C₁₆	D₅₃	1	1	yes	characteristic
42	C₂×C₅₃	#1370	1	1	yes	characteristic
43	C₂×C₅₃	#1372	1	1	yes	characteristic
44	C₂×C₅₃	#1373	1	1	yes	characteristic
45	C₄⋊C₁₆	—	106	2	no	—
46	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
47	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
48	C₈²	C₂×D₅₃	1	1	yes	characteristic
49	C₅₃	#7101	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	212	2	no	—
52	C₂×C₁₆	#88322	1	1	yes	characteristic
53	C₂×C₁₆	#88322	1	1	yes	characteristic
54	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
55	C₄×C₈	D₂₁₂	1	1	yes	characteristic
56	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	D₄₂₄	1	1	yes	characteristic
59	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
60	C₂×C₈	C₂×#88322	1	1	yes	characteristic
61	C₂×C₈	#11719917	1	1	yes	characteristic
62	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	424	2	no	—
66	C₁₆	#11719908	1	2	yes	complemented
67	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
68	C₂×C₄	#13133151	1	1	yes	characteristic
69	C₂×C₄	#13133153	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	C₂×#11719908	1	1	yes	characteristic
74	C₈	#13133157	1	1	yes	characteristic
75	C₈	#13133162	1	2	yes	—
76	C₂²	#16810376	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	C₄⋊#11719908	1	1	yes	characteristic
79	C₄	#16810444	1	1	yes	characteristic
80	C₄	#16810451	1	1	yes	characteristic
81	C₄	#16810453	1	1	yes	characteristic
82	C₂	#20272785	1	1	yes	characteristic
83	C₂	#20272789	1	1	yes	characteristic
84	C₂	#20272791	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₂₄
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	#335
cosocle	C₈²×C₅₃	C₂²
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#7101
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
22. C₁₆×C₅₃	C₁₆	semidirect
49. C₅₃	#7101	semidirect
66. C₁₆	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20272785	2
C₈×C₁₆×C₅₃	2
#7101	53