Subgroups — #27298833

Order 13568.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276997	C₂	1	2	yes	maximal
2	#20276990	C₂	1	2	yes	maximal
3	#20274325	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811028	C₂²	1	4	yes	—
6	#16810974	—	2	4	no	—
7	#16810451	—	2	4	no	—
8	#16810449	—	2	4	no	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	#13133217	—	2	4	no	—
13	#13133151	—	2	4	no	—
14	#13133151	—	2	4	no	—
15	#13133142	—	4	8	no	—
16	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
17	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
18	C₄×C₈×C₅₃	D₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₀₆	—	4	4	no	—
21	C₂×C₈×C₅₃	D₈	1	4	yes	—
22	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₄⋊Dic₅₃	—	4	4	no	—
26	C₄⋊Dic₅₃	—	4	4	no	—
27	C₁₆×C₅₃	Q₁₆	1	4	yes	—
28	C₁₆×C₅₃	Q₁₆	1	4	yes	—
29	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
31	C₂×Dic₅₃	—	8	4	no	—
32	Dic₁₀₆	—	8	8	no	—
33	C₈×C₅₃	—	2	4	no	—
34	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
35	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
36	#12522	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	#493	1	1	yes	characteristic
38	C₄×C₅₃	#494	1	1	yes	characteristic
39	C₄×C₅₃	#494	1	1	yes	characteristic
40	C₄×C₅₃	#508	1	4	yes	—
41	Dic₅₃	—	16	4	no	—
42	#2254	—	53	2	no	—
43	#2252	—	53	2	no	—
44	#1710	—	53	2	no	—
45	C₈×C₁₆	D₅₃	1	1	yes	characteristic
46	C₂×C₅₃	#1713	1	1	yes	characteristic
47	C₂×C₅₃	#2249	1	2	yes	—
48	#500	—	53	4	no	—
49	#487	—	106	4	no	—
50	#367	—	106	4	no	—
51	#366	—	106	4	no	—
52	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
53	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
54	C₈²	C₂×D₅₃	1	1	yes	characteristic
55	C₅₃	#12522	1	1	yes	characteristic, complemented, Sylow
56	C₄⋊Q₈	—	106	4	no	—
57	C₂×C₁₆	D₂₁₂	1	2	yes	—
58	C₂×C₁₆	D₂₁₂	1	2	yes	—
59	#107	—	106	4	no	—
60	#107	—	106	4	no	—
61	#103	—	212	8	no	—
62	C₄×C₈	D₂₁₂	1	2	yes	—
63	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	212	4	no	—
65	C₂×C₈	D₄₂₄	1	4	yes	—
66	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
67	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
68	C₄⋊C₄	—	212	4	no	—
69	C₄⋊C₄	—	212	4	no	—
70	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
71	C₁₆	Dic₂₁₂	1	4	yes	—
72	C₁₆	Dic₂₁₂	1	4	yes	—
73	Q₈	—	424	8	no	—
74	C₂×C₄	—	424	4	no	—
75	C₂×C₄	#13133214	1	1	yes	characteristic
76	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Dic₂₁₂	1	2	yes	—
79	C₈	C₂×Dic₂₁₂	1	2	yes	—
80	C₂²	#16811000	1	1	yes	characteristic
81	C₄	—	848	4	no	—
82	C₄	#16811003	1	1	yes	characteristic
83	C₄	#16811003	1	1	yes	characteristic
84	C₄	#16811055	1	4	yes	—
85	C₂	#20274333	1	1	yes	characteristic
86	C₂	#20276976	1	2	yes	—

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12522
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
55. C₅₃	#12522	semidirect

Maximal subgroups

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

subgroup	index
#20276997	2
#20276990	2
#20274325	2
C₈×C₁₆×C₅₃	2
#12522	53