Subgroups — #28299131

Order 13568.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289875	C₂	1	1	yes	maximal, characteristic
2	#20289751	C₂	1	1	yes	maximal, characteristic
3	#20276996	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	1	no	—
5	#16811545	—	2	1	no	—
6	#16811027	—	2	1	no	—
7	#16810451	C₄	1	1	yes	characteristic
8	#16810450	—	2	1	no	—
9	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#367	C₄	1	1	yes	characteristic
11	#13133291	—	4	1	no	—
12	#13133267	—	4	1	no	—
13	#13133216	—	4	1	no	—
14	C₂×#11719908	D₄	1	1	yes	characteristic
15	#13133151	—	2	1	no	—
16	#13133150	—	4	1	no	—
17	C₈×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#107×C₅₃	—	2	1	no	—
20	C₂×Dic₁₀₆	—	8	1	no	—
21	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	4	1	no	—
23	C₄⋊Dic₅₃	—	4	1	no	—
24	#11719919	—	8	1	no	—
25	C₄×Dic₅₃	SD₁₆	1	1	yes	characteristic
26	C₂×#88322	D₈	1	1	yes	characteristic
27	#11719910	—	8	1	no	—
28	C₁₆×C₅₃	—	2	1	no	—
29	#11719908	SD₁₆	1	1	yes	characteristic
30	#11719908	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	8	1	no	—
32	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	1	no	—
34	C₂×Dic₅₃	—	8	1	no	—
35	Dic₁₀₆	—	16	2	no	—
36	C₈×C₅₃	#101	1	1	yes	characteristic
37	C₈×C₅₃	#103	1	1	yes	characteristic
38	#88322	SD₃₂	1	1	yes	characteristic
39	#88322	Q₃₂	1	1	yes	characteristic
40	#7453	—	53	1	no	maximal, Sylow
41	C₂²×C₅₃	#361	1	1	yes	characteristic
42	C₄×C₅₃	—	16	1	no	—
43	C₄×C₅₃	#332	1	1	yes	characteristic
44	C₄×C₅₃	#358	1	1	yes	characteristic
45	Dic₅₃	—	2	2	no	—
46	Dic₅₃	—	16	1	no	—
47	#2254	—	53	1	no	—
48	#1422	—	53	1	no	—
49	#1373	—	53	1	no	—
50	C₂×C₅₃	—	2	1	no	—
51	C₂×C₅₃	#1365	1	1	yes	characteristic
52	#500	—	106	1	no	—
53	M₆₄	—	106	1	no	—
54	#367	—	53	1	no	—
55	#367	—	106	1	no	—
56	#367	#6157	1	1	yes	characteristic
57	#364	—	106	1	no	—
58	C₄×C₁₆	—	53	1	no	—
59	C₅₃	#7453	1	1	yes	characteristic, complemented, Sylow
60	C₄⋊Q₈	—	212	1	no	—
61	M₃₂	—	212	1	no	—
62	C₂×C₁₆	—	53	1	no	—
63	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
64	#107	—	2	1	no	—
65	#107	—	106	1	no	—
66	#107	—	212	1	no	—
67	C₄×C₈	—	53	1	no	—
68	C₃₂	—	212	1	no	—
69	C₂×Q₈	—	424	1	no	—
70	C₂×C₈	—	53	1	no	—
71	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
72	C₄⋊C₄	—	4	1	no	—
73	C₄⋊C₄	—	424	1	no	—
74	C₄⋊C₄	—	212	1	no	—
75	C₄²	—	53	1	no	—
76	C₁₆	—	2	1	no	—
77	C₁₆	—	53	1	no	—
78	C₁₆	—	53	1	no	—
79	C₁₆	—	424	1	no	—
80	Q₈	—	848	2	no	—
81	C₂×C₄	—	8	1	no	—
82	C₂×C₄	—	106	1	no	—
83	C₂×C₄	—	424	1	no	—
84	C₂×C₄	#13133279	1	1	yes	characteristic
85	C₈	—	53	1	no	—
86	C₈	—	53	1	no	—
87	C₈	#13133278	1	1	yes	characteristic
88	C₈	#13133280	1	1	yes	characteristic
89	C₂²	#16811541	1	1	yes	characteristic
90	C₄	—	16	1	no	—
91	C₄	—	848	1	no	—
92	C₄	—	106	2	no	—
93	C₄	#16811482	1	1	yes	characteristic
94	C₄	#16811536	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	#20289734	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289734
commutator	C₂×C₁₆×C₅₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₅₃×#367	C₄
socle	C₂×C₅₃	#1365
prime radical	C₂²×C₅₃	#361
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	#367	#6157
hypocenter	C₅₃	#7453
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
59. C₅₃	#7453	semidirect

Maximal subgroups

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

subgroup	index
#20289875	2
#20289751	2
#20276996	2
#7453	53