Subgroups — #28299379

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#20289878	C₂	1	1	yes	maximal, characteristic, complemented
3	#20277025	C₂	1	1	yes	maximal, characteristic
4	#16811551	—	2	1	no	—
5	#16811462	—	2	1	no	—
6	#16811056	—	2	1	no	—
7	#16810460	—	2	1	no	—
8	D₁₆₉₆	C₄	1	1	yes	characteristic
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×Q₆₄	C₄	1	1	yes	characteristic
11	#13133288	—	4	1	no	—
12	#13133240	—	4	1	no	—
13	#13133160	—	4	1	no	—
14	D₈₄₈	—	2	1	no	—
15	C₁₆×D₅₃	D₄	1	1	yes	characteristic
16	#13133123	D₄	1	1	yes	characteristic
17	Q₃₂×C₅₃	—	2	1	no	—
18	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#11719949	—	8	1	no	—
20	#11719936	—	8	1	no	—
21	Q₁₆×C₅₃	—	4	1	no	—
22	#11719924	—	8	1	no	—
23	D₄₂₄	—	4	1	no	—
24	C₈×D₅₃	D₈	1	1	yes	characteristic
25	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	#11719908	SD₁₆	1	1	yes	characteristic
27	Q₈×C₅₃	—	8	1	no	—
28	D₂₁₂	—	8	1	no	—
29	D₂₁₂	—	16	1	no	—
30	C₄×D₅₃	—	16	1	no	—
31	C₄×D₅₃	D₁₆	1	1	yes	characteristic
32	#88324	—	16	1	no	—
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	#88322	SD₃₂	1	1	yes	characteristic
35	#7539	—	53	1	no	maximal, Sylow
36	C₂×D₅₃	—	16	1	no	—
37	C₂×D₅₃	SD₆₄	1	1	yes	characteristic
38	C₄×C₅₃	—	16	1	no	—
39	C₄×C₅₃	#357	1	1	yes	characteristic
40	Dic₅₃	D₃₂	1	1	yes	characteristic
41	#2262	—	53	1	no	—
42	C₂×C₆₄	—	53	1	no	—
43	#1425	—	53	1	no	—
44	C₂×C₅₃	#1415	1	1	yes	characteristic
45	D₅₃	—	2	1	no	—
46	D₅₃	—	32	1	no	—
47	#508	—	106	1	no	—
48	Q₆₄	#6157	1	1	yes	characteristic
49	SD₆₄	—	106	1	no	—
50	D₃₂	—	53	1	no	—
51	C₂×C₃₂	—	53	1	no	—
52	#368	—	106	1	no	—
53	C₆₄	—	106	1	no	—
54	C₅₃	#7539	1	1	yes	characteristic, complemented, Sylow
55	#135	—	212	1	no	—
56	Q₃₂	—	2	1	no	—
57	SD₃₂	—	212	1	no	—
58	D₁₆	—	106	1	no	—
59	C₂×C₁₆	—	53	1	no	—
60	#108	—	212	1	no	—
61	C₃₂	—	53	1	no	—
62	C₃₂	C₂×#6157	1	1	yes	characteristic
63	#41	—	424	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	424	1	no	—
66	D₈	—	212	1	no	—
67	M₁₆	—	424	1	no	—
68	C₂×C₈	—	53	1	no	—
69	C₁₆	—	53	1	no	—
70	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
71	Q₈	—	8	1	no	—
72	D₄	—	848	1	no	—
73	D₄	—	424	1	no	—
74	C₂×C₄	—	53	1	no	—
75	C₂×C₄	—	848	1	no	—
76	C₈	—	53	1	no	—
77	C₈	—	848	1	no	—
78	C₈	#13133279	1	1	yes	characteristic
79	C₂²	—	53	1	no	—
80	C₂²	—	848	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	53	1	no	—
83	C₄	#16811535	1	1	yes	characteristic
84	C₂	—	106	1	no	—
85	C₂	—	1696	1	no	—
86	C₂	#20289864	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289864
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₃₂	C₂×#6157
Fitting	C₅₃×Q₆₄	C₄
socle	C₂×C₅₃	#1415
prime radical	#20277025	C₂
cosocle	C₃₂×D₅₃	C₂²
hypercenter	Q₆₄	#6157
hypocenter	C₅₃	#7539
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. D₅₃⋊C₆₄	C₂	semidirect
2. #20289878	C₂	semidirect
54. C₅₃	#7539	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊C₆₄	2
#20289878	2
#20277025	2
#7539	53