Subgroups — #28299376

Order 13568.

98 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	#20289876	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811549	—	2	1	no	—
5	#16811462	—	2	1	no	—
6	D₁₆×D₅₃	—	2	1	no	—
7	D₁₆₉₆	C₄	1	1	yes	characteristic, complemented
8	#16810458	—	2	1	no	—
9	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×D₃₂	C₄	1	1	yes	characteristic, complemented
11	#13133286	—	4	1	no	—
12	D₈×D₅₃	—	4	1	no	—
13	D₈₄₈	—	2	1	no	—
14	#13133158	—	4	1	no	—
15	C₁₆×D₅₃	D₄	1	1	yes	characteristic
16	#13133123	D₄	1	1	yes	characteristic
17	D₁₆×C₅₃	—	2	1	no	—
18	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
19	D₄×D₅₃	—	8	1	no	—
20	C₄⋊#6157	—	8	1	no	—
21	D₈×C₅₃	—	4	1	no	—
22	#11719922	—	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	C₂²×D₅₃	—	16	1	no	—
28	C₂×#6157	—	16	1	no	—
29	D₄×C₅₃	—	8	1	no	—
30	C₂²⋊D₅₃	—	16	1	no	—
31	D₂₁₂	—	8	1	no	—
32	C₄×D₅₃	D₁₆	1	1	yes	characteristic
33	C₈×C₅₃	#102	1	1	yes	characteristic
34	#88322	SD₃₂	1	1	yes	characteristic
35	#7537	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	—	16	1	no	—
37	C₂×D₅₃	—	16	1	no	—
38	C₂×D₅₃	—	32	1	no	—
39	C₂×D₅₃	—	32	1	no	—
40	C₂×D₅₃	D₃₂	1	1	yes	characteristic
41	#6157	—	32	1	no	—
42	C₄×C₅₃	#357	1	1	yes	characteristic
43	Dic₅₃	SD₆₄	1	1	yes	characteristic
44	C₂×D₃₂	—	53	1	no	—
45	C₂×C₆₄	—	53	1	no	—
46	#1423	—	53	1	no	—
47	C₂×C₅₃	—	32	1	no	—
48	C₂×C₅₃	#1415	1	1	yes	characteristic
49	D₅₃	—	32	1	no	—
50	D₅₃	D₆₄	1	1	yes	characteristic
51	D₅₃	SD₁₂₈	1	1	yes	characteristic
52	C₂×D₁₆	—	106	1	no	—
53	D₃₂	—	53	1	no	—
54	D₃₂	—	106	1	no	—
55	D₃₂	#6157	1	1	yes	characteristic, complemented
56	C₂×C₃₂	—	53	1	no	—
57	#366	—	106	1	no	—
58	C₆₄	—	106	1	no	—
59	C₅₃	#7537	1	1	yes	characteristic, complemented, Sylow
60	C₂×D₈	—	212	1	no	—
61	D₁₆	—	2	1	no	—
62	D₁₆	—	106	1	no	—
63	D₁₆	—	212	1	no	—
64	C₂×C₁₆	—	53	1	no	—
65	#107	—	212	1	no	—
66	C₃₂	—	53	1	no	—
67	C₃₂	C₂×#6157	1	1	yes	characteristic
68	C₂×D₄	—	424	1	no	—
69	D₈	—	4	1	no	—
70	D₈	—	424	1	no	—
71	D₈	—	212	1	no	—
72	C₂×C₈	—	53	1	no	—
73	C₄⋊C₄	—	424	1	no	—
74	C₁₆	—	53	1	no	—
75	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
76	C₂³	—	848	1	no	—
77	D₄	—	8	1	no	—
78	D₄	—	848	1	no	—
79	D₄	—	424	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	848	1	no	—
82	C₈	—	53	1	no	—
83	C₈	#13133279	1	1	yes	characteristic
84	C₂²	—	16	1	no	—
85	C₂²	—	53	1	no	—
86	C₂²	—	848	1	no	—
87	C₂²	—	1696	1	no	—
88	C₂²	—	1696	1	no	—
89	C₄	—	53	1	no	—
90	C₄	—	1696	1	no	—
91	C₄	#16811535	1	1	yes	characteristic
92	C₂	—	32	1	no	—
93	C₂	—	53	1	no	—
94	C₂	—	53	1	no	—
95	C₂	—	1696	1	no	—
96	C₂	#20289864	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289864
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₃₂	C₂×#6157
Fitting	C₅₃×D₃₂	C₄
socle	C₂×C₅₃	#1415
prime radical	D₃₂×D₅₃	C₂
cosocle	C₃₂×D₅₃	C₂²
hypercenter	D₃₂	#6157
hypocenter	C₅₃	#7537
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ D₃₂

Complements

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

N	H ≅ G/N	type
1. D₅₃⋊C₆₄	C₂	semidirect
2. #20289876	C₂	semidirect
7. D₁₆₉₆	C₄	semidirect
10. C₅₃×D₃₂	C₄	semidirect
55. D₃₂	#6157	semidirect
59. C₅₃	#7537	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
#20289876	2
D₃₂×D₅₃	2
#7537	53