Subgroups — #28299311

Order 13568.

82 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
2	C₃₂×#6157	C₂	1	1	yes	maximal, characteristic
3	C₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊C₃₂	C₄	1	1	yes	characteristic
5	C₁₆×#6157	C₄	1	1	yes	characteristic
6	#16811462	C₄	1	2	yes	complemented
7	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
8	#16810342	C₄	1	1	yes	characteristic, complemented
9	C₅₃×C₆₄	C₄	1	1	yes	characteristic, complemented
10	D₅₃⋊C₁₆	C₈	1	1	yes	characteristic
11	C₈×#6157	C₈	1	1	yes	characteristic
12	#13133267	C₈	1	2	yes	—
13	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
14	#13133123	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄×#6157	C₁₆	1	1	yes	characteristic
17	D₅₃⋊C₈	C₁₆	1	1	yes	characteristic
18	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
19	#11719910	C₁₆	1	2	yes	—
20	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
21	#11719908	C₂×C₈	1	1	yes	characteristic
22	C₂×#6157	C₃₂	1	2	yes	—
23	C₄×D₅₃	C₂×C₁₆	1	1	yes	characteristic
24	#88324	C₃₂	1	2	yes	—
25	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
26	#88322	C₂×C₁₆	1	1	yes	characteristic
27	#7510	—	53	1	no	maximal, Sylow
28	C₂×D₅₃	C₂×C₃₂	1	1	yes	characteristic
29	#6157	—	2	2	no	—
30	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
31	Dic₅₃	C₂×C₃₂	1	1	yes	characteristic
32	C₂×C₆₄	—	53	1	no	—
33	C₂×C₆₄	—	53	1	no	—
34	C₄×C₃₂	—	53	1	no	—
35	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
36	D₅₃	M₁₂₈	1	2	yes	—
37	C₂×C₃₂	—	53	1	no	—
38	C₂×C₃₂	—	53	1	no	—
39	C₄×C₁₆	—	53	1	no	—
40	C₆₄	—	53	2	no	—
41	C₆₄	—	53	1	no	—
42	C₆₄	#6157	1	1	yes	characteristic, complemented
43	C₅₃	#7510	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₁₆	—	53	1	no	—
45	C₂×C₁₆	—	53	1	no	—
46	C₄×C₈	—	53	1	no	—
47	C₃₂	—	53	2	no	—
48	C₃₂	—	53	1	no	—
49	C₃₂	C₂×#6157	1	1	yes	characteristic
50	C₂×C₈	—	53	1	no	—
51	C₂×C₈	—	53	1	no	—
52	C₄²	—	53	1	no	—
53	C₁₆	—	53	2	no	—
54	C₁₆	—	53	1	no	—
55	C₁₆	C₄×#6157	1	1	yes	characteristic
56	C₂×C₄	—	53	1	no	—
57	C₂×C₄	—	53	2	no	—
58	C₈	—	53	2	no	—
59	C₈	—	53	1	no	—
60	C₈	C₈×#6157	1	1	yes	characteristic
61	C₂²	—	53	1	no	—
62	C₄	—	53	1	no	—
63	C₄	—	106	2	no	—
64	C₄	C₁₆×#6157	1	1	yes	characteristic
65	C₂	—	53	2	no	—
66	C₂	C₃₂×#6157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂×#6157
commutator, socle	C₂×C₅₃	C₄×C₃₂
Fitting	C₅₃×C₆₄	C₄
prime radical	C₂×D₅₃	C₂×C₃₂
cosocle	C₃₂×D₅₃	C₂²
hypercenter	C₆₄	#6157
hypocenter	C₅₃	#7510
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₆₄

Complements

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

N	H ≅ G/N	type
6. #16811462	C₄	semidirect
8. #16810342	C₄	semidirect
9. C₅₃×C₆₄	C₄	semidirect
42. C₆₄	#6157	semidirect
43. C₅₃	#7510	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
C₃₂×#6157	2
C₆₄×D₅₃	2
#7510	53