Subgroups — #16810406

Order 3392.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133223	C₂	1	1	yes	maximal, characteristic
2	#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719945	C₄	1	1	yes	characteristic
5	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	#11719917	C₄	1	1	yes	characteristic
7	#11719912	—	2	1	no	—
8	C₈×D₅₃	—	2	1	no	—
9	C₁₆×C₅₃	—	2	1	no	—
10	#11719908	—	2	1	no	—
11	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²⋊D₅₃	—	2	1	no	—
13	D₂₁₂	C₈	1	1	yes	characteristic
14	C₄×D₅₃	—	2	1	no	—
15	Dic₁₀₆	C₈	1	1	yes	characteristic
16	C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₈×C₅₃	D₄	1	1	yes	characteristic
18	#88322	—	2	1	no	—
19	C₂²×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂×D₅₃	—	2	1	no	—
21	C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
23	Dic₅₃	—	2	1	no	—
24	C₂×C₅₃	—	2	1	no	—
25	C₂×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
26	D₅₃	—	4	1	no	—
27	#350	—	53	1	no	maximal, Sylow
28	C₅₃	#350	1	1	yes	characteristic, complemented, Sylow
29	#131	—	53	1	no	—
30	M₃₂	—	53	1	no	—
31	C₂×C₁₆	D₅₃	1	1	yes	characteristic, complemented
32	#41	—	53	1	no	—
33	M₁₆	—	53	1	no	—
34	M₁₆	—	106	1	no	—
35	C₂×C₈	—	106	1	no	—
36	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	106	1	no	—
39	Q₈	—	53	1	no	—
40	D₄	—	53	1	no	—
41	D₄	—	106	1	no	—
42	C₂×C₄	—	106	1	no	—
43	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
44	C₈	—	106	1	no	—
45	C₈	D₂₁₂	1	1	yes	characteristic
46	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
47	C₂²	—	106	1	no	—
48	C₂²	#11719912	1	1	yes	characteristic
49	C₄	—	106	1	no	—
50	C₄	C₈×D₅₃	1	1	yes	characteristic
51	C₄	#11719921	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	212	1	no	—
54	C₂	#13133129	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₁₂
commutator	C₄×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₅₃
Fitting	C₂×C₁₆×C₅₃	C₂
socle	C₂×C₅₃	C₂²⋊C₈
prime radical	#11719945	C₄
cosocle	C₂×C₈×C₅₃	C₂²
hypercenter	C₂×C₁₆	D₅₃
hypocenter	C₅₃	#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13133157	C₂	semidirect
3. C₂×C₁₆×C₅₃	C₂	semidirect
28. C₅₃	#350	semidirect
31. C₂×C₁₆	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133223	2
#13133157	2
C₂×C₁₆×C₅₃	2
#350	53