Subgroups — #20291138

Order 6784.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811555	C₂	1	3	yes	maximal, complemented
2	C₂×#13133267	C₂	1	3	yes	maximal, complemented
3	#16811040	C₂	1	1	yes	maximal, characteristic
4	#13133267	C₂²	1	1	yes	characteristic
5	#13133267	C₂²	1	3	yes	—
6	#13133157	C₄	1	3	yes	—
7	C₂×#11719908	C₂²	1	3	yes	—
8	#131×C₅₃	C₄	1	1	yes	characteristic
9	#41×C₅₃	C₈	1	1	yes	characteristic
10	M₁₆×C₅₃	C₈	1	3	yes	—
11	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
12	#11719908	C₂×C₄	1	3	yes	—
13	#11719908	C₂³	1	1	yes	characteristic
14	Q₈×C₅₃	C₁₆	1	1	yes	characteristic
15	D₄×C₅₃	C₁₆	1	3	yes	—
16	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
17	C₈×C₅₃	C₂×C₈	1	3	yes	—
18	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₅₃	C₂×C₁₆	1	3	yes	—
20	C₄×C₅₃	C₂×C₁₆	1	3	yes	—
21	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
22	#2258	—	53	1	no	maximal, Sylow
23	C₂×C₅₃	—	2	3	no	—
24	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
25	#504	—	53	1	no	—
26	M₆₄	—	53	3	no	—
27	C₂×C₃₂	—	53	3	no	—
28	C₅₃	#2258	1	1	yes	characteristic, complemented, Sylow
29	#131	#6157	1	1	yes	characteristic
30	M₃₂	—	53	3	no	—
31	C₂×C₁₆	—	53	3	no	—
32	C₃₂	—	53	1	no	—
33	C₃₂	—	53	3	no	—
34	#41	#88324	1	1	yes	characteristic
35	M₁₆	#88324	1	3	yes	—
36	C₂×C₈	C₂×#6157	1	3	yes	—
37	C₁₆	—	53	1	no	—
38	C₁₆	—	53	3	no	—
39	Q₈	#11719910	1	1	yes	characteristic
40	D₄	#11719910	1	3	yes	—
41	C₂×C₄	C₂×#88324	1	3	yes	—
42	C₈	C₂×#88324	1	3	yes	—
43	C₈	C₂²×#6157	1	1	yes	characteristic
44	C₂²	C₂×#11719910	1	3	yes	—
45	C₄	C₂×#11719910	1	3	yes	—
46	C₄	C₂²×#88324	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂²×#11719910	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×#6157
commutator, socle	C₂×C₅₃	C₂²×C₁₆
Fitting	#131×C₅₃	C₄
hypercenter	#131	#6157
hypocenter	C₅₃	#2258
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #16811555	C₂	semidirect
2. C₂×#13133267	C₂	semidirect
28. C₅₃	#2258	semidirect