Subgroups — #16810407

Order 3392.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₅₃
Fitting	#131×C₅₃	C₂
socle	C₂×C₅₃	C₂²⋊C₈
prime radical	#41×C₅₃	C₄
cosocle	C₂×C₈×C₅₃	C₂²
hypercenter	#131	D₅₃
hypocenter	C₅₃	#350
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #13133157	C₂	semidirect
2. C₂×#11719908	C₂	semidirect
28. C₅₃	#350	semidirect

Maximal subgroups

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

subgroup	index
#13133157	2
C₂×#11719908	2
#131×C₅₃	2
#350	53