Subgroups — #16810435

Order 3392.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₂	—	2	1	no	—
5	M₁₆×C₅₃	—	2	1	no	—
6	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	D₄₂₄	—	2	1	no	—
8	D₄₂₄	C₄	1	2	yes	—
9	#11719908	—	2	1	no	—
10	C₂²×D₅₃	—	4	1	no	—
11	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
12	D₂₁₂	—	2	2	no	—
13	D₂₁₂	—	4	1	no	—
14	C₈×C₅₃	—	4	1	no	—
15	C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₅₃	SD₁₆	1	1	yes	characteristic
18	C₂×D₅₃	—	4	2	no	—
19	C₂×D₅₃	—	8	2	no	—
20	C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₅₃	D₈	1	1	yes	characteristic
22	C₂×C₅₃	—	2	1	no	—
23	C₂×C₅₃	#102	1	1	yes	characteristic
24	D₅₃	—	8	2	no	—
25	#361	—	53	1	no	maximal, Sylow
26	C₅₃	#361	1	1	yes	characteristic, complemented, Sylow
27	C₂×D₈	—	53	1	no	—
28	M₃₂	—	53	1	no	—
29	#108	D₅₃	1	1	yes	characteristic, complemented
30	C₂×D₄	—	106	1	no	—
31	D₈	—	53	2	no	—
32	D₈	—	106	1	no	—
33	M₁₆	—	2	1	no	—
34	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
35	C₁₆	—	106	1	no	—
36	C₂³	—	212	1	no	—
37	D₄	—	106	2	no	—
38	D₄	—	212	1	no	—
39	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
40	C₈	—	4	1	no	—
41	C₈	C₄×D₅₃	1	1	yes	characteristic
42	C₈	D₂₁₂	1	1	yes	characteristic
43	C₂²	—	424	2	no	—
44	C₂²	—	212	2	no	—
45	C₂²	#11719924	1	1	yes	characteristic
46	C₄	#11719921	1	1	yes	characteristic
47	C₄	#11719922	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	424	2	no	—
50	C₂	#13133137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13133137
commutator	C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₅₃
Fitting	#108×C₅₃	C₂
socle	C₂×C₅₃	#102
prime radical	C₂×D₄₂₄	C₂
cosocle	C₂×C₈×C₅₃	C₂²
hypercenter	#108	D₅₃
hypocenter	C₅₃	#361
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #13133157	C₂	semidirect
3. #108×C₅₃	C₂	semidirect
26. C₅₃	#361	semidirect
29. #108	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₂₄	2
#13133157	2
#108×C₅₃	2
#361	53