Subgroups — #13133243

Order 1696.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719945	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719923	C₂	1	2	yes	maximal, complemented
4	#11719922	C₂	1	2	yes	maximal, complemented
5	#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₅₃	—	2	1	no	—
7	D₄×C₅₃	—	2	1	no	—
8	D₄×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²⋊D₅₃	—	2	1	no	—
11	D₂₁₂	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₅₃	—	2	1	no	—
13	Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
14	#88322	C₂²	1	2	yes	complemented
15	C₂²×C₅₃	—	2	2	no	—
16	C₂²×C₅₃	—	4	2	no	—
17	C₂²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×D₅₃	—	2	1	no	—
19	C₄×C₅₃	D₄	1	1	yes	characteristic
20	C₄×C₅₃	C₂³	1	1	yes	characteristic
21	Dic₅₃	—	2	1	no	—
22	C₂×C₅₃	—	2	1	no	—
23	C₂×C₅₃	—	4	2	no	—
24	C₂×C₅₃	C₂×D₄	1	1	yes	characteristic
25	D₅₃	—	4	1	no	—
26	C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	53	1	no	maximal, Sylow
28	#41	—	53	1	no	—
29	C₂×D₄	D₅₃	1	1	yes	characteristic, complemented
30	SD₁₆	—	53	2	no	—
31	D₈	—	53	2	no	—
32	M₁₆	—	53	1	no	—
33	C₂³	—	2	1	no	—
34	Q₈	—	53	1	no	—
35	D₄	—	2	1	no	—
36	D₄	—	53	1	no	—
37	D₄	—	106	1	no	—
38	D₄	C₂×D₅₃	1	2	yes	—
39	C₂×C₄	—	106	1	no	—
40	C₂×C₄	C₂×D₅₃	1	1	yes	characteristic
41	C₈	—	53	2	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	4	2	no	—
44	C₂²	—	106	1	no	—
45	C₂²	C₂²⋊D₅₃	1	1	yes	characteristic
46	C₄	—	106	1	no	—
47	C₄	C₂²⋊D₅₃	1	1	yes	characteristic
48	C₄	C₂²×D₅₃	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	212	1	no	—
52	C₂	C₂×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₅₃
commutator, cosocle	C₄×C₅₃	C₂³
Frattini	C₄	C₂²×D₅₃
Fitting	C₂×D₄×C₅₃	C₂
socle	C₂×C₅₃	C₂×D₄
hypercenter	C₂×D₄	D₅₃
hypocenter	C₅₃	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₅₃	C₂	semidirect
2. #11719945	C₂	semidirect
3. #11719923	C₂	semidirect
4. #11719922	C₂	semidirect
5. #11719917	C₂	semidirect
11. D₂₁₂	C₂²	semidirect
13. Dic₁₀₆	C₂²	semidirect
14. #88322	C₂²	semidirect
26. C₅₃	AGL(1,ℤ/8)	semidirect
29. C₂×D₄	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₅₃	2
#11719945	2
#11719923	2
#11719922	2
#11719917	2
AGL(1,ℤ/8)	53