Subgroups — #13133248

Order 1696.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719945	C₂	1	1	yes	maximal, characteristic
3	#11719925	C₂	1	2	yes	maximal, complemented
4	#11719924	C₂	1	2	yes	maximal, complemented
5	#11719917	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₅₃	—	2	1	no	—
7	Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₅₃	—	2	1	no	—
9	C₂×C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²⋊D₅₃	—	2	1	no	—
11	D₂₁₂	C₂²	1	1	yes	characteristic
12	C₄×D₅₃	—	2	1	no	—
13	Dic₁₀₆	C₂²	1	1	yes	characteristic
14	#88322	C₂²	1	2	yes	—
15	C₂²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×D₅₃	—	2	1	no	—
17	C₄×C₅₃	—	2	2	no	—
18	C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₄×C₅₃	C₂³	1	1	yes	characteristic
20	Dic₅₃	—	2	1	no	—
21	C₂×C₅₃	—	2	1	no	—
22	C₂×C₅₃	C₂×D₄	1	1	yes	characteristic
23	D₅₃	—	4	1	no	—
24	C₅₃	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	53	1	no	maximal, Sylow
26	#41	—	53	1	no	—
27	C₂×Q₈	D₅₃	1	1	yes	characteristic, complemented
28	Q₁₆	—	53	2	no	—
29	SD₁₆	—	53	2	no	—
30	M₁₆	—	53	1	no	—
31	Q₈	—	2	1	no	—
32	Q₈	—	53	1	no	—
33	Q₈	C₂×D₅₃	1	2	yes	—
34	D₄	—	53	1	no	—
35	D₄	—	106	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	106	1	no	—
38	C₂×C₄	C₂×D₅₃	1	1	yes	characteristic
39	C₈	—	53	2	no	—
40	C₂²	—	106	1	no	—
41	C₂²	C₂²⋊D₅₃	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	106	1	no	—
44	C₄	C₂²⋊D₅₃	1	1	yes	characteristic
45	C₄	C₂²×D₅₃	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	212	1	no	—
48	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×Q₈×C₅₃	C₂
socle	C₂×C₅₃	C₂×D₄
prime radical	#11719945	C₂
hypercenter	C₂×Q₈	D₅₃
hypocenter	C₅₃	#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₅₃	C₂	semidirect
3. #11719925	C₂	semidirect
4. #11719924	C₂	semidirect
5. #11719917	C₂	semidirect
24. C₅₃	#137	semidirect
27. C₂×Q₈	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₅₃	2
#11719945	2
#11719925	2
#11719924	2
#11719917	2
#137	53