Subgroups — #16810404

Order 3392.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719912	C₂	1	1	yes	maximal, characteristic
2	#13133157	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₅₃	C₄	1	1	yes	characteristic
5	C₂×C₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#88322	C₄	1	1	yes	characteristic
7	#11719912	—	2	1	no	—
8	#11719912	—	2	1	no	—
9	C₁₆×C₅₃	—	2	1	no	—
10	#11719908	—	2	1	no	—
11	C₂²×D₅₃	C₈	1	1	yes	characteristic
12	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×Dic₅₃	C₈	1	1	yes	characteristic
14	C₄×D₅₃	—	2	1	no	—
15	C₄×D₅₃	—	2	1	no	—
16	C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₈×C₅₃	D₄	1	1	yes	characteristic
18	#88322	—	2	1	no	—
19	C₂²×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂×D₅₃	—	2	1	no	—
21	C₂×D₅₃	—	4	1	no	—
22	C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₅₃	M₁₆	1	1	yes	characteristic
24	Dic₅₃	—	2	1	no	—
25	C₂×C₅₃	—	2	1	no	—
26	C₂×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
27	D₅₃	—	4	1	no	—
28	#349	—	53	1	no	maximal, Sylow
29	C₅₃	#349	1	1	yes	characteristic, complemented, Sylow
30	C₂×M₁₆	—	53	1	no	—
31	M₃₂	—	53	1	no	—
32	M₃₂	D₅₃	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	53	1	no	—
34	M₁₆	—	106	1	no	—
35	M₁₆	—	106	1	no	—
36	C₂×C₈	—	53	1	no	—
37	C₂×C₈	C₂×D₅₃	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	—	106	1	no	—
40	C₂³	—	53	1	no	—
41	C₂×C₄	—	53	1	no	—
42	C₂×C₄	—	106	1	no	—
43	C₂×C₄	—	106	1	no	—
44	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
45	C₈	—	106	1	no	—
46	C₈	D₂₁₂	1	1	yes	characteristic
47	C₈	C₂²⋊D₅₃	1	1	yes	characteristic
48	C₂²	—	106	1	no	—
49	C₂²	—	212	1	no	—
50	C₂²	C₈×D₅₃	1	1	yes	characteristic
51	C₄	—	106	1	no	—
52	C₄	#11719912	1	1	yes	characteristic
53	C₄	#11719921	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	212	1	no	—
56	C₂	#13133129	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11719921
commutator	C₂²×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×D₅₃
Fitting	M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂²⋊C₈
prime radical	C₂²×D₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂²
hypercenter	M₃₂	D₅₃
hypocenter	C₅₃	#349
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #13133157	C₂	semidirect
3. M₃₂×C₅₃	C₂	semidirect
29. C₅₃	#349	semidirect
32. M₃₂	D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719912	2
#13133157	2
M₃₂×C₅₃	2
#349	53