Subgroups — #16810364

Order 3392.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13133209	C₂	1	1	yes	maximal, characteristic
2	#13133128	C₂	1	1	yes	maximal, characteristic
3	#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₅₃	—	2	1	no	—
5	C₄²×C₅₃	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
7	C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
8	#11719919	—	2	1	no	—
9	#11719919	—	2	1	no	—
10	C₂×#88322	—	2	1	no	—
11	C₂×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂×Dic₅₃	—	2	1	no	—
15	C₂×Dic₅₃	—	2	1	no	—
16	C₈×C₅₃	—	2	2	no	—
17	#88322	—	2	2	no	—
18	C₂²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₄×C₅₃	—	2	1	no	—
20	C₄×C₅₃	—	2	1	no	—
21	C₄×C₅₃	—	2	1	no	—
22	Dic₅₃	—	4	1	no	—
23	Dic₅₃	—	4	1	no	—
24	C₂×C₅₃	#101	1	1	yes	characteristic
25	C₂×C₅₃	#104	1	1	yes	characteristic
26	C₂×C₅₃	#104	1	1	yes	characteristic
27	#330	—	53	1	no	maximal, Sylow
28	C₅₃	#330	1	1	yes	characteristic, complemented, Sylow
29	#125	—	53	1	no	—
30	#97	—	53	1	no	—
31	#97	D₅₃	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	106	1	no	—
34	C₄⋊C₄	—	53	1	no	—
35	C₄⋊C₄	—	53	1	no	—
36	C₄⋊C₄	—	106	1	no	—
37	C₄⋊C₄	—	106	1	no	—
38	C₄²	C₂×D₅₃	1	1	yes	characteristic
39	C₂×C₄	—	106	1	no	—
40	C₂×C₄	—	106	1	no	—
41	C₂×C₄	C₄×D₅₃	1	1	yes	characteristic
42	C₂×C₄	D₂₁₂	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	—	106	2	no	—
46	C₂²	#11719921	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	212	1	no	—
51	C₄	—	212	1	no	—
52	C₂	#13133135	1	1	yes	characteristic
53	C₂	#13133143	1	1	yes	characteristic
54	C₂	#13133145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄
Frattini	C₄²	C₂×D₅₃
Fitting	#97×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₂²⋊C₄
cosocle	C₄²×C₅₃	C₂²
hypercenter	#97	D₅₃
hypocenter	C₅₃	#330
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
28. C₅₃	#330	semidirect

Maximal subgroups

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

subgroup	index
#13133209	2
#13133128	2
#97×C₅₃	2
#330	53