Subgroups — #20273014

Order 6784.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810674	C₂	1	3	yes	maximal
2	#16810345	C₂	1	3	yes	maximal, complemented
3	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₅₃	C₂²	1	1	yes	characteristic
5	#13133146	C₄	1	6	yes	—
6	#13133129	C₄	1	6	yes	—
7	C₄×#88322	C₂²	1	3	yes	—
8	C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	—
10	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
11	C₄²×C₅₃	C₂³	1	1	yes	characteristic
12	C₄×Dic₅₃	C₂×C₄	1	3	yes	—
13	C₂×#88322	C₂×C₄	1	6	yes	—
14	C₂²×D₅₃	C₄²	1	1	yes	characteristic
15	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
16	C₂×Dic₅₃	C₄²	1	3	yes	—
17	C₄×D₅₃	—	2	6	no	—
18	C₈×C₅₃	—	2	6	no	—
19	#88322	—	2	6	no	—
20	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
21	C₂×D₅₃	—	2	3	no	—
22	C₄×C₅₃	#131	1	6	yes	—
23	Dic₅₃	—	2	3	no	—
24	#1454	—	53	1	no	maximal, Sylow
25	C₂×C₅₃	#405	1	3	yes	—
26	D₅₃	—	4	1	no	—
27	#431	—	53	3	no	—
28	#322	—	53	3	no	—
29	C₈²	D₅₃	1	1	yes	characteristic, complemented
30	C₅₃	#1454	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₄²	—	53	1	no	—
32	C₄⋊C₈	—	53	6	no	—
33	C₂²⋊C₈	—	53	6	no	—
34	C₄×C₈	—	53	3	no	—
35	C₄×C₈	C₂×D₅₃	1	3	yes	—
36	C₂²×C₄	—	53	3	no	—
37	C₂×C₈	—	53	6	no	—
38	C₂×C₈	C₄×D₅₃	1	6	yes	—
39	C₄²	—	53	3	no	—
40	C₄²	C₂²×D₅₃	1	1	yes	characteristic
41	C₂³	—	53	1	no	—
42	C₂×C₄	—	53	3	no	—
43	C₂×C₄	—	106	6	no	—
44	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
45	C₈	—	2	6	no	—
46	C₈	—	106	6	no	—
47	C₂²	—	106	3	no	—
48	C₂²	C₄²×D₅₃	1	1	yes	characteristic
49	C₄	—	106	3	no	—
50	C₄	#13133223	1	6	yes	—
51	C₂	—	212	1	no	—
52	C₂	#16810577	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₈²×C₅₃	C₂
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1454
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #16810345	C₂	semidirect
3. C₈²×C₅₃	C₂	semidirect
29. C₈²	D₅₃	semidirect
30. C₅₃	#1454	semidirect