Subgroups — #20274402

Order 6784.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133128	C₂	1	3	yes	maximal
2	C₂×C₄×#88322	C₂	1	3	yes	maximal
3	C₅₃×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₂²	1	1	yes	characteristic
5	C₂²×#88322	C₂²	1	3	yes	—
6	#13133128	C₄	1	12	yes	complemented
7	C₄×#88322	C₄	1	12	yes	complemented
8	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
9	#95×C₅₃	C₄	1	4	yes	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₄²×C₅₃	C₂×C₄	1	6	yes	—
13	C₄²×C₅₃	C₂×C₄	1	6	yes	—
14	C₂×#88322	C₂×C₄	1	6	yes	—
15	C₂×#88322	C₂×C₄	1	6	yes	—
16	C₂×#88322	C₂×C₄	1	12	yes	—
17	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	12	no	—
19	C₂×C₄×C₅₃	C₄²	1	12	yes	—
20	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
21	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
22	#88322	C₄²	1	4	yes	—
23	#88322	C₄²	1	12	yes	—
24	C₂²×C₅₃	C₂×C₄²	1	3	yes	—
25	C₂²×C₅₃	#131	1	4	yes	—
26	C₄×C₅₃	—	2	12	no	—
27	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
28	C₄×C₅₃	C₂×C₄²	1	3	yes	—
29	#1726	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	C₄³	1	1	yes	characteristic
31	C₂×C₅₃	#405	1	6	yes	—
32	C₂×#97	—	53	3	no	—
33	C₂×C₄×C₈	—	53	3	no	—
34	#376	D₅₃	1	1	yes	characteristic
35	C₅₃	#1726	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₈	—	53	1	no	—
37	C₂²×C₈	—	53	3	no	—
38	C₂×C₄²	C₂×D₅₃	1	3	yes	—
39	#97	—	53	12	no	—
40	C₄×C₈	—	53	12	no	—
41	#95	Dic₅₃	1	4	yes	—
42	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
43	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
44	C₂×C₈	—	53	6	no	—
45	C₂×C₈	—	53	6	no	—
46	C₂×C₈	—	53	12	no	—
47	C₄²	C₄×D₅₃	1	6	yes	—
48	C₄²	C₄×D₅₃	1	6	yes	—
49	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
50	C₂×C₄	—	2	12	no	—
51	C₂×C₄	C₄×Dic₅₃	1	12	yes	—
52	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
53	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
54	C₈	—	53	4	no	—
55	C₈	—	53	12	no	—
56	C₂²	C₂×C₄×Dic₅₃	1	3	yes	—
57	C₂²	#13133224	1	4	yes	—
58	C₄	—	2	12	no	—
59	C₄	C₄²×D₅₃	1	1	yes	characteristic
60	C₄	C₄²×D₅₃	1	3	yes	—
61	C₂	C₄²×Dic₅₃	1	1	yes	characteristic
62	C₂	#16810579	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₄³
Fitting	C₅₃×#376	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	#376	D₅₃
hypocenter	C₅₃	#1726
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

Complements

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

N	H ≅ G/N	type
6. #13133128	C₄	semidirect
7. C₄×#88322	C₄	semidirect
35. C₅₃	#1726	semidirect