Subgroups — #20290642

Order 6784.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811489	C₂	1	6	yes	maximal
2	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88324	C₂²	1	4	yes	—
4	C₂×C₄×Dic₅₃	C₄	1	3	yes	—
5	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
6	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
8	C₂²×Dic₅₃	C₂×C₄	1	3	yes	—
9	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
10	C₂×#88324	—	2	12	no	—
11	C₄²×C₅₃	C₈	1	4	yes	complemented
12	C₄×Dic₅₃	—	2	6	no	—
13	C₄×Dic₅₃	—	2	6	no	—
14	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₃	—	2	3	no	—
16	C₂×C₄×C₅₃	—	2	3	no	—
17	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
18	C₂×Dic₅₃	—	2	3	no	—
19	C₂×Dic₅₃	—	2	3	no	—
20	C₂×Dic₅₃	M₁₆	1	6	yes	—
21	C₂×Dic₅₃	#41	1	6	yes	—
22	#88324	—	4	4	no	—
23	C₂²×C₅₃	#117	1	3	yes	—
24	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₅₃	C₂×M₁₆	1	3	yes	—
26	C₄×C₅₃	—	2	6	no	—
27	Dic₅₃	—	2	6	no	—
28	Dic₅₃	#126	1	4	yes	—
29	#1839	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	#383	1	1	yes	characteristic
31	C₂×C₅₃	#431	1	3	yes	—
32	C₂×C₅₃	#432	1	3	yes	—
33	C₄³	—	53	1	no	—
34	#336	—	53	6	no	—
35	C₅₃	#1839	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₈	—	53	4	no	—
37	C₂×C₄²	—	53	3	no	—
38	C₂×C₄²	—	53	3	no	—
39	C₂×C₄²	#6157	1	1	yes	characteristic
40	C₂²×C₄	—	53	1	no	—
41	C₂²×C₄	—	53	3	no	—
42	C₂²×C₄	C₂×#6157	1	3	yes	—
43	C₂×C₈	—	106	12	no	—
44	C₄²	—	106	6	no	—
45	C₄²	—	106	6	no	—
46	C₄²	#88324	1	4	yes	complemented
47	C₂³	C₂²×#6157	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	53	6	no	—
51	C₂×C₄	—	53	6	no	—
52	C₂×C₄	—	106	3	no	—
53	C₂×C₄	—	106	3	no	—
54	C₂×C₄	C₂×#88324	1	6	yes	—
55	C₈	—	212	4	no	—
56	C₂²	#13133296	1	3	yes	—
57	C₂²	C₂²×#88324	1	1	yes	characteristic
58	C₂²	D₅₃⋊M₁₆	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	53	4	no	—
61	C₄	—	106	6	no	—
62	C₂	#16811568	1	1	yes	characteristic
63	C₂	#16811639	1	3	yes	—
64	C₂	#16811646	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄²	#6157
hypocenter	C₅₃	#1839
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
11. C₄²×C₅₃	C₈	semidirect
35. C₅₃	#1839	semidirect
46. C₄²	#88324	semidirect