Subgroups — #20272585

Order 6784.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810680	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	—	2	2	no	—
4	C₂²⋊#88322	—	2	2	no	—
5	#13133128	C₄	1	4	yes	—
6	C₂²×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	2	no	—
13	C₂×C₈×C₅₃	—	4	2	no	—
14	C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₄²×C₅₃	D₄	1	2	yes	—
16	C₂×#88322	—	2	4	no	—
17	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	2	no	—
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₈×C₅₃	—	4	2	no	—
27	#88322	—	4	4	no	—
28	C₂²×C₅₃	—	2	1	no	—
29	C₂²×C₅₃	—	2	1	no	—
30	C₂²×C₅₃	#95	1	1	yes	characteristic
31	C₂²×C₅₃	#108	1	2	yes	—
32	C₄×C₅₃	—	2	2	no	—
33	C₄×C₅₃	—	2	2	no	—
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	#100	1	1	yes	characteristic
36	C₄×C₅₃	#101	1	1	yes	characteristic
37	#1290	—	53	1	no	maximal, Sylow
38	C₂×C₅₃	—	2	2	no	—
39	C₂×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
40	C₂×C₅₃	#341	1	1	yes	characteristic
41	C₂×C₅₃	#343	1	1	yes	characteristic
42	#432	—	53	2	no	—
43	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
44	C₅₃	#1290	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	Dic₅₃	1	2	yes	—
46	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	106	2	no	—
49	C₂²⋊C₈	—	106	2	no	—
50	#97	—	53	4	no	—
51	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
52	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
53	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	106	4	no	—
57	C₄²	C₄×D₅₃	1	2	yes	—
58	C₄²	C₂²⋊D₅₃	1	2	yes	—
59	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
65	C₂×C₄	#11719919	1	2	yes	—
66	C₂×C₄	#11719921	1	2	yes	—
67	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
68	C₈	—	4	2	no	—
69	C₈	—	212	4	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	#13133124	1	1	yes	characteristic
73	C₂²	#13133153	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	#13133134	1	1	yes	characteristic
78	C₄	#13133136	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	#16810380	1	1	yes	characteristic
81	C₂	#16810388	1	1	yes	characteristic
82	C₂	#16810392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1290
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₅₃	#1290	semidirect