Subgroups — #20272666

Order 6784.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810405	C₂	1	2	yes	maximal
2	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	2	2	no	—
4	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₅₃	—	2	2	no	—
8	C₂²×C₄×C₅₃	—	2	1	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	M₁₆×C₅₃	—	4	1	no	—
11	M₁₆×C₅₃	—	4	1	no	—
12	C₂×C₈×C₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	D₄	1	2	yes	—
14	C₄²×C₅₃	—	2	1	no	—
15	C₄²×C₅₃	C₈	1	1	yes	characteristic
16	C₄²×C₅₃	C₈	1	1	yes	characteristic
17	#11719908	—	4	2	no	—
18	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	—	4	2	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₈×C₅₃	—	2	2	no	—
27	C₈×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	—	2	1	no	—
29	C₂²×C₅₃	—	4	1	no	—
30	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₅₃	—	2	2	no	—
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	—	4	1	no	—
34	C₄×C₅₃	#99	1	1	yes	characteristic
35	C₄×C₅₃	#101	1	1	yes	characteristic
36	#1327	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	—	4	1	no	—
39	C₂×C₅₃	#324	1	1	yes	characteristic
40	#423	D₅₃	1	1	yes	characteristic
41	#349	—	53	2	no	—
42	C₅₃	#1327	1	1	yes	characteristic, complemented, Sylow
43	C₂×M₁₆	Dic₅₃	1	1	yes	characteristic
44	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
45	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
46	M₃₂	—	106	2	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
50	M₁₆	—	4	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
54	C₄²	—	2	1	no	—
55	C₄²	#88322	1	1	yes	characteristic
56	C₄²	#88322	1	1	yes	characteristic
57	C₁₆	—	212	2	no	—
58	C₂³	#11719917	1	1	yes	characteristic
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	C₂×#88322	1	1	yes	characteristic
65	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₈	—	4	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	C₂²⋊#88322	1	1	yes	characteristic
71	C₄	—	2	2	no	—
72	C₄	—	2	1	no	—
73	C₄	—	4	1	no	—
74	C₄	#13133132	1	1	yes	characteristic
75	C₄	#13133136	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	#16810349	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂×C₅₃	#324
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1327
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #423

Complements

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

N	H ≅ G/N	type
42. C₅₃	#1327	semidirect