Subgroups — #20272665

Order 6784.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810646	C₂	1	1	yes	maximal, characteristic
2	#16810405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#349	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
6	#13133157	—	2	1	no	—
7	#13133146	—	2	1	no	—
8	#13133146	—	2	1	no	—
9	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	M₃₂×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×Dic₅₃	—	2	1	no	—
13	M₁₆×C₅₃	—	4	1	no	—
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄×Dic₅₃	—	2	1	no	—
17	C₄×Dic₅₃	C₈	1	1	yes	characteristic
18	C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	#11719917	—	4	1	no	—
20	C₂×#88322	—	2	1	no	—
21	C₁₆×C₅₃	—	4	1	no	—
22	#11719908	—	4	1	no	—
23	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₅₃	—	2	1	no	—
29	C₂×Dic₅₃	—	2	1	no	—
30	C₂×Dic₅₃	—	4	2	no	—
31	C₈×C₅₃	—	2	1	no	—
32	C₈×C₅₃	—	2	1	no	—
33	#88322	—	4	1	no	—
34	C₂²×C₅₃	—	2	1	no	—
35	C₂²×C₅₃	—	4	1	no	—
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	#99	1	1	yes	characteristic
39	C₄×C₅₃	#101	1	1	yes	characteristic
40	Dic₅₃	—	2	2	no	—
41	Dic₅₃	—	4	1	no	—
42	#1327	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	—	4	1	no	—
45	C₂×C₅₃	#324	1	1	yes	characteristic
46	#423	—	53	1	no	—
47	#349	—	53	1	no	—
48	#349	D₅₃	1	1	yes	characteristic
49	C₅₃	#1327	1	1	yes	characteristic, complemented, Sylow
50	C₂×M₁₆	—	53	1	no	—
51	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
52	C₂×C₄²	—	53	1	no	—
53	M₃₂	—	2	1	no	—
54	M₃₂	—	106	1	no	—
55	C₄⋊C₈	—	106	1	no	—
56	C₄⋊C₈	—	106	1	no	—
57	C₂²×C₄	—	106	1	no	—
58	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
59	M₁₆	—	4	1	no	—
60	M₁₆	—	212	1	no	—
61	C₂×C₈	—	106	1	no	—
62	C₂×C₈	D₂₁₂	1	1	yes	characteristic
63	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₄²	—	53	1	no	—
65	C₄²	—	53	1	no	—
66	C₄²	—	106	1	no	—
67	C₁₆	—	4	1	no	—
68	C₁₆	—	212	1	no	—
69	C₂³	#11719912	1	1	yes	characteristic
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	106	1	no	—
73	C₂×C₄	—	106	1	no	—
74	C₂×C₄	—	212	2	no	—
75	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
76	C₂×C₄	#11719921	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	212	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	#13133129	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	106	2	no	—
85	C₄	—	212	1	no	—
86	C₄	#13133131	1	1	yes	characteristic
87	C₄	#13133135	1	1	yes	characteristic
88	C₂	—	2	1	no	—
89	C₂	—	4	1	no	—
90	C₂	#16810348	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133131
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×#349	C₂
socle	C₂×C₅₃	#324
hypercenter	#349	D₅₃
hypocenter	C₅₃	#1327
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

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