Subgroups — #20276601

Order 6784.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810740	C₂	1	2	yes	maximal, complemented
2	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810430	C₂	1	2	yes	maximal, complemented
4	#16810398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#508	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	2	no	—
7	C₂×#11719908	C₂²	1	1	yes	characteristic
8	#13133153	C₂²	1	2	yes	—
9	#13133144	—	2	2	no	—
10	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	#135×C₅₃	C₂²	1	2	yes	—
12	Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
13	SD₃₂×C₅₃	C₄	1	2	yes	complemented
14	D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#41×C₅₃	—	2	2	no	—
17	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
18	SD₁₆×C₅₃	—	2	2	no	—
19	D₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	#11719917	—	2	2	no	—
23	#11719917	—	4	2	no	—
24	C₂×#88322	—	4	2	no	—
25	C₂×#88322	D₄	1	1	yes	characteristic
26	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#11719908	—	2	1	no	—
29	Q₈×C₅₃	—	2	2	no	—
30	D₄×C₅₃	—	2	2	no	—
31	D₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₅₃	#41	1	1	yes	characteristic
37	#88322	—	4	2	no	—
38	#88322	D₈	1	2	yes	—
39	C₂²×C₅₃	—	4	2	no	—
40	C₂²×C₅₃	#135	1	1	yes	characteristic
41	C₄×C₅₃	—	4	2	no	—
42	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	Dic₅₃	—	4	1	no	—
45	#2179	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	—	8	2	no	—
48	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
49	#508	D₅₃	1	1	yes	characteristic
50	#443	—	53	2	no	—
51	#368	—	53	1	no	—
52	#359	—	53	2	no	—
53	#346	—	53	1	no	—
54	C₅₃	#2179	1	1	yes	characteristic, complemented, Sylow
55	#135	C₂×D₅₃	1	2	yes	—
56	#131	—	106	2	no	—
57	Q₃₂	Dic₅₃	1	1	yes	characteristic, complemented
58	SD₃₂	Dic₅₃	1	2	yes	complemented
59	D₁₆	Dic₅₃	1	1	yes	characteristic, complemented
60	C₂×C₁₆	—	53	1	no	—
61	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
62	#108	—	53	2	no	—
63	#104	—	106	2	no	—
64	C₄×C₈	—	53	1	no	—
65	#41	—	2	2	no	—
66	Q₁₆	C₂×Dic₅₃	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×Dic₅₃	1	2	yes	—
69	M₁₆	—	106	2	no	—
70	M₁₆	—	212	2	no	—
71	C₂×C₈	—	53	1	no	—
72	C₂×C₈	—	212	2	no	—
73	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
74	C₄²	—	53	1	no	—
75	C₁₆	—	106	1	no	—
76	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
77	C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	106	1	no	—
83	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
84	C₈	—	53	2	no	—
85	C₈	—	212	2	no	—
86	C₈	#11719947	1	1	yes	characteristic
87	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#13133236	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	212	1	no	—
92	C₄	D₄×Dic₅₃	1	1	yes	characteristic
93	C₄	D₈×D₅₃	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	D₈×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#508	C₂
socle	C₂×C₅₃	C₄×D₈
hypercenter	#508	D₅₃
hypocenter	C₅₃	#2179
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #508

Complements

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

N	H ≅ G/N	type
1. #16810740	C₂	semidirect
2. #16810453	C₂	semidirect
3. #16810430	C₂	semidirect
4. #16810398	C₂	semidirect
12. Q₃₂×C₅₃	C₄	semidirect
13. SD₃₂×C₅₃	C₄	semidirect
14. D₁₆×C₅₃	C₄	semidirect
54. C₅₃	#2179	semidirect
57. Q₃₂	Dic₅₃	semidirect
58. SD₃₂	Dic₅₃	semidirect
59. D₁₆	Dic₅₃	semidirect