Subgroups — #20276508

Order 6784.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133156	C₂	1	1	yes	maximal, characteristic
2	#108×D₅₃	C₂	1	2	yes	maximal
3	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810452	C₂	1	2	yes	maximal, complemented
5	C₅₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×D₅₃	—	2	2	no	—
7	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	#13133156	C₄	1	4	yes	—
9	C₂×#11719908	C₂²	1	1	yes	characteristic
10	#13133153	C₂²	1	2	yes	—
11	#13133152	—	2	2	no	—
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#108×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
15	M₁₆×C₅₃	—	2	2	no	—
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	#11719917	—	2	2	no	—
18	C₂×#88322	D₄	1	1	yes	characteristic
19	#11719912	—	4	2	no	—
20	C₈×D₅₃	—	4	2	no	—
21	C₈×D₅₃	C₂×C₄	1	2	yes	—
22	C₈×D₅₃	Q₈	1	2	yes	—
23	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	#11719908	C₂×C₄	1	2	yes	—
25	C₂²×D₅₃	Q₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	Q₁₆	1	1	yes	characteristic
28	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
29	C₄×D₅₃	D₈	1	2	yes	—
30	C₈×C₅₃	—	4	2	no	—
31	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	#88322	—	4	2	no	—
34	#88322	C₄⋊C₄	1	2	yes	—
35	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×D₅₃	—	4	1	no	—
37	C₂×D₅₃	#107	1	2	yes	—
38	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
40	Dic₅₃	#107	1	2	yes	—
41	#2161	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
44	D₅₃	—	2	2	no	—
45	C₂×M₃₂	—	53	1	no	—
46	C₂×#108	—	53	2	no	—
47	#368	—	53	1	no	—
48	#368	—	53	2	no	—
49	#368	D₅₃	1	1	yes	characteristic, complemented
50	C₅₃	#2161	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₁₆	—	106	2	no	—
52	C₂²×C₈	—	53	1	no	—
53	M₃₂	—	53	4	no	—
54	C₂×C₁₆	—	53	1	no	—
55	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
56	#108	—	53	2	no	—
57	#108	—	106	2	no	—
58	#108	C₂×D₅₃	1	2	yes	—
59	C₂²×C₄	—	53	1	no	—
60	M₁₆	—	2	2	no	—
61	M₁₆	—	106	2	no	—
62	M₁₆	—	212	2	no	—
63	C₂×C₈	—	53	1	no	—
64	C₂×C₈	—	53	2	no	—
65	C₂×C₈	—	53	2	no	—
66	C₂×C₈	—	212	2	no	—
67	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
68	C₁₆	—	53	2	no	—
69	C₁₆	C₄×D₅₃	1	2	yes	—
70	C₂³	—	53	1	no	—
71	C₂×C₄	—	53	1	no	—
72	C₂×C₄	—	53	2	no	—
73	C₂×C₄	—	53	2	no	—
74	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
75	C₈	—	4	2	no	—
76	C₈	—	53	2	no	—
77	C₈	—	212	2	no	—
78	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
79	C₈	Q₈×D₅₃	1	1	yes	characteristic
80	C₂²	—	53	2	no	—
81	C₂²	—	212	1	no	—
82	C₂²	Q₁₆×D₅₃	1	1	yes	characteristic
83	C₄	—	53	2	no	—
84	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
85	C₄	D₈×D₅₃	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	106	2	no	—
88	C₂	#107×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#368	C₂
socle	C₂×C₅₃	C₂×#107
hypercenter	#368	D₅₃
hypocenter	C₅₃	#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
3. #16810453	C₂	semidirect
4. #16810452	C₂	semidirect
5. C₅₃×#368	C₂	semidirect
49. #368	D₅₃	semidirect
50. C₅₃	#2161	semidirect