Subgroups — #20272875

Order 6784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810420	C₂	1	1	yes	maximal, characteristic
2	#16810418	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#497	C₂	1	1	yes	maximal, characteristic
4	#13133136	—	2	1	no	—
5	#13133132	—	2	1	no	—
6	C₂×Q₁₆×C₅₃	—	2	1	no	—
7	C₂×SD₁₆×C₅₃	—	2	1	no	—
8	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
9	#124×C₅₃	C₂²	1	1	yes	characteristic
10	#97×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×Q₈×C₅₃	—	2	1	no	—
12	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
14	Q₁₆×C₅₃	—	4	2	no	—
15	SD₁₆×C₅₃	—	4	2	no	—
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₄⋊C₄×C₅₃	—	4	1	no	—
18	C₂²⋊C₄×C₅₃	—	4	1	no	—
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²⋊Dic₅₃	—	4	1	no	—
21	#11719917	—	4	1	no	—
22	C₂³×C₅₃	—	2	1	no	—
23	Q₈×C₅₃	—	4	1	no	—
24	Q₈×C₅₃	—	4	1	no	—
25	Q₈×C₅₃	—	4	1	no	—
26	D₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	8	1	no	—
32	C₈×C₅₃	—	2	2	no	—
33	#88322	—	8	1	no	—
34	C₂²×C₅₃	—	4	1	no	—
35	C₂²×C₅₃	—	8	1	no	—
36	C₂²×C₅₃	#99	1	1	yes	characteristic
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	—	4	1	no	—
39	C₄×C₅₃	—	4	1	no	—
40	C₄×C₅₃	—	8	1	no	—
41	Dic₅₃	—	16	1	no	—
42	#1407	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	—	8	1	no	—
45	C₂×C₅₃	#353	1	1	yes	characteristic
46	#497	D₅₃	1	1	yes	characteristic
47	#355	—	53	1	no	—
48	#354	—	53	1	no	—
49	C₅₃	#1407	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×SD₁₆	—	2	1	no	—
52	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic, complemented
53	#124	C₂×D₅₃	1	1	yes	characteristic
54	#101	—	106	1	no	—
55	#99	—	106	1	no	—
56	#97	Dic₅₃	1	1	yes	characteristic, complemented
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
60	Q₁₆	—	4	2	no	—
61	SD₁₆	—	4	2	no	—
62	M₁₆	—	212	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	212	1	no	—
67	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
68	C₂³	—	2	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	424	1	no	—
77	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	424	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	#13133132	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	848	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#16810416	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810416
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#124	C₂×D₅₃
Fitting	C₅₃×#497	C₂
socle	C₂×C₅₃	#353
hypercenter	#497	D₅₃
hypocenter	C₅₃	#1407
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #497

Complements

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

N	H ≅ G/N	type
8. C₄⋊Q₈×C₅₃	C₄	semidirect
10. #97×C₅₃	C₄	semidirect
49. C₅₃	#1407	semidirect
52. C₄⋊Q₈	Dic₅₃	semidirect
56. #97	Dic₅₃	semidirect