Subgroups — #20272703

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810405	C₂	1	1	yes	maximal, characteristic
2	#16810380	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#470	C₂	1	1	yes	maximal, characteristic
4	#13133176	—	2	1	no	—
5	#13133157	—	2	1	no	—
6	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
7	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂²⋊Q₈×C₅₃	—	2	1	no	—
9	#106×C₅₃	C₄	1	1	yes	characteristic, complemented
10	#103×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	—	2	1	no	—
12	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
13	Q₁₆×C₅₃	—	4	1	no	—
14	M₁₆×C₅₃	—	4	1	no	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₄⋊C₄×C₅₃	—	4	1	no	—
19	C₂²⋊C₄×C₅₃	—	4	1	no	—
20	C₂²⋊Dic₅₃	—	4	1	no	—
21	C₄⋊Dic₅₃	—	4	1	no	—
22	C₄×Dic₅₃	—	2	2	no	—
23	#11719908	—	4	1	no	—
24	C₂³×C₅₃	D₈	1	1	yes	characteristic
25	Q₈×C₅₃	—	4	1	no	—
26	Q₈×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	—	4	1	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	2	no	—
34	C₈×C₅₃	—	2	1	no	—
35	C₈×C₅₃	—	2	1	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	#102	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	8	1	no	—
41	C₄×C₅₃	—	8	1	no	—
42	C₄×C₅₃	#99	1	1	yes	characteristic
43	C₄×C₅₃	#104	1	1	yes	characteristic
44	Dic₅₃	—	8	2	no	—
45	#1340	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	—	4	1	no	—
48	C₂×C₅₃	#327	1	1	yes	characteristic
49	#470	D₅₃	1	1	yes	characteristic
50	#349	—	53	1	no	—
51	H(3,ℤ/4)	—	53	1	no	—
52	C₅₃	#1340	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	Dic₅₃	1	1	yes	characteristic, complemented
54	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
55	C₂²⋊Q₈	—	2	1	no	—
56	#117	—	106	1	no	—
57	M₃₂	—	106	1	no	—
58	#106	Dic₅₃	1	1	yes	characteristic, complemented
59	#103	—	2	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
62	Q₁₆	—	4	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	4	1	no	—
68	C₄⋊C₄	—	212	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	212	1	no	—
71	C₄²	—	106	2	no	—
72	C₁₆	—	212	1	no	—
73	C₂³	#11719922	1	1	yes	characteristic
74	Q₈	—	4	1	no	—
75	Q₈	—	4	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	212	2	no	—
81	C₂×C₄	#11719923	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#13133138	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	1	no	—
90	C₄	—	8	1	no	—
91	C₄	—	424	2	no	—
92	C₄	#13133132	1	1	yes	characteristic
93	C₄	#13133144	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#16810357	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810357
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×#470	C₂
socle	C₂×C₅₃	#327
hypercenter	#470	D₅₃
hypocenter	C₅₃	#1340
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
6. C₂×Q₁₆×C₅₃	C₄	semidirect
9. #106×C₅₃	C₄	semidirect
52. C₅₃	#1340	semidirect
53. C₂×Q₁₆	Dic₅₃	semidirect
58. #106	Dic₅₃	semidirect