Subgroups — #20272702

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810881	C₂	1	1	yes	maximal, characteristic
2	#16810405	C₂	1	1	yes	maximal, characteristic
3	C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
5	#13133196	—	2	1	no	—
6	#13133157	—	2	1	no	—
7	#13133149	C₄	1	1	yes	characteristic, complemented
8	#13133142	—	2	1	no	—
9	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	#117×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
12	C₂×Dic₁₀₆	—	2	1	no	—
13	M₁₆×C₅₃	—	4	1	no	—
14	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₂²⋊C₄×C₅₃	—	4	1	no	—
18	C₄²×C₅₃	—	2	2	no	—
19	C₂²⋊Dic₅₃	—	4	1	no	—
20	C₄⋊Dic₅₃	—	2	1	no	—
21	#11719919	—	4	1	no	—
22	Dic₂₁₂	—	4	1	no	—
23	#11719908	—	4	1	no	—
24	C₂³×C₅₃	D₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₂×Dic₅₃	—	4	1	no	—
31	C₂×Dic₅₃	—	4	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	Dic₁₀₆	—	4	1	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	2	no	—
41	C₄×C₅₃	#99	1	1	yes	characteristic
42	C₄×C₅₃	#104	1	1	yes	characteristic
43	Dic₅₃	—	8	1	no	—
44	Dic₅₃	—	8	1	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	—	53	1	no	—
50	#349	—	53	1	no	—
51	H(3,ℤ/4)	D₅₃	1	1	yes	characteristic
52	C₅₃	#1340	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	—	53	1	no	—
54	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
55	C₂²⋊Q₈	—	106	1	no	—
56	#117	—	2	1	no	—
57	M₃₂	—	106	1	no	—
58	#106	—	53	1	no	—
59	#103	—	106	1	no	—
60	C₂×Q₈	—	106	1	no	—
61	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
62	Q₁₆	—	212	1	no	—
63	M₁₆	—	4	1	no	—
64	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
65	C₂×C₈	D₂₁₂	1	1	yes	characteristic
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	106	1	no	—
68	C₄⋊C₄	—	212	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	212	1	no	—
71	C₄²	—	2	2	no	—
72	C₁₆	—	212	1	no	—
73	C₂³	#11719922	1	1	yes	characteristic
74	Q₈	—	212	1	no	—
75	Q₈	—	212	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	212	1	no	—
80	C₂×C₄	—	212	1	no	—
81	C₂×C₄	#11719921	1	1	yes	characteristic
82	C₂×C₄	#11719924	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	#13133137	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	8	2	no	—
90	C₄	—	424	1	no	—
91	C₄	—	424	1	no	—
92	C₄	#13133131	1	1	yes	characteristic
93	C₄	#13133145	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	4	1	no	—
96	C₂	#16810356	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810356
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₅₃
Fitting	C₅₃×H(3,ℤ/4)	C₂
socle	C₂×C₅₃	#327
hypercenter	H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	#1340
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₂₁₂	C₄	semidirect
7. #13133149	C₄	semidirect
52. C₅₃	#1340	semidirect