Subgroups — #20272755

Order 6784.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810996	C₂	1	1	yes	maximal, characteristic
2	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11719913	—	2	1	no	—
5	C₂×#11719913	—	2	1	no	—
6	#13133217	C₄	1	1	yes	characteristic
7	#13133214	C₄	1	1	yes	characteristic
8	#13133157	—	2	1	no	—
9	M₃₂×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₂	—	2	1	no	—
12	C₂×D₂₁₂	—	4	1	no	—
13	C₂×Dic₁₀₆	—	2	1	no	—
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄⋊Dic₅₃	—	4	1	no	—
18	#11719913	—	4	1	no	—
19	#11719913	—	4	1	no	—
20	#11719913	—	4	1	no	—
21	#11719913	—	4	1	no	—
22	C₁₆×C₅₃	—	4	1	no	—
23	#11719908	—	4	1	no	—
24	C₂²×D₅₃	—	4	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	1	no	—
28	D₂₁₂	—	4	1	no	—
29	D₂₁₂	—	4	1	no	—
30	D₂₁₂	—	8	2	no	—
31	Dic₁₀₆	—	4	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	C₈×C₅₃	D₈	1	1	yes	characteristic
34	C₈×C₅₃	D₈	1	1	yes	characteristic
35	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₅₃	#100	1	1	yes	characteristic
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	8	1	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	#102	1	1	yes	characteristic
42	C₄×C₅₃	#102	1	1	yes	characteristic
43	Dic₅₃	—	8	1	no	—
44	#1359	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#331	1	1	yes	characteristic
47	D₅₃	—	16	1	no	—
48	#492	—	53	1	no	—
49	#364	—	53	1	no	—
50	#364	D₅₃	1	1	yes	characteristic, complemented
51	C₅₃	#1359	1	1	yes	characteristic, complemented, Sylow
52	C₂×SD₁₆	—	106	1	no	—
53	C₂×SD₁₆	—	106	1	no	—
54	C₄⋊Q₈	—	53	1	no	—
55	#127	—	53	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	106	1	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×Q₈	—	106	1	no	—
60	C₂×D₄	—	106	1	no	—
61	C₂×D₄	—	212	1	no	—
62	SD₁₆	—	212	1	no	—
63	SD₁₆	—	212	1	no	—
64	SD₁₆	—	212	1	no	—
65	SD₁₆	—	212	1	no	—
66	C₂×C₈	D₂₁₂	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₄⋊C₄	—	212	1	no	—
69	C₄²	C₄×D₅₃	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	212	1	no	—
72	C₂³	—	212	1	no	—
73	Q₈	—	212	1	no	—
74	Q₈	—	212	1	no	—
75	D₄	—	424	2	no	—
76	D₄	—	212	1	no	—
77	D₄	—	212	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	212	1	no	—
80	C₂×C₄	#11719921	1	1	yes	characteristic
81	C₈	#11719913	1	1	yes	characteristic
82	C₈	D₄₂₄	1	1	yes	characteristic
83	C₈	#11719922	1	1	yes	characteristic
84	C₈	#11719924	1	1	yes	characteristic
85	C₂²	—	424	2	no	—
86	C₂²	—	424	1	no	—
87	C₂²	#13133133	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	424	1	no	—
90	C₄	#13133137	1	1	yes	characteristic
91	C₄	#13133139	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	848	1	no	—
94	C₂	#16810366	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810366
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#364	C₂
socle	C₂×C₅₃	#331
hypercenter	#364	D₅₃
hypocenter	C₅₃	#1359
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #16810444	C₂	semidirect
3. C₅₃×#364	C₂	semidirect
50. #364	D₅₃	semidirect
51. C₅₃	#1359	semidirect