Subgroups — #20272762

Order 6784.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄₂₄	—	2	1	no	—
5	C₂×D₄₂₄	C₄	1	2	yes	—
6	#13133214	—	2	1	no	—
7	C₄⋊#88322	—	2	1	no	—
8	C₂×C₁₆×C₅₃	—	2	1	no	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₂₁₂	—	2	2	no	—
11	C₂×D₂₁₂	—	4	1	no	—
12	C₂×D₂₁₂	—	4	1	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	D₄	1	1	yes	characteristic
16	C₂×#88322	—	4	1	no	—
17	D₄₂₄	—	4	2	no	—
18	D₄₂₄	—	4	1	no	—
19	D₄₂₄	—	4	1	no	—
20	C₁₆×C₅₃	—	4	1	no	—
21	C₂²×D₅₃	—	4	2	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
25	D₂₁₂	—	4	2	no	—
26	D₂₁₂	—	4	2	no	—
27	D₂₁₂	—	8	2	no	—
28	D₂₁₂	—	8	1	no	—
29	D₂₁₂	—	8	1	no	—
30	C₈×C₅₃	—	2	1	no	—
31	C₈×C₅₃	D₈	1	1	yes	characteristic
32	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
33	#88322	—	8	1	no	—
34	C₂²×C₅₃	#102	1	1	yes	characteristic
35	C₂×D₅₃	—	8	2	no	—
36	C₂×D₅₃	—	8	2	no	—
37	C₂×D₅₃	—	8	2	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	#100	1	1	yes	characteristic
40	C₄×C₅₃	#102	1	1	yes	characteristic
41	C₄×C₅₃	D₁₆	1	1	yes	characteristic
42	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
43	#1361	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	#331	1	1	yes	characteristic
45	C₂×C₅₃	#357	1	1	yes	characteristic
46	C₂×C₅₃	#361	1	1	yes	characteristic
47	D₅₃	—	16	2	no	—
48	#493	—	53	1	no	—
49	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
50	#335	—	53	1	no	—
51	C₅₃	#1361	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₈	—	53	2	no	—
53	C₂×D₈	—	106	1	no	—
54	#127	—	106	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₄⋊C₈	—	106	1	no	—
57	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
58	C₂×D₄	—	106	2	no	—
59	C₂×D₄	—	212	1	no	—
60	C₂×D₄	—	212	1	no	—
61	D₈	—	212	2	no	—
62	D₈	—	212	1	no	—
63	D₈	—	212	1	no	—
64	C₂×C₈	—	212	1	no	—
65	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₄²	D₂₁₂	1	1	yes	characteristic
68	C₁₆	—	4	1	no	—
69	C₂³	—	212	2	no	—
70	D₄	—	424	2	no	—
71	D₄	—	424	1	no	—
72	D₄	—	424	1	no	—
73	D₄	—	212	2	no	—
74	D₄	—	212	2	no	—
75	C₂×C₄	#11719913	1	1	yes	characteristic
76	C₂×C₄	D₄₂₄	1	1	yes	characteristic
77	C₂×C₄	#11719921	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	424	1	no	—
80	C₈	#11719922	1	1	yes	characteristic
81	C₈	#11719924	1	1	yes	characteristic
82	C₂²	—	424	2	no	—
83	C₂²	—	424	2	no	—
84	C₂²	—	424	2	no	—
85	C₂²	#13133139	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#13133133	1	1	yes	characteristic
88	C₄	#13133137	1	1	yes	characteristic
89	C₄	D₈₄₈	1	1	yes	characteristic
90	C₄	#13133162	1	1	yes	characteristic
91	C₂	—	848	2	no	—
92	C₂	#16810366	1	1	yes	characteristic
93	C₂	#16810425	1	1	yes	characteristic
94	C₂	#16810437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	#102
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#1361
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #16810376	C₂	semidirect
3. C₅₃×C₄⋊C₁₆	C₂	semidirect
49. C₄⋊C₁₆	D₅₃	semidirect
51. C₅₃	#1361	semidirect