Subgroups — #20272674

Order 6784.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810702	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11719912	—	2	1	no	—
5	C₄×D₂₁₂	C₄	1	1	yes	characteristic
6	C₂×#11719908	—	2	1	no	—
7	C₄⋊#88322	C₄	1	1	yes	characteristic
8	#13133129	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₂	C₈	1	1	yes	characteristic
12	C₂×C₄×D₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#11719921	—	2	1	no	—
17	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
18	C₂×#88322	—	2	1	no	—
19	#11719912	—	4	1	no	—
20	#11719912	—	4	1	no	—
21	C₁₆×C₅₃	—	2	2	no	—
22	#11719908	—	4	1	no	—
23	C₂²×D₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×Dic₅₃	—	2	1	no	—
28	D₂₁₂	—	4	1	no	—
29	C₄×D₅₃	—	4	1	no	—
30	C₄×D₅₃	—	4	1	no	—
31	C₈×C₅₃	—	2	1	no	—
32	C₈×C₅₃	D₈	1	1	yes	characteristic
33	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
34	#88322	—	4	1	no	—
35	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
36	C₂×D₅₃	—	4	1	no	—
37	C₂×D₅₃	—	4	1	no	—
38	C₂×D₅₃	—	4	1	no	—
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	#102	1	1	yes	characteristic
42	C₄×C₅₃	#104	1	1	yes	characteristic
43	Dic₅₃	—	4	1	no	—
44	#1330	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	#349	1	1	yes	characteristic
47	C₂×C₅₃	#350	1	1	yes	characteristic
48	D₅₃	—	8	1	no	—
49	#436	—	53	1	no	—
50	C₄⋊C₁₆	—	53	1	no	—
51	#346	D₅₃	1	1	yes	characteristic, complemented
52	C₅₃	#1330	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₁₆	—	106	1	no	—
54	C₄×D₄	—	53	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	106	1	no	—
57	C₄⋊C₈	—	53	1	no	—
58	C₂²⋊C₈	—	106	1	no	—
59	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂×D₄	—	53	1	no	—
61	C₂²×C₄	—	106	1	no	—
62	M₁₆	—	212	1	no	—
63	M₁₆	—	212	1	no	—
64	C₂×C₈	—	106	1	no	—
65	C₂×C₈	D₂₁₂	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₄⋊C₄	—	53	1	no	—
68	C₂²⋊C₄	—	106	1	no	—
69	C₄²	C₄×D₅₃	1	1	yes	characteristic
70	C₁₆	—	2	2	no	—
71	C₁₆	—	212	1	no	—
72	C₂³	—	106	1	no	—
73	D₄	—	212	1	no	—
74	C₂×C₄	—	106	1	no	—
75	C₂×C₄	—	212	1	no	—
76	C₂×C₄	—	212	1	no	—
77	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
78	C₂×C₄	#11719912	1	1	yes	characteristic
79	C₂×C₄	#11719921	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	212	1	no	—
82	C₈	#11719913	1	1	yes	characteristic
83	C₈	D₄₂₄	1	1	yes	characteristic
84	C₂²	—	212	1	no	—
85	C₂²	—	212	1	no	—
86	C₂²	—	212	1	no	—
87	C₂²	#13133129	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	212	1	no	—
91	C₄	#13133139	1	1	yes	characteristic
92	C₄	#13133145	1	1	yes	characteristic
93	C₂	—	424	1	no	—
94	C₂	#16810352	1	1	yes	characteristic
95	C₂	#16810404	1	1	yes	characteristic
96	C₂	#16810408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1330
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄⋊#11719908	C₂	semidirect
3. C₅₃×#346	C₂	semidirect
51. #346	D₅₃	semidirect
52. C₅₃	#1330	semidirect