Subgroups — #20276602

Order 6784.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811055	C₂	1	1	yes	maximal, characteristic
2	#16810741	C₂	1	2	yes	maximal, complemented
3	#16810453	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810431	C₂	1	2	yes	maximal, complemented
5	C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133239	C₂²	1	2	yes	—
7	#13133223	—	2	2	no	—
8	Dic₄₂₄	C₄	1	1	yes	characteristic, complemented
9	#13133162	C₄	1	2	yes	complemented
10	D₈₄₈	C₄	1	1	yes	characteristic, complemented
11	#13133153	C₂²	1	2	yes	—
12	#13133145	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#11719945	—	2	2	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	#11719917	—	2	2	no	—
21	Dic₂₁₂	C₂×C₄	1	2	yes	—
22	D₄₂₄	C₂×C₄	1	2	yes	—
23	#11719913	—	2	2	no	—
24	#11719912	—	4	2	no	—
25	C₈×D₅₃	—	4	2	no	—
26	C₁₆×C₅₃	—	2	1	no	—
27	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂²⋊D₅₃	—	4	2	no	—
32	D₂₁₂	—	2	2	no	—
33	C₄×D₅₃	—	4	2	no	—
34	Dic₁₀₆	—	2	2	no	—
35	C₈×C₅₃	D₈	1	2	yes	—
36	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₈×C₅₃	#41	1	1	yes	characteristic
38	#88322	—	4	2	no	—
39	C₂²×C₅₃	#135	1	1	yes	characteristic
40	C₂×D₅₃	—	4	2	no	—
41	C₄×C₅₃	—	4	1	no	—
42	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
43	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	Dic₅₃	—	4	2	no	—
45	#2179	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
48	D₅₃	—	8	2	no	—
49	#508	—	53	1	no	—
50	#443	—	53	2	no	—
51	#368	—	53	1	no	—
52	#359	—	53	2	no	—
53	#346	D₅₃	1	1	yes	characteristic, complemented
54	C₅₃	#2179	1	1	yes	characteristic, complemented, Sylow
55	#135	—	53	2	no	—
56	#131	—	106	2	no	—
57	Q₃₂	—	53	1	no	—
58	SD₃₂	—	53	2	no	—
59	D₁₆	—	53	1	no	—
60	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
61	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
62	#108	—	53	2	no	—
63	#104	—	106	2	no	—
64	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
65	#41	—	106	2	no	—
66	Q₁₆	—	53	2	no	—
67	SD₁₆	—	106	2	no	—
68	D₈	—	53	2	no	—
69	M₁₆	—	106	2	no	—
70	M₁₆	—	212	2	no	—
71	C₂×C₈	—	212	2	no	—
72	C₂×C₈	D₂₁₂	1	1	yes	characteristic
73	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
74	C₄²	D₂₁₂	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₄×D₅₃	1	1	yes	characteristic
77	C₁₆	C₄×D₅₃	1	1	yes	characteristic
78	Q₈	—	106	2	no	—
79	D₄	—	106	2	no	—
80	D₄	—	212	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	212	2	no	—
83	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
84	C₈	—	212	2	no	—
85	C₈	D₄₂₄	1	2	yes	—
86	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
87	C₈	#11719945	1	1	yes	characteristic
88	C₂²	—	212	2	no	—
89	C₂²	#13133239	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	212	2	no	—
92	C₄	C₄×D₂₁₂	1	1	yes	characteristic
93	C₄	C₂×D₄₂₄	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	424	2	no	—
96	C₂	C₄×D₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂×C₅₃	C₄×D₈
hypercenter	#346	D₅₃
hypocenter	C₅₃	#2179
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. #16810741	C₂	semidirect
3. #16810453	C₂	semidirect
4. #16810431	C₂	semidirect
5. C₅₃×#346	C₂	semidirect
8. Dic₄₂₄	C₄	semidirect
9. #13133162	C₄	semidirect
10. D₈₄₈	C₄	semidirect
53. #346	D₅₃	semidirect
54. C₅₃	#2179	semidirect