Subgroups — #20276958

Order 6784.

152 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	#16810423	C₂	1	4	yes	maximal, complemented
3	#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄₂₄	—	2	1	no	—
6	C₂×D₄₂₄	C₂²	1	2	yes	—
7	#13133214	—	2	1	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#125×C₅₃	—	2	1	no	—
10	#107×C₅₃	C₂²	1	2	yes	—
11	#106×C₅₃	—	2	1	no	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₂₁₂	—	2	2	no	—
14	C₂×D₂₁₂	—	4	2	no	—
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₄⋊C₄×C₅₃	—	4	2	no	—
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	D₄₂₄	—	2	4	no	—
21	D₄₂₄	—	4	2	no	—
22	#11719908	—	2	2	no	—
23	C₂²×D₅₃	—	4	2	no	—
24	C₂×C₄×C₅₃	—	4	2	no	—
25	C₂×C₄×C₅₃	D₈	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	D₂₁₂	—	4	2	no	—
28	D₂₁₂	—	4	2	no	—
29	D₂₁₂	—	8	4	no	—
30	C₈×C₅₃	SD₁₆	1	2	yes	—
31	C₈×C₅₃	#41	1	2	yes	—
32	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
33	C₂×D₅₃	—	8	2	no	—
34	C₂×D₅₃	—	8	4	no	—
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	—	8	2	no	—
37	C₄×C₅₃	#124	1	1	yes	characteristic
38	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
39	#2244	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	#486	1	1	yes	characteristic
41	C₂×C₅₃	#509	1	2	yes	—
42	D₅₃	—	16	2	no	—
43	#499	D₅₃	1	1	yes	characteristic, complemented
44	#493	—	53	1	no	—
45	#357	—	53	4	no	—
46	#346	—	53	1	no	—
47	C₅₃	#2244	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₈	—	53	2	no	—
49	C₂×D₈	—	106	1	no	—
50	#127	—	106	1	no	—
51	#125	—	2	1	no	—
52	C₂×C₁₆	—	53	2	no	—
53	#107	C₂×D₅₃	1	2	yes	—
54	#106	—	2	1	no	—
55	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂×D₄	—	106	2	no	—
57	C₂×D₄	—	212	2	no	—
58	D₈	—	106	4	no	—
59	D₈	—	212	2	no	—
60	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
65	C₁₆	—	106	2	no	—
66	C₂³	—	212	2	no	—
67	D₄	—	424	4	no	—
68	D₄	—	212	2	no	—
69	D₄	—	212	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#11719922	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
73	C₈	#11719924	1	2	yes	—
74	C₈	#11719949	1	2	yes	—
75	C₂²	—	424	4	no	—
76	C₂²	—	424	2	no	—
77	C₂²	C₂×#11719922	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	#13133206	1	1	yes	characteristic
81	C₄	C₂×#11719924	1	1	yes	characteristic
82	C₂	—	848	2	no	—
83	C₂	#16810971	1	1	yes	characteristic
84	C₂	#16811058	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#499	D₅₃
hypocenter	C₅₃	#2244
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
2. #16810423	C₂	semidirect
3. #16810399	C₂	semidirect
4. C₅₃×#499	C₂	semidirect
43. #499	D₅₃	semidirect
47. C₅₃	#2244	semidirect