Subgroups — #20272690

Order 6784.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810741	C₂	1	1	yes	maximal, characteristic
2	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133239	C₄	1	1	yes	characteristic
5	#13133223	—	2	1	no	—
6	#13133153	C₄	1	1	yes	characteristic
7	#13133145	—	2	1	no	—
8	C₄×#88322	—	2	1	no	—
9	M₃₂×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	#11719945	—	2	1	no	—
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	D₄	1	1	yes	characteristic
15	#11719917	—	2	1	no	—
16	C₂×#88322	—	2	2	no	—
17	Dic₂₁₂	C₈	1	1	yes	characteristic, complemented
18	D₄₂₄	C₈	1	1	yes	characteristic, complemented
19	#11719913	—	2	1	no	—
20	#11719912	—	4	1	no	—
21	C₈×D₅₃	—	4	1	no	—
22	C₁₆×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²⋊D₅₃	—	4	1	no	—
26	D₂₁₂	—	2	1	no	—
27	C₄×D₅₃	—	4	1	no	—
28	Dic₁₀₆	—	2	1	no	—
29	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₈×C₅₃	M₁₆	1	1	yes	characteristic
31	C₈×C₅₃	D₈	1	1	yes	characteristic
32	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
33	#88322	—	4	2	no	—
34	#88322	—	4	1	no	—
35	C₂²×C₅₃	#104	1	1	yes	characteristic
36	C₂×D₅₃	—	4	1	no	—
37	C₄×C₅₃	—	2	2	no	—
38	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₅₃	#102	1	1	yes	characteristic
40	Dic₅₃	—	4	1	no	—
41	#1336	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
44	D₅₃	—	8	1	no	—
45	#443	—	53	1	no	—
46	#364	D₅₃	1	1	yes	characteristic, complemented
47	#322	—	53	1	no	—
48	C₅₃	#1336	1	1	yes	characteristic, complemented, Sylow
49	#135	—	53	1	no	—
50	#131	—	106	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	53	1	no	—
53	#104	—	106	1	no	—
54	C₄×C₈	—	106	1	no	—
55	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
56	#41	—	106	1	no	—
57	Q₁₆	—	53	1	no	—
58	SD₁₆	—	106	1	no	—
59	D₈	—	53	1	no	—
60	M₁₆	—	106	1	no	—
61	M₁₆	—	212	1	no	—
62	C₂×C₈	—	106	2	no	—
63	C₂×C₈	—	212	1	no	—
64	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₄²	D₂₁₂	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	106	1	no	—
69	D₄	—	106	1	no	—
70	D₄	—	212	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	212	1	no	—
73	C₂×C₄	#11719921	1	1	yes	characteristic
74	C₈	—	212	2	no	—
75	C₈	—	212	1	no	—
76	C₈	C₈×D₅₃	1	1	yes	characteristic
77	C₈	#11719912	1	1	yes	characteristic
78	C₈	#11719922	1	1	yes	characteristic
79	C₈	#11719924	1	1	yes	characteristic
80	C₂²	—	212	1	no	—
81	C₂²	#13133143	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	212	1	no	—
84	C₄	#13133129	1	1	yes	characteristic
85	C₄	#13133137	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	424	1	no	—
88	C₂	#16810350	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133137
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#364	C₂
socle	C₂×C₅₃	D₄⋊C₈
hypercenter	#364	D₅₃
hypocenter	C₅₃	#1336
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #16810344	C₂	semidirect
3. C₅₃×#364	C₂	semidirect
17. Dic₂₁₂	C₈	semidirect
18. D₄₂₄	C₈	semidirect
46. #364	D₅₃	semidirect
48. C₅₃	#1336	semidirect