Subgroups — #20277008

Order 6784.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811039	C₂	1	1	yes	maximal, characteristic
2	#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810456	C₂	1	2	yes	maximal, complemented
4	C₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133223	C₄	1	1	yes	characteristic
7	#13133157	C₄	1	1	yes	characteristic
8	#13133156	C₄	1	2	yes	—
9	C₁₆×D₅₃	C₂²	1	2	yes	—
10	#13133123	C₂²	1	2	yes	—
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₅₃	C₂²	1	1	yes	characteristic
13	C₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	#11719945	C₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#11719917	C₈	1	1	yes	characteristic
17	#11719912	C₈	1	2	yes	—
18	C₈×D₅₃	C₂×C₄	1	2	yes	—
19	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₅₃	C₂³	1	1	yes	characteristic
21	#11719908	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂²⋊D₅₃	C₁₆	1	2	yes	—
24	D₂₁₂	C₁₆	1	1	yes	characteristic
25	C₄×D₅₃	C₂×C₈	1	2	yes	—
26	Dic₁₀₆	C₁₆	1	1	yes	characteristic
27	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	#88322	C₂×C₈	1	2	yes	—
30	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₅₃	C₂×C₁₆	1	2	yes	—
32	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
34	Dic₅₃	C₂×C₁₆	1	2	yes	—
35	#2258	—	53	1	no	maximal, Sylow
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
38	D₅₃	—	2	2	no	—
39	#504	—	53	1	no	—
40	M₆₄	—	53	1	no	—
41	M₆₄	—	53	2	no	—
42	C₂×C₃₂	—	53	2	no	—
43	C₂×C₃₂	D₅₃	1	1	yes	characteristic, complemented
44	C₅₃	#2258	1	1	yes	characteristic, complemented, Sylow
45	#131	—	53	1	no	—
46	M₃₂	—	53	1	no	—
47	M₃₂	—	53	2	no	—
48	C₂×C₁₆	—	53	2	no	—
49	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
50	C₃₂	—	53	2	no	—
51	C₃₂	C₂×D₅₃	1	1	yes	characteristic
52	C₃₂	C₂×D₅₃	1	1	yes	characteristic
53	#41	—	53	1	no	—
54	M₁₆	—	53	1	no	—
55	M₁₆	—	53	2	no	—
56	C₂×C₈	—	53	2	no	—
57	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
58	C₁₆	—	53	2	no	—
59	C₁₆	C₄×D₅₃	1	1	yes	characteristic
60	C₁₆	C₂²×D₅₃	1	1	yes	characteristic
61	Q₈	—	53	1	no	—
62	D₄	—	53	2	no	—
63	D₄	—	53	1	no	—
64	C₂×C₄	—	53	2	no	—
65	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
66	C₈	—	53	2	no	—
67	C₈	C₈×D₅₃	1	1	yes	characteristic
68	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
69	C₂²	—	53	2	no	—
70	C₂²	C₁₆×D₅₃	1	1	yes	characteristic
71	C₄	—	53	2	no	—
72	C₄	C₁₆×D₅₃	1	1	yes	characteristic
73	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	106	2	no	—
76	C₂	C₂×C₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₅₃
Fitting	C₂×C₃₂×C₅₃	C₂
hypercenter	C₂×C₃₂	D₅₃
hypocenter	C₅₃	#2258
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #16810457	C₂	semidirect
3. #16810456	C₂	semidirect
4. C₃₂×D₅₃	C₂	semidirect
5. C₂×C₃₂×C₅₃	C₂	semidirect
43. C₂×C₃₂	D₅₃	semidirect
44. C₅₃	#2258	semidirect