Subgroups — #20272817

Order 6784.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810661	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133176	—	2	2	no	—
4	#13133151	—	2	2	no	—
5	#13133149	C₄	1	4	yes	complemented
6	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	M₃₂×C₅₃	—	2	2	no	—
8	C₂×C₁₆×C₅₃	C₄	1	2	yes	complemented
9	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
10	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₅₃	D₄	1	2	yes	—
13	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂²⋊Dic₅₃	—	4	2	no	—
15	C₄⋊Dic₅₃	—	2	4	no	—
16	C₄×Dic₅₃	—	4	2	no	—
17	C₁₆×C₅₃	—	2	2	no	—
18	C₂³×C₅₃	D₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	4	no	—
24	C₈×C₅₃	C₄²	1	1	yes	characteristic
25	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	#102	1	2	yes	—
29	C₂²×C₅₃	#107	1	1	yes	characteristic
30	C₄×C₅₃	#95	1	1	yes	characteristic
31	C₄×C₅₃	#103	1	2	yes	—
32	C₄×C₅₃	#106	1	1	yes	characteristic
33	Dic₅₃	—	8	4	no	—
34	#1384	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	2	no	—
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	#340	1	1	yes	characteristic
38	C₂×M₃₂	D₅₃	1	1	yes	characteristic
39	#427	—	53	2	no	—
40	C₅₃	#1384	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
42	#117	—	106	2	no	—
43	M₃₂	—	2	2	no	—
44	C₂×C₁₆	Dic₅₃	1	2	yes	complemented
45	#107	—	106	2	no	—
46	#106	—	53	4	no	—
47	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
48	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
49	C₂×C₈	C₄×D₅₃	1	2	yes	—
50	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
52	C₄⋊C₄	—	106	4	no	—
53	C₂²⋊C₄	—	212	2	no	—
54	C₄²	—	212	2	no	—
55	C₁₆	—	2	2	no	—
56	C₂³	D₄₂₄	1	1	yes	characteristic
57	C₂×C₄	—	212	4	no	—
58	C₂×C₄	#11719913	1	2	yes	—
59	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
60	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
61	C₂×C₄	#11719921	1	2	yes	—
62	C₈	C₄×Dic₅₃	1	1	yes	characteristic
63	C₈	#11719919	1	2	yes	—
64	C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
65	C₂²	—	4	1	no	—
66	C₂²	#13133139	1	2	yes	—
67	C₂²	#13133151	1	1	yes	characteristic
68	C₄	—	424	4	no	—
69	C₄	#13133124	1	1	yes	characteristic
70	C₄	#13133142	1	2	yes	—
71	C₄	#13133149	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	#16810386	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133149
commutator	C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	#340
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#1384
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
5. #13133149	C₄	semidirect
8. C₂×C₁₆×C₅₃	C₄	semidirect
40. C₅₃	#1384	semidirect
44. C₂×C₁₆	Dic₅₃	semidirect