Subgroups — #20274309

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	1	yes	maximal, characteristic
2	#16811019	C₂	1	2	yes	maximal
3	#16810975	C₂	1	2	yes	maximal
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	#13133217	C₂²	1	2	yes	—
7	#13133151	—	2	2	no	—
8	#13133148	—	2	4	no	—
9	#13133140	—	2	4	no	—
10	C₄×#88322	C₂²	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₅₃	—	2	2	no	—
13	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×Dic₁₀₆	—	2	2	no	—
17	Q₁₆×C₅₃	—	4	4	no	—
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×#88322	D₄	1	2	yes	—
25	C₂×#88322	D₄	1	2	yes	—
26	Q₈×C₅₃	—	4	4	no	—
27	C₂×C₄×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	2	no	—
32	Dic₁₀₆	—	4	4	no	—
33	C₈×C₅₃	SD₁₆	1	4	yes	—
34	#88322	—	2	2	no	—
35	#88322	Q₁₆	1	4	yes	complemented
36	C₂²×C₅₃	#127	1	1	yes	characteristic
37	C₄×C₅₃	—	8	2	no	—
38	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
39	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
40	C₄×C₅₃	#135	1	2	yes	—
41	Dic₅₃	—	8	2	no	—
42	#1707	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	#492	1	1	yes	characteristic
44	C₂×C₅₃	#494	1	1	yes	characteristic
45	C₂×C₅₃	#495	1	1	yes	characteristic
46	#500	—	53	1	no	—
47	#498	—	53	2	no	—
48	#494	D₅₃	1	1	yes	characteristic
49	#487	—	53	2	no	—
50	C₈²	—	53	1	no	—
51	C₅₃	#1707	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	2	2	no	—
53	C₄⋊Q₈	—	53	2	no	—
54	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
55	#107	—	106	2	no	—
56	#106	—	106	4	no	—
57	#103	—	106	4	no	—
58	C₄×C₈	—	53	1	no	—
59	C₄×C₈	—	53	1	no	—
60	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	106	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	53	2	no	—
65	C₂×C₈	—	53	2	no	—
66	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	106	2	no	—
70	C₄⋊C₄	—	106	2	no	—
71	C₄²	C₂²×D₅₃	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	212	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	212	2	no	—
76	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
77	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₈	—	53	4	no	—
80	C₈	—	106	2	no	—
81	C₈	#11719923	1	4	yes	—
82	C₂²	#13133213	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	424	2	no	—
85	C₄	C₂×#11719923	1	2	yes	—
86	C₄	Q₁₆×D₅₃	1	2	yes	—
87	C₄	#13133240	1	2	yes	—
88	C₂	#16810995	1	1	yes	characteristic
89	C₂	#16811001	1	1	yes	characteristic
90	C₂	#16811008	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	#127
hypercenter	#494	D₅₃
hypocenter	C₅₃	#1707
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
35. #88322	Q₁₆	semidirect
51. C₅₃	#1707	semidirect