Subgroups — #20274324

Order 6784.

168 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	#16810975	C₂	1	2	yes	maximal
3	#16810973	C₂	1	2	yes	maximal
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#500	C₂	1	1	yes	maximal, characteristic
6	#13133217	C₂²	1	2	yes	—
7	#13133151	—	2	2	no	—
8	#13133141	—	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	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
13	#107×C₅₃	—	2	2	no	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×Dic₁₀₆	—	2	2	no	—
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₂×#88322	D₄	1	2	yes	—
24	C₂×#88322	D₄	1	2	yes	—
25	Q₈×C₅₃	—	4	4	no	—
26	C₂×C₄×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×Dic₅₃	—	4	2	no	—
31	Dic₁₀₆	—	4	4	no	—
32	C₈×C₅₃	Q₁₆	1	4	yes	—
33	#88322	—	2	2	no	—
34	#88322	—	2	2	no	—
35	C₂²×C₅₃	#127	1	1	yes	characteristic
36	C₄×C₅₃	—	8	2	no	—
37	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
38	C₄×C₅₃	#135	1	2	yes	—
39	C₄×C₅₃	#135	1	2	yes	—
40	Dic₅₃	—	8	2	no	—
41	#1710	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	#494	1	1	yes	characteristic
43	C₂×C₅₃	#495	1	1	yes	characteristic
44	C₂×C₅₃	#495	1	1	yes	characteristic
45	#500	—	53	1	no	—
46	#500	D₅₃	1	1	yes	characteristic
47	#487	—	53	2	no	—
48	#487	—	53	2	no	—
49	C₈²	—	53	1	no	—
50	C₅₃	#1710	1	1	yes	characteristic, complemented, Sylow
51	C₄⋊Q₈	—	53	2	no	—
52	C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
53	#107	—	2	2	no	—
54	#107	—	106	2	no	—
55	#103	—	106	4	no	—
56	#103	—	106	4	no	—
57	C₄×C₈	—	53	1	no	—
58	C₄×C₈	—	53	1	no	—
59	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	106	2	no	—
62	C₂×C₈	—	53	2	no	—
63	C₂×C₈	—	53	2	no	—
64	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	106	2	no	—
68	C₄⋊C₄	—	106	2	no	—
69	C₄²	C₂²×D₅₃	1	1	yes	characteristic
70	Q₈	—	4	4	no	—
71	Q₈	—	212	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	212	2	no	—
74	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
75	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
77	C₈	—	106	2	no	—
78	C₈	—	106	2	no	—
79	C₈	#11719925	1	4	yes	—
80	C₂²	#13133213	1	1	yes	characteristic
81	C₄	—	8	2	no	—
82	C₄	—	424	2	no	—
83	C₄	C₂×#11719925	1	2	yes	—
84	C₄	#13133236	1	2	yes	—
85	C₄	#13133240	1	2	yes	—
86	C₂	#16811002	1	1	yes	characteristic
87	C₂	#16811006	1	1	yes	characteristic
88	C₂	#16811008	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#500	C₂
socle	C₂²×C₅₃	#127
hypercenter	#500	D₅₃
hypocenter	C₅₃	#1710
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
50. C₅₃	#1710	semidirect