Subgroups — #20283112

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133216	C₂	1	6	yes	maximal
2	#16810493	C₂	1	1	yes	maximal, characteristic
3	#16810492	C₂	1	6	yes	maximal
4	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
7	#13133216	—	2	24	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
9	C₂×#11719919	C₂²	1	12	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
16	C₂²×Dic₅₃	C₂³	1	4	yes	—
17	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂³	1	3	yes	—
19	C₂×Dic₁₀₆	—	2	24	no	—
20	C₄⋊C₄×C₅₃	—	2	12	no	—
21	C₄²×C₅₃	Q₈	1	4	yes	complemented
22	C₄⋊Dic₅₃	—	2	12	no	—
23	#11719919	—	2	24	no	—
24	C₄×Dic₅₃	D₄	1	12	yes	—
25	C₄×Dic₅₃	Q₈	1	12	yes	complemented
26	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	12	no	—
28	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
29	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
30	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
31	C₂×Dic₅₃	—	2	12	no	—
32	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
33	C₂×Dic₅₃	C₂×D₄	1	6	yes	—
34	C₂×Dic₅₃	C₂×Q₈	1	12	yes	—
35	Dic₁₀₆	—	4	16	no	—
36	C₂²×C₅₃	C₂²×D₄	1	3	yes	—
37	C₂²×C₅₃	C₂²×Q₈	1	3	yes	—
38	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
39	C₄×C₅₃	—	4	4	no	—
40	C₄×C₅₃	C₄⋊Q₈	1	12	yes	complemented
41	Dic₅₃	—	4	4	no	—
42	Dic₅₃	#127	1	4	yes	—
43	Dic₅₃	C₄⋊Q₈	1	12	yes	complemented
44	#2868	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
46	C₂×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
47	C₂×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
48	C₂×C₄⋊Q₈	—	53	6	no	—
49	#384	—	53	6	no	—
50	#384	—	53	1	no	—
51	#384	D₅₃	1	1	yes	characteristic
52	C₄³	—	53	1	no	—
53	C₅₃	#2868	1	1	yes	characteristic, complemented, Sylow
54	C₂²×Q₈	—	53	4	no	—
55	C₄⋊Q₈	—	106	24	no	—
56	C₂×C₄⋊C₄	—	53	12	no	—
57	C₂×C₄⋊C₄	—	53	6	no	—
58	C₂×C₄⋊C₄	C₂×D₅₃	1	6	yes	—
59	C₂×C₄²	—	53	3	no	—
60	C₂×C₄²	—	53	3	no	—
61	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
62	C₂×Q₈	—	106	24	no	—
63	C₂²×C₄	—	53	3	no	—
64	C₂²×C₄	—	53	1	no	—
65	C₂²×C₄	—	53	4	no	—
66	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
67	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
68	C₄⋊C₄	—	2	12	no	—
69	C₄⋊C₄	—	106	24	no	—
70	C₄⋊C₄	—	106	12	no	—
71	C₄²	—	53	12	no	—
72	C₄²	—	53	12	no	—
73	C₄²	Dic₁₀₆	1	4	yes	complemented
74	C₂³	C₂³×D₅₃	1	1	yes	characteristic
75	Q₈	—	212	16	no	—
76	C₂×C₄	—	2	12	no	—
77	C₂×C₄	—	53	6	no	—
78	C₂×C₄	—	53	6	no	—
79	C₂×C₄	—	53	12	no	—
80	C₂×C₄	—	106	12	no	—
81	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
82	C₂×C₄	D₄×D₅₃	1	6	yes	—
83	C₂×C₄	Q₈×D₅₃	1	6	yes	—
84	C₂²	C₂×D₄×D₅₃	1	3	yes	—
85	C₂²	C₂×Q₈×D₅₃	1	3	yes	—
86	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
87	C₄	—	4	4	no	—
88	C₄	—	53	4	no	—
89	C₄	—	53	12	no	—
90	C₄	—	212	4	no	—
91	C₄	#13133216	1	12	yes	complemented
92	C₂	#127×D₅₃	1	1	yes	characteristic
93	C₂	C₄⋊Q₈×D₅₃	1	3	yes	—
94	C₂	C₂×#13133216	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#384	C₂
hypercenter	#384	D₅₃
hypocenter	C₅₃	#2868
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #384

Complements

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

N	H ≅ G/N	type
21. C₄²×C₅₃	Q₈	semidirect
25. C₄×Dic₅₃	Q₈	semidirect
40. C₄×C₅₃	C₄⋊Q₈	semidirect
43. Dic₅₃	C₄⋊Q₈	semidirect
53. C₅₃	#2868	semidirect
73. C₄²	Dic₁₀₆	semidirect
91. C₄	#13133216	semidirect