Subgroups — #20274312

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	#16811021	C₂	1	2	yes	maximal
3	#16811003	C₂	1	1	yes	maximal, characteristic
4	#16810974	C₂	1	2	yes	maximal
5	C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	2	no	—
7	#13133217	C₂²	1	2	yes	—
8	#13133217	C₂²	1	2	yes	—
9	#13133151	—	2	2	no	—
10	#13133149	—	2	4	no	—
11	#13133142	—	2	4	no	—
12	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀₆	—	2	2	no	—
16	C₂×Dic₁₀₆	—	2	2	no	—
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₂×C₈×C₅₃	D₄	1	2	yes	—
20	C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	Dic₂₁₂	—	4	4	no	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×Dic₅₃	—	4	2	no	—
30	C₂×Dic₅₃	—	4	2	no	—
31	Dic₁₀₆	—	4	4	no	—
32	Dic₁₀₆	—	4	4	no	—
33	C₈×C₅₃	—	2	2	no	—
34	C₈×C₅₃	SD₁₆	1	4	yes	—
35	C₈×C₅₃	Q₁₆	1	4	yes	complemented
36	C₂²×C₅₃	#127	1	1	yes	characteristic
37	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
38	C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
39	C₄×C₅₃	#135	1	2	yes	—
40	Dic₅₃	—	8	2	no	—
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	—	53	1	no	—
49	#487	—	53	2	no	—
50	C₈²	D₅₃	1	1	yes	characteristic
51	C₅₃	#1707	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₁₆	—	106	2	no	—
53	C₄⋊Q₈	—	53	2	no	—
54	C₄⋊Q₈	—	53	2	no	—
55	#107	—	106	2	no	—
56	#106	—	106	4	no	—
57	#103	—	106	4	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₂×Q₈	—	106	2	no	—
62	C₂×Q₈	—	106	2	no	—
63	Q₁₆	—	212	4	no	—
64	C₂×C₈	D₂₁₂	1	2	yes	—
65	C₂×C₈	D₂₁₂	1	2	yes	—
66	C₂×C₈	D₂₁₂	1	2	yes	—
67	C₄⋊C₄	—	106	2	no	—
68	C₄⋊C₄	—	106	2	no	—
69	C₄⋊C₄	—	106	2	no	—
70	C₄⋊C₄	—	106	2	no	—
71	C₄²	C₂²×D₅₃	1	1	yes	characteristic
72	Q₈	—	212	4	no	—
73	Q₈	—	212	4	no	—
74	C₂×C₄	—	212	2	no	—
75	C₂×C₄	—	212	2	no	—
76	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
77	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
78	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#11719913	1	4	yes	—
81	C₈	Dic₂₁₂	1	4	yes	complemented
82	C₂²	#13133214	1	1	yes	characteristic
83	C₄	—	424	2	no	—
84	C₄	—	424	2	no	—
85	C₄	C₂×#11719913	1	2	yes	—
86	C₄	C₂×Dic₂₁₂	1	2	yes	—
87	C₄	#13133239	1	2	yes	—
88	C₂	#16810996	1	1	yes	characteristic
89	C₂	#16811003	1	1	yes	characteristic
90	C₂	#16811007	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₈²×C₅₃	C₂
socle	C₂²×C₅₃	#127
hypercenter	C₈²	D₅₃
hypocenter	C₅₃	#1707
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
35. C₈×C₅₃	Q₁₆	semidirect
51. C₅₃	#1707	semidirect
81. C₈	Dic₂₁₂	semidirect