Subgroups — #20272749

Order 6784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810992	C₂	1	1	yes	maximal, characteristic
2	#16810401	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#347	C₂	1	1	yes	maximal, characteristic
4	#13133217	C₄	1	1	yes	characteristic
5	#13133209	C₄	1	1	yes	characteristic
6	#13133157	—	2	1	no	—
7	#13133142	—	2	1	no	—
8	#13133142	—	2	1	no	—
9	M₃₂×C₅₃	—	2	1	no	—
10	#97×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₆	—	2	1	no	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄⋊Dic₅₃	—	2	1	no	—
16	C₄⋊Dic₅₃	—	4	1	no	—
17	#11719919	—	4	1	no	—
18	C₁₆×C₅₃	—	2	2	no	—
19	#11719908	—	2	2	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×Dic₅₃	—	4	1	no	—
23	C₂×Dic₅₃	—	4	1	no	—
24	Dic₁₀₆	—	4	1	no	—
25	Dic₁₀₆	—	4	1	no	—
26	C₈×C₅₃	—	2	1	no	—
27	C₈×C₅₃	—	2	1	no	—
28	C₂²×C₅₃	#100	1	1	yes	characteristic
29	C₄×C₅₃	—	4	1	no	—
30	C₄×C₅₃	#104	1	1	yes	characteristic
31	C₄×C₅₃	#104	1	1	yes	characteristic
32	Dic₅₃	—	8	1	no	—
33	Dic₅₃	—	8	1	no	—
34	#1356	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	1	no	—
36	C₂×C₅₃	#329	1	1	yes	characteristic
37	#491	—	53	1	no	—
38	#347	—	53	1	no	—
39	#347	D₅₃	1	1	yes	characteristic
40	C₅₃	#1356	1	1	yes	characteristic, complemented, Sylow
41	C₄⋊Q₈	—	53	1	no	—
42	#125	—	53	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	106	1	no	—
45	#103	—	106	1	no	—
46	#103	—	106	1	no	—
47	#97	C₂×D₅₃	1	1	yes	characteristic
48	C₂×Q₈	—	106	1	no	—
49	C₂×C₈	D₂₁₂	1	1	yes	characteristic
50	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
51	C₄⋊C₄	—	106	1	no	—
52	C₄⋊C₄	—	212	1	no	—
53	C₄⋊C₄	—	212	1	no	—
54	C₄²	C₄×D₅₃	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	106	2	no	—
57	Q₈	—	212	1	no	—
58	Q₈	—	212	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	212	1	no	—
61	C₂×C₄	—	212	1	no	—
62	C₂×C₄	#11719921	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#13133133	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	424	1	no	—
68	C₄	—	424	1	no	—
69	C₄	#13133143	1	1	yes	characteristic
70	C₄	#13133145	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#16810362	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810362
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#97	C₂×D₅₃
Fitting	C₅₃×#347	C₂
socle	C₂×C₅₃	#329
hypercenter	#347	D₅₃
hypocenter	C₅₃	#1356
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
40. C₅₃	#1356	semidirect