Subgroups — #20272747

Order 6784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810988	C₂	1	1	yes	maximal, characteristic
2	#16810401	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133214	C₄	1	1	yes	characteristic
5	#13133209	C₄	1	1	yes	characteristic
6	#13133157	—	2	1	no	—
7	#13133139	—	2	1	no	—
8	#13133139	—	2	1	no	—
9	M₃₂×C₅₃	—	2	1	no	—
10	#97×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₂₁₂	—	2	1	no	—
12	C₂×D₂₁₂	—	4	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₅₃	—	2	1	no	—
17	#11719919	—	4	1	no	—
18	C₁₆×C₅₃	—	2	2	no	—
19	#11719908	—	2	2	no	—
20	C₂²×D₅₃	—	4	1	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₅₃	—	4	1	no	—
24	D₂₁₂	—	4	1	no	—
25	D₂₁₂	—	4	1	no	—
26	D₂₁₂	—	8	2	no	—
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	—	2	1	no	—
29	C₂²×C₅₃	#100	1	1	yes	characteristic
30	C₂×D₅₃	—	8	2	no	—
31	C₂×D₅₃	—	8	1	no	—
32	C₄×C₅₃	—	4	1	no	—
33	C₄×C₅₃	#104	1	1	yes	characteristic
34	C₄×C₅₃	#104	1	1	yes	characteristic
35	Dic₅₃	—	8	1	no	—
36	#1355	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#329	1	1	yes	characteristic
39	D₅₃	—	16	1	no	—
40	#490	—	53	1	no	—
41	#347	—	53	1	no	—
42	#347	D₅₃	1	1	yes	characteristic, complemented
43	C₅₃	#1355	1	1	yes	characteristic, complemented, Sylow
44	#127	—	53	1	no	—
45	#125	—	53	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	106	1	no	—
48	#102	—	106	1	no	—
49	#102	—	106	1	no	—
50	#97	C₂×D₅₃	1	1	yes	characteristic
51	C₂×D₄	—	106	1	no	—
52	C₂×D₄	—	212	1	no	—
53	C₂×C₈	D₂₁₂	1	1	yes	characteristic
54	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
55	C₄⋊C₄	—	106	1	no	—
56	C₄⋊C₄	—	212	1	no	—
57	C₄²	C₄×D₅₃	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	106	2	no	—
60	C₂³	—	212	1	no	—
61	D₄	—	424	2	no	—
62	D₄	—	212	1	no	—
63	D₄	—	212	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	212	1	no	—
66	C₂×C₄	#11719921	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	424	2	no	—
70	C₂²	—	424	1	no	—
71	C₂²	#13133133	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	424	1	no	—
74	C₄	#13133143	1	1	yes	characteristic
75	C₄	#13133145	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	848	1	no	—
78	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	#1355
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
2. #16810401	C₂	semidirect
3. C₅₃×#347	C₂	semidirect
42. #347	D₅₃	semidirect
43. C₅₃	#1355	semidirect