Subgroups — #20276117

Order 6784.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810671	C₂	1	3	yes	maximal
2	C₂×#13133135	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
5	C₂×#11719917	C₂²	1	3	yes	—
6	#13133152	—	2	6	no	—
7	#13133135	—	2	6	no	—
8	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
9	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
10	C₂²×Dic₅₃	Q₈	1	2	yes	—
11	C₂×Dic₁₀₆	—	2	3	no	—
12	C₂×Dic₁₀₆	—	2	3	no	—
13	M₁₆×C₅₃	—	2	3	no	—
14	M₁₆×C₅₃	—	2	3	no	—
15	C₂×C₈×C₅₃	—	2	3	no	—
16	#11719917	—	2	3	no	—
17	#11719917	D₄	1	6	yes	—
18	C₂×#88322	—	2	3	no	—
19	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
21	C₂×C₄×C₅₃	#41	1	3	yes	—
22	C₂×Dic₅₃	—	2	6	no	—
23	Dic₁₀₆	—	4	3	no	—
24	Dic₁₀₆	—	4	1	no	—
25	C₈×C₅₃	—	4	3	no	—
26	#88322	—	2	6	no	—
27	C₂²×C₅₃	—	4	1	no	—
28	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₅₃	#121	1	3	yes	—
30	C₄×C₅₃	#126	1	1	yes	characteristic
31	Dic₅₃	—	4	2	no	—
32	#2070	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	—	2	3	no	—
34	C₂×C₅₃	#396	1	1	yes	characteristic
35	#430	—	53	3	no	—
36	C₂×#101	—	53	3	no	—
37	#338	D₅₃	1	1	yes	characteristic
38	C₅₃	#2070	1	1	yes	characteristic, complemented, Sylow
39	C₂²×Q₈	—	53	1	no	—
40	C₂×M₁₆	—	53	3	no	—
41	C₂×M₁₆	C₂×D₅₃	1	3	yes	—
42	#108	—	106	6	no	—
43	#101	—	106	6	no	—
44	C₂×Q₈	—	106	3	no	—
45	C₂×Q₈	—	106	3	no	—
46	C₂²×C₄	—	53	2	no	—
47	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	53	6	no	—
51	M₁₆	—	106	3	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	106	3	no	—
54	C₂³	Q₈×D₅₃	1	1	yes	characteristic
55	Q₈	—	212	3	no	—
56	Q₈	—	212	1	no	—
57	C₂×C₄	—	106	6	no	—
58	C₂×C₄	#11719945	1	3	yes	—
59	C₂×C₄	D₄×D₅₃	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	106	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#13133195	1	3	yes	—
64	C₄	—	212	2	no	—
65	C₄	#13133190	1	3	yes	—
66	C₄	#13133211	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#16810547	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810547
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#396
hypercenter	#338	D₅₃
hypocenter	C₅₃	#2070
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
38. C₅₃	#2070	semidirect