Subgroups — #20276232

Order 6784.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810672	C₂	1	3	yes	maximal, complemented
2	#16810586	C₂	1	3	yes	maximal
3	C₅₃×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
5	C₂×#11719917	C₂²	1	3	yes	—
6	#13133153	—	2	6	no	—
7	#13133129	—	2	6	no	—
8	C₂×M₁₆×C₅₃	C₂²	1	3	yes	—
9	C₂³×D₅₃	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	C₂²×Dic₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₄×D₅₃	—	2	3	no	—
13	C₂×C₄×D₅₃	—	2	3	no	—
14	M₁₆×C₅₃	—	2	6	no	—
15	C₂×C₈×C₅₃	D₄	1	6	yes	—
16	#11719917	—	2	6	no	—
17	C₂×#88322	—	2	3	no	—
18	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
19	C₂²×D₅₃	—	2	3	no	—
20	C₂²×D₅₃	—	4	2	no	—
21	C₂×C₄×C₅₃	C₂×D₄	1	3	yes	—
22	C₂×C₄×C₅₃	#41	1	3	yes	—
23	C₂×Dic₅₃	—	2	3	no	—
24	C₄×D₅₃	—	4	3	no	—
25	C₄×D₅₃	—	4	1	no	—
26	C₈×C₅₃	—	2	6	no	—
27	#88322	—	4	3	no	—
28	C₂²×C₅₃	—	4	1	no	—
29	C₂²×C₅₃	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₅₃	—	2	6	no	—
31	C₂×D₅₃	—	4	3	no	—
32	C₂×D₅₃	—	4	1	no	—
33	C₄×C₅₃	#123	1	3	yes	—
34	C₄×C₅₃	#127	1	1	yes	characteristic
35	Dic₅₃	—	4	1	no	—
36	#2091	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	3	no	—
38	C₂×C₅₃	#399	1	1	yes	characteristic
39	D₅₃	—	4	2	no	—
40	#430	—	53	3	no	—
41	#407	—	53	3	no	—
42	#338	D₅₃	1	1	yes	characteristic, complemented
43	C₅₃	#2091	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₄	—	53	1	no	—
45	C₂×M₁₆	—	53	3	no	—
46	C₂×M₁₆	C₂×D₅₃	1	3	yes	—
47	#108	—	106	6	no	—
48	C₂²⋊C₈	—	106	6	no	—
49	C₂⁴	—	53	1	no	—
50	C₂²×C₄	—	53	1	no	—
51	C₂²×C₄	—	106	3	no	—
52	C₂²×C₄	—	106	3	no	—
53	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	106	6	no	—
56	C₂×C₈	—	106	3	no	—
57	C₂×C₈	D₂₁₂	1	6	yes	—
58	C₂³	—	106	3	no	—
59	C₂³	—	212	2	no	—
60	C₂³	Q₈×D₅₃	1	1	yes	characteristic
61	C₂×C₄	—	106	3	no	—
62	C₂×C₄	—	212	3	no	—
63	C₂×C₄	—	212	1	no	—
64	C₂×C₄	C₂×D₂₁₂	1	3	yes	—
65	C₂×C₄	#11719947	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	212	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	106	6	no	—
70	C₂²	—	212	3	no	—
71	C₂²	—	212	1	no	—
72	C₂²	#13133197	1	3	yes	—
73	C₄	—	212	1	no	—
74	C₄	#13133201	1	3	yes	—
75	C₄	#13133214	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	212	2	no	—
78	C₂	#16810560	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133214
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#399
hypercenter	#338	D₅₃
hypocenter	C₅₃	#2091
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
3. C₅₃×#338	C₂	semidirect
42. #338	D₅₃	semidirect
43. C₅₃	#2091	semidirect