Subgroups — #20290656

Order 6784.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133288	C₂	1	4	yes	maximal
2	#16811627	C₂	1	2	yes	maximal
3	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719936	C₂²	1	4	yes	—
5	#13133288	—	2	8	no	—
6	C₄⋊#88324	—	2	4	no	—
7	C₂×C₈×D₅₃	C₂²	1	2	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic
10	C₈×Dic₅₃	C₄	1	2	yes	—
11	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×D₅₃	Q₈	1	2	yes	—
13	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
14	C₂×#88324	—	2	4	no	—
15	#11719936	—	2	8	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	D₄	1	2	yes	—
20	C₂×#88322	C₂×C₄	1	2	yes	—
21	C₈×D₅₃	D₄	1	4	yes	—
22	C₈×D₅₃	Q₈	1	4	yes	—
23	C₂²×D₅₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×Dic₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
28	C₄×D₅₃	—	2	4	no	—
29	C₄×D₅₃	C₂×D₄	1	2	yes	—
30	C₄×D₅₃	C₂×Q₈	1	2	yes	—
31	#88324	—	4	4	no	—
32	C₈×C₅₃	C₄⋊C₄	1	4	yes	—
33	#88322	C₄⋊C₄	1	4	yes	—
34	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×D₅₃	—	2	2	no	—
36	C₂×D₅₃	C₄⋊Q₈	1	2	yes	—
37	C₄×C₅₃	#108	1	4	yes	complemented
38	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
39	Dic₅₃	—	2	2	no	—
40	Dic₅₃	#127	1	1	yes	characteristic
41	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
42	#1848	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	#384	1	1	yes	characteristic
44	C₂×C₅₃	C₂×#108	1	2	yes	—
45	D₅₃	—	2	2	no	—
46	C₂×#108	—	53	4	no	—
47	#423	—	53	2	no	—
48	C₂×C₄×C₈	—	53	1	no	—
49	C₅₃	#1848	1	1	yes	characteristic, complemented, Sylow
50	C₂×M₁₆	—	53	4	no	—
51	C₂²×C₈	—	53	2	no	—
52	C₂×C₄²	—	53	1	no	—
53	#108	—	106	8	no	—
54	C₄⋊C₈	—	106	4	no	—
55	C₄×C₈	—	53	2	no	—
56	C₄×C₈	—	53	1	no	—
57	C₄×C₈	#6157	1	1	yes	characteristic
58	C₂²×C₄	—	53	1	no	—
59	C₂²×C₄	—	53	2	no	—
60	M₁₆	—	106	8	no	—
61	C₂×C₈	—	53	2	no	—
62	C₂×C₈	—	53	4	no	—
63	C₂×C₈	—	53	4	no	—
64	C₂×C₈	—	106	4	no	—
65	C₂×C₈	C₂×#6157	1	2	yes	—
66	C₄²	—	53	2	no	—
67	C₄²	—	53	1	no	—
68	C₄²	C₂×#6157	1	1	yes	characteristic
69	C₂³	—	53	1	no	—
70	C₂×C₄	—	53	1	no	—
71	C₂×C₄	—	53	2	no	—
72	C₂×C₄	—	53	2	no	—
73	C₂×C₄	—	53	2	no	—
74	C₂×C₄	—	106	4	no	—
75	C₂×C₄	C₄⋊#6157	1	2	yes	—
76	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
77	C₈	—	53	4	no	—
78	C₈	—	212	4	no	—
79	C₈	C₄⋊#6157	1	4	yes	—
80	C₂²	—	53	2	no	—
81	C₂²	—	106	2	no	—
82	C₂²	C₂×C₄⋊#6157	1	1	yes	characteristic
83	C₄	—	53	1	no	—
84	C₄	—	53	1	no	—
85	C₄	—	106	2	no	—
86	C₄	#13133288	1	4	yes	complemented
87	C₄	C₂×C₄⋊#6157	1	2	yes	—
88	C₂	—	106	2	no	—
89	C₂	#16811569	1	1	yes	characteristic
90	C₂	C₂×#13133288	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
37. C₄×C₅₃	#108	semidirect
49. C₅₃	#1848	semidirect
86. C₄	#13133288	semidirect