Subgroups — #20274887

Order 6784.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133153	C₂	1	4	yes	maximal
2	#16810647	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	4	yes	—
5	#13133153	—	2	8	no	—
6	C₄⋊#88322	—	2	4	no	—
7	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
12	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₈×C₅₃	D₄	1	4	yes	—
15	C₂×C₈×C₅₃	Q₈	1	4	yes	—
16	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	D₄	1	2	yes	—
19	#11719917	—	2	8	no	—
20	C₂×#88322	—	2	4	no	—
21	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
22	C₂×C₄×C₅₃	—	2	4	no	—
23	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
29	C₈×C₅₃	C₄⋊C₄	1	8	yes	—
30	#88322	—	4	4	no	—
31	C₂²×C₅₃	—	2	2	no	—
32	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	#108	1	4	yes	complemented
36	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₅₃	#127	1	1	yes	characteristic
38	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
39	#1848	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	2	no	—
41	C₂×C₅₃	#384	1	1	yes	characteristic
42	C₂×C₅₃	C₂×#108	1	2	yes	—
43	C₂×#108	—	53	4	no	—
44	#423	—	53	2	no	—
45	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
46	C₅₃	#1848	1	1	yes	characteristic, complemented, Sylow
47	C₂×M₁₆	—	53	4	no	—
48	C₂²×C₈	C₂×D₅₃	1	2	yes	—
49	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
50	#108	—	106	8	no	—
51	C₄⋊C₈	—	106	4	no	—
52	C₄×C₈	Dic₅₃	1	2	yes	—
53	C₄×C₈	Dic₅₃	1	2	yes	—
54	C₂²×C₄	Dic₁₀₆	1	2	yes	—
55	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
56	M₁₆	—	106	8	no	—
57	C₂×C₈	—	106	4	no	—
58	C₂×C₈	Dic₁₀₆	1	4	yes	—
59	C₂×C₈	D₂₁₂	1	4	yes	—
60	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
61	C₄²	D₂₁₂	1	2	yes	—
62	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
63	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
64	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
68	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
69	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
70	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
72	C₈	—	212	4	no	—
73	C₈	C₄⋊Dic₅₃	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
76	C₂²	#13133217	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#13133153	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
80	C₄	#13133214	1	1	yes	characteristic
81	C₄	#13133217	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#16810493	1	1	yes	characteristic
84	C₂	C₂×#13133153	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₅₃	#108	semidirect
46. C₅₃	#1848	semidirect
78. C₄	#13133153	semidirect