Subgroups — #20276587

Order 6784.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₁₂	C₂	1	2	yes	maximal
3	#16810451	C₂	1	1	yes	maximal, characteristic
4	#16810428	C₂	1	2	yes	maximal
5	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	C₄×Dic₁₀₆	—	2	2	no	—
8	Dic₄₂₄	C₄	1	4	yes	complemented
9	#13133151	C₂²	1	2	yes	—
10	#13133142	—	2	2	no	—
11	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₆	—	2	2	no	—
15	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₄⋊Dic₅₃	—	2	2	no	—
19	#11719919	—	4	2	no	—
20	C₄×Dic₅₃	—	4	2	no	—
21	Dic₂₁₂	—	2	2	no	—
22	Dic₂₁₂	C₂×C₄	1	4	yes	—
23	C₁₆×C₅₃	—	2	1	no	—
24	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₅₃	D₈	1	2	yes	—
26	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₅₃	—	4	2	no	—
28	C₂×Dic₅₃	—	4	2	no	—
29	Dic₁₀₆	—	2	4	no	—
30	Dic₁₀₆	—	4	2	no	—
31	C₈×C₅₃	—	2	1	no	—
32	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₅₃	#41	1	1	yes	characteristic
34	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
35	C₄×C₅₃	—	2	2	no	—
36	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
37	C₄×C₅₃	#135	1	1	yes	characteristic
38	Dic₅₃	—	4	4	no	—
39	Dic₅₃	—	8	2	no	—
40	#2176	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
42	C₂×C₅₃	#510	1	2	yes	—
43	C₂×Q₃₂	—	53	1	no	—
44	C₄×Q₁₆	—	53	2	no	—
45	#367	—	53	1	no	—
46	#358	—	53	2	no	—
47	#346	D₅₃	1	1	yes	characteristic
48	C₅₃	#2176	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₁₆	—	53	2	no	—
50	C₄×Q₈	—	106	2	no	—
51	Q₃₂	—	53	4	no	—
52	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
54	#107	—	53	2	no	—
55	#103	—	106	2	no	—
56	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
57	C₂×Q₈	—	106	2	no	—
58	Q₁₆	—	53	4	no	—
59	Q₁₆	—	106	2	no	—
60	C₂×C₈	D₂₁₂	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
62	C₄⋊C₄	—	106	2	no	—
63	C₄⋊C₄	—	212	2	no	—
64	C₄²	—	212	2	no	—
65	C₄²	D₂₁₂	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₅₃	1	2	yes	—
68	Q₈	—	106	4	no	—
69	Q₈	—	212	2	no	—
70	C₂×C₄	—	212	2	no	—
71	C₂×C₄	—	212	2	no	—
72	C₂×C₄	D₄₂₄	1	2	yes	—
73	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
76	C₈	#11719945	1	1	yes	characteristic
77	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	424	2	no	—
80	C₄	—	212	4	no	—
81	C₄	C₄×D₂₁₂	1	1	yes	characteristic
82	C₄	#13133239	1	1	yes	characteristic
83	C₂	C₄×D₄₂₄	1	1	yes	characteristic
84	C₂	#16811066	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#346	D₅₃
hypocenter	C₅₃	#2176
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₄₂₄	C₄	semidirect
48. C₅₃	#2176	semidirect