Subgroups — #20272686

Order 6784.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#16810374	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
5	C₄×Dic₁₀₆	—	2	1	no	—
6	#13133151	C₄	1	1	yes	characteristic
7	C₄⋊#88322	—	2	1	no	—
8	#13133142	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₆	—	2	1	no	—
12	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₄⋊Dic₅₃	—	2	1	no	—
16	#11719919	—	4	1	no	—
17	C₄×Dic₅₃	—	4	1	no	—
18	C₂×#88322	—	4	1	no	—
19	Dic₂₁₂	—	2	1	no	—
20	Dic₂₁₂	C₈	1	2	yes	complemented
21	C₁₆×C₅₃	—	2	2	no	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₅₃	—	4	1	no	—
26	C₂×Dic₅₃	—	4	1	no	—
27	Dic₁₀₆	—	2	2	no	—
28	Dic₁₀₆	—	4	1	no	—
29	C₈×C₅₃	—	2	1	no	—
30	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₅₃	M₁₆	1	1	yes	characteristic
32	#88322	—	8	1	no	—
33	C₂²×C₅₃	#102	1	1	yes	characteristic
34	C₄×C₅₃	—	2	1	no	—
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₅₃	#104	1	1	yes	characteristic
38	Dic₅₃	—	4	2	no	—
39	Dic₅₃	—	8	1	no	—
40	#1334	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₅₃	#360	1	1	yes	characteristic
43	C₂×C₅₃	#362	1	1	yes	characteristic
44	C₄×Q₁₆	—	53	1	no	—
45	#346	D₅₃	1	1	yes	characteristic
46	#334	—	53	1	no	—
47	C₅₃	#1334	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₁₆	—	53	1	no	—
49	C₄×Q₈	—	106	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	53	1	no	—
52	C₄⋊C₈	—	106	1	no	—
53	#103	—	106	1	no	—
54	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
55	C₂×Q₈	—	106	1	no	—
56	Q₁₆	—	53	2	no	—
57	Q₁₆	—	106	1	no	—
58	C₂×C₈	—	212	1	no	—
59	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
60	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
61	C₄⋊C₄	—	106	1	no	—
62	C₄⋊C₄	—	212	1	no	—
63	C₄²	—	212	1	no	—
64	C₄²	D₂₁₂	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	Q₈	—	106	2	no	—
67	Q₈	—	212	1	no	—
68	C₂×C₄	—	212	1	no	—
69	C₂×C₄	—	212	1	no	—
70	C₂×C₄	#11719913	1	1	yes	characteristic
71	C₂×C₄	D₄₂₄	1	1	yes	characteristic
72	C₂×C₄	#11719921	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	424	1	no	—
75	C₈	C₈×D₅₃	1	1	yes	characteristic
76	C₈	#11719912	1	1	yes	characteristic
77	C₂²	#13133139	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	2	1	no	—
80	C₄	—	424	1	no	—
81	C₄	—	212	2	no	—
82	C₄	#13133129	1	1	yes	characteristic
83	C₄	#13133145	1	1	yes	characteristic
84	C₂	#16810352	1	1	yes	characteristic
85	C₂	#16810434	1	1	yes	characteristic
86	C₂	#16810440	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133139
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#346	C₂
socle	C₂²×C₅₃	#102
hypercenter	#346	D₅₃
hypocenter	C₅₃	#1334
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
20. Dic₂₁₂	C₈	semidirect
47. C₅₃	#1334	semidirect