Subgroups — #20272695

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
5	C₂×#11719908	—	2	1	no	—
6	C₄⋊#88322	C₄	1	1	yes	characteristic
7	#13133146	—	2	1	no	—
8	C₈×Dic₅₃	—	2	1	no	—
9	C₂×C₁₆×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
16	#11719919	—	2	1	no	—
17	C₄×Dic₅₃	—	2	1	no	—
18	C₂×#88322	—	2	1	no	—
19	C₁₆×C₅₃	—	2	2	no	—
20	#11719908	—	4	1	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×Dic₅₃	—	2	1	no	—
25	C₂×Dic₅₃	—	2	1	no	—
26	Dic₁₀₆	—	2	1	no	—
27	Dic₁₀₆	C₁₆	1	2	yes	complemented
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
31	#88322	—	4	1	no	—
32	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	#103	1	1	yes	characteristic
35	C₄×C₅₃	#104	1	1	yes	characteristic
36	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₅₃	M₃₂	1	1	yes	characteristic
38	Dic₅₃	—	2	2	no	—
39	Dic₅₃	—	4	1	no	—
40	Q₈⋊C₁₆	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
42	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₂×C₅₃	#350	1	1	yes	characteristic
44	C₈×Q₈	—	53	1	no	—
45	C₄⋊C₁₆	—	53	1	no	—
46	C₄×C₁₆	D₅₃	1	1	yes	characteristic
47	C₅₃	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
48	C₄×Q₈	—	53	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	106	1	no	—
51	C₄⋊C₈	—	53	1	no	—
52	C₄⋊C₈	—	106	1	no	—
53	C₄×C₈	—	106	1	no	—
54	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
55	C₂×Q₈	—	53	1	no	—
56	C₂×C₈	—	106	1	no	—
57	C₂×C₈	D₂₁₂	1	1	yes	characteristic
58	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
59	C₄⋊C₄	—	53	1	no	—
60	C₄⋊C₄	—	106	1	no	—
61	C₄²	—	106	1	no	—
62	C₄²	C₄×D₅₃	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	212	1	no	—
65	Q₈	—	53	2	no	—
66	Q₈	—	106	1	no	—
67	C₂×C₄	—	106	1	no	—
68	C₂×C₄	—	106	1	no	—
69	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
70	C₂×C₄	#11719912	1	1	yes	characteristic
71	C₂×C₄	#11719921	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	212	1	no	—
74	C₈	#11719913	1	1	yes	characteristic
75	C₈	Dic₂₁₂	1	1	yes	characteristic
76	C₂²	#13133129	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	106	2	no	—
79	C₄	—	212	1	no	—
80	C₄	#13133142	1	1	yes	characteristic
81	C₄	#13133145	1	1	yes	characteristic
82	C₄	C₁₆×D₅₃	1	1	yes	characteristic
83	C₄	#13133156	1	1	yes	characteristic
84	C₂	#16810355	1	1	yes	characteristic
85	C₂	#16810402	1	1	yes	characteristic
86	C₂	#16810406	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	Q₈⋊C₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
27. Dic₁₀₆	C₁₆	semidirect
47. C₅₃	Q₈⋊C₁₆	semidirect