Subgroups — #20276623

Order 6784.

126 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	#16810441	C₂	1	2	yes	maximal
4	#16810398	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
7	C₂×#11719908	C₂²	1	2	yes	—
8	C₄⋊#88322	C₄	1	1	yes	characteristic
9	#13133146	C₄	1	2	yes	—
10	C₈×Dic₅₃	C₂²	1	2	yes	—
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	C₈	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
19	#11719919	C₈	1	2	yes	—
20	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×#88322	C₂×C₄	1	2	yes	—
22	C₁₆×C₅₃	—	2	1	no	—
23	C₁₆×C₅₃	Q₈	1	2	yes	complemented
24	#11719908	—	2	2	no	—
25	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
29	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
30	Dic₁₀₆	—	2	2	no	—
31	C₈×C₅₃	—	2	1	no	—
32	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
33	C₈×C₅₃	#41	1	1	yes	characteristic
34	#88322	—	2	2	no	—
35	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	M₃₂	1	2	yes	—
38	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₅₃	#131	1	1	yes	characteristic
40	Dic₅₃	—	2	2	no	—
41	Dic₅₃	—	2	2	no	—
42	#2183	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	C₈×Q₈	1	1	yes	characteristic
44	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
45	C₂×C₅₃	#504	1	1	yes	characteristic
46	C₈×Q₈	—	53	1	no	—
47	C₄⋊C₁₆	—	53	2	no	—
48	C₄⋊C₁₆	—	53	1	no	—
49	#346	—	53	2	no	—
50	C₄×C₁₆	D₅₃	1	1	yes	characteristic
51	C₅₃	#2183	1	1	yes	characteristic, complemented, Sylow
52	C₄×Q₈	—	53	1	no	—
53	C₂×C₁₆	—	53	2	no	—
54	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
55	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
56	C₄⋊C₈	—	53	2	no	—
57	C₄⋊C₈	—	53	1	no	—
58	C₄×C₈	—	53	2	no	—
59	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
60	C₂×Q₈	—	53	1	no	—
61	C₂×C₈	—	53	2	no	—
62	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
64	C₄⋊C₄	—	53	2	no	—
65	C₄⋊C₄	—	53	1	no	—
66	C₄²	—	53	2	no	—
67	C₄²	C₄×D₅₃	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	106	2	no	—
70	C₁₆	Dic₁₀₆	1	2	yes	complemented
71	Q₈	—	106	2	no	—
72	C₂×C₄	—	53	2	no	—
73	C₂×C₄	—	53	2	no	—
74	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
75	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	106	2	no	—
79	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
80	C₈	#11719945	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	106	2	no	—
84	C₄	—	106	2	no	—
85	C₄	#13133156	1	2	yes	—
86	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
87	C₄	#13133223	1	1	yes	characteristic
88	C₂	C₈×Dic₁₀₆	1	1	yes	characteristic
89	C₂	C₂×#13133156	1	1	yes	characteristic
90	C₂	#16811039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×C₁₆×C₅₃	C₂
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2183
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
23. C₁₆×C₅₃	Q₈	semidirect
51. C₅₃	#2183	semidirect
70. C₁₆	Dic₁₀₆	semidirect