Subgroups — #20276967

Order 6784.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133163	C₂	1	4	yes	maximal
2	#16811003	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719908	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₂₁₂	—	2	1	no	—
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	#13133217	—	2	1	no	—
8	#13133163	—	2	8	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₅₃	—	2	1	no	—
11	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₅₃	—	2	2	no	—
15	C₂×Dic₁₀₆	—	2	2	no	—
16	Q₁₆×C₅₃	—	2	4	no	—
17	Q₁₆×C₅₃	—	4	2	no	—
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₅₃	—	4	2	no	—
21	C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	4	2	no	—
23	Dic₂₁₂	—	2	4	no	—
24	Dic₂₁₂	—	4	2	no	—
25	#11719908	D₄	1	4	yes	—
26	Q₈×C₅₃	—	4	2	no	—
27	Q₈×C₅₃	—	4	2	no	—
28	C₂×C₄×C₅₃	—	4	2	no	—
29	C₂×C₄×C₅₃	D₈	1	2	yes	—
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	2	no	—
32	Dic₁₀₆	—	4	2	no	—
33	Dic₁₀₆	—	4	2	no	—
34	C₈×C₅₃	D₈	1	2	yes	—
35	C₈×C₅₃	C₂×D₄	1	2	yes	—
36	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
37	C₄×C₅₃	—	8	2	no	—
38	C₄×C₅₃	Q₃₂	1	4	yes	complemented
39	C₄×C₅₃	#127	1	1	yes	characteristic
40	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
41	Dic₅₃	—	8	2	no	—
42	#2247	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	#493	1	1	yes	characteristic
44	C₂×C₅₃	C₂×Q₃₂	1	2	yes	—
45	C₂×Q₃₂	—	53	4	no	—
46	#494	—	53	1	no	—
47	#494	D₅₃	1	1	yes	characteristic
48	C₄×C₁₆	—	53	1	no	—
49	C₅₃	#2247	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×Q₁₆	—	53	2	no	—
52	C₂×Q₁₆	—	106	1	no	—
53	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
54	C₄⋊Q₈	—	2	1	no	—
55	C₄⋊Q₈	—	106	1	no	—
56	Q₃₂	—	106	8	no	—
57	C₂×C₁₆	—	53	2	no	—
58	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	106	2	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	Q₁₆	—	106	4	no	—
64	Q₁₆	—	212	2	no	—
65	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	212	2	no	—
69	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
70	C₁₆	—	53	4	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	212	2	no	—
74	Q₈	—	212	2	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	212	2	no	—
77	C₂×C₄	#11719922	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
79	C₈	#11719922	1	2	yes	—
80	C₈	D₄×D₅₃	1	2	yes	—
81	C₂²	C₂×#11719922	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	424	2	no	—
84	C₄	#13133163	1	4	yes	complemented
85	C₄	#13133213	1	1	yes	characteristic
86	C₄	C₂×#11719922	1	1	yes	characteristic
87	C₂	#16810999	1	1	yes	characteristic
88	C₂	C₂×#13133163	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#494	D₅₃
hypocenter	C₅₃	#2247
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
38. C₄×C₅₃	Q₃₂	semidirect
49. C₅₃	#2247	semidirect
84. C₄	#13133163	semidirect