Subgroups — #20272697

Order 6784.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	#16810399	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#446	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₂×#11719908	—	2	1	no	—
6	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₅₃	—	2	1	no	—
8	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
9	#97×C₅₃	—	2	1	no	—
10	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₃	C₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
17	C₄²×C₅₃	—	2	1	no	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#11719908	—	2	2	no	—
20	#11719908	—	4	1	no	—
21	Q₈×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
27	C₈×C₅₃	—	2	1	no	—
28	C₈×C₅₃	—	4	1	no	—
29	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	—	4	1	no	—
35	C₄×C₅₃	—	4	1	no	—
36	C₄×C₅₃	#103	1	1	yes	characteristic
37	C₄×C₅₃	#104	1	1	yes	characteristic
38	#1338	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
40	C₂×C₅₃	#349	1	1	yes	characteristic
41	C₂×C₅₃	#350	1	1	yes	characteristic
42	#446	D₅₃	1	1	yes	characteristic
43	C₄⋊C₁₆	—	53	1	no	—
44	#346	—	53	1	no	—
45	C₅₃	#1338	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₈	Dic₅₃	1	1	yes	characteristic
47	C₂×C₁₆	—	106	1	no	—
48	C₂×C₁₆	—	106	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
51	#97	—	2	1	no	—
52	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
53	C₂×Q₈	#88322	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
56	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	#88322	1	1	yes	characteristic
59	C₄²	—	2	1	no	—
60	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
61	C₁₆	—	106	2	no	—
62	C₁₆	—	212	1	no	—
63	Q₈	—	4	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₂×#88322	1	1	yes	characteristic
67	C₂×C₄	#11719917	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	4	1	no	—
71	C₈	#11719924	1	1	yes	characteristic
72	C₈	#11719925	1	1	yes	characteristic
73	C₂²	C₂²⋊#88322	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	#13133141	1	1	yes	characteristic
79	C₄	#13133144	1	1	yes	characteristic
80	C₂	Q₈⋊#88322	1	1	yes	characteristic
81	C₂	#16810405	1	1	yes	characteristic
82	C₂	#16810407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#446	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	#446	D₅₃
hypocenter	C₅₃	#1338
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #446

Complements

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

N	H ≅ G/N	type
45. C₅₃	#1338	semidirect