Subgroups — #20272656

Order 6784.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810649	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
3	C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₄	1	1	yes	characteristic
5	#13133176	C₄	1	1	yes	characteristic
6	C₂×#11719908	—	2	1	no	—
7	#13133146	—	2	1	no	—
8	#13133146	—	2	1	no	—
9	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	4	1	no	—
13	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
15	C₂²⋊Dic₅₃	—	2	1	no	—
16	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
17	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
18	C₄×Dic₅₃	—	2	1	no	—
19	#11719917	—	4	1	no	—
20	C₂×#88322	—	2	1	no	—
21	C₁₆×C₅₃	—	4	1	no	—
22	#11719908	—	4	1	no	—
23	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×Dic₅₃	—	2	1	no	—
29	C₂×Dic₅₃	—	2	1	no	—
30	C₈×C₅₃	—	2	1	no	—
31	C₈×C₅₃	—	2	1	no	—
32	#88322	—	4	1	no	—
33	C₂²×C₅₃	—	2	1	no	—
34	C₂²×C₅₃	—	2	1	no	—
35	C₂²×C₅₃	—	2	1	no	—
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	#99	1	1	yes	characteristic
39	C₄×C₅₃	#101	1	1	yes	characteristic
40	Dic₅₃	—	4	1	no	—
41	Dic₅₃	—	4	1	no	—
42	#1323	—	53	1	no	maximal, Sylow
43	C₂×C₅₃	—	4	1	no	—
44	C₂×C₅₃	#324	1	1	yes	characteristic
45	C₂×C₅₃	#350	1	1	yes	characteristic
46	C₂×C₅₃	#350	1	1	yes	characteristic
47	#424	—	53	1	no	—
48	C₂²⋊C₁₆	—	53	1	no	—
49	C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
50	C₅₃	#1323	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₁₆	—	53	1	no	—
52	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
53	#117	—	53	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	106	1	no	—
56	C₄⋊C₈	—	106	1	no	—
57	C₄⋊C₈	—	106	1	no	—
58	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
59	M₁₆	—	212	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	106	1	no	—
62	C₂×C₈	D₂₁₂	1	1	yes	characteristic
63	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
64	C₄⋊C₄	—	53	1	no	—
65	C₄⋊C₄	—	53	1	no	—
66	C₂²⋊C₄	—	106	1	no	—
67	C₄²	—	106	1	no	—
68	C₁₆	—	4	1	no	—
69	C₁₆	—	212	1	no	—
70	C₂³	#11719912	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	106	1	no	—
74	C₂×C₄	—	106	1	no	—
75	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
76	C₂×C₄	#11719921	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	212	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#13133129	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	212	1	no	—
86	C₄	—	212	1	no	—
87	C₄	#13133131	1	1	yes	characteristic
88	C₄	#13133135	1	1	yes	characteristic
89	C₂	—	4	1	no	—
90	C₂	#16810348	1	1	yes	characteristic
91	C₂	#16810406	1	1	yes	characteristic
92	C₂	#16810408	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#1323
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
50. C₅₃	#1323	semidirect