Subgroups — #20276496

Order 6784.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#16810660	C₂	1	2	yes	maximal
3	#16810450	C₂	1	2	yes	maximal, complemented
4	#16810449	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
7	#13133175	—	2	2	no	—
8	C₂×#11719908	C₂²	1	1	yes	characteristic
9	C₁₆×D₅₃	C₄	1	4	yes	complemented
10	#13133151	C₂²	1	2	yes	—
11	#13133148	—	2	2	no	—
12	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	#107×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	—	2	2	no	—
17	#11719921	—	4	2	no	—
18	C₄⋊Dic₅₃	—	2	2	no	—
19	C₄×Dic₅₃	—	4	2	no	—
20	C₂×#88322	D₄	1	1	yes	characteristic
21	C₈×D₅₃	C₂×C₄	1	2	yes	—
22	C₈×D₅₃	Q₈	1	2	yes	—
23	C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	#11719908	C₂×C₄	1	2	yes	—
25	C₂²×D₅₃	D₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	2	no	—
29	C₂×Dic₅₃	D₈	1	1	yes	characteristic
30	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
31	C₄×D₅₃	Q₁₆	1	2	yes	—
32	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	#88322	C₄⋊C₄	1	2	yes	—
35	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
36	C₂×D₅₃	—	2	1	no	—
37	C₂×D₅₃	—	2	1	no	—
38	C₂×D₅₃	#107	1	2	yes	—
39	C₄×C₅₃	—	8	2	no	—
40	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₅₃	—	8	2	no	—
43	Dic₅₃	#107	1	2	yes	—
44	#2158	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
46	C₂×C₅₃	#508	1	1	yes	characteristic
47	C₂×C₅₃	#508	1	1	yes	characteristic
48	D₅₃	—	2	2	no	—
49	C₂²×C₁₆	—	53	1	no	—
50	#427	—	53	2	no	—
51	#367	—	53	2	no	—
52	#366	—	53	1	no	—
53	#366	D₅₃	1	1	yes	characteristic, complemented
54	C₅₃	#2158	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₈	—	53	1	no	—
56	#117	—	106	2	no	—
57	C₂×C₁₆	—	53	1	no	—
58	C₂×C₁₆	—	53	4	no	—
59	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
60	#107	—	53	2	no	—
61	#107	C₂×D₅₃	1	2	yes	—
62	#106	—	106	2	no	—
63	C₂²×C₄	—	53	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	53	2	no	—
66	C₂×C₈	—	53	2	no	—
67	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	106	2	no	—
70	C₂²⋊C₄	—	212	2	no	—
71	C₄²	—	212	2	no	—
72	C₁₆	—	53	2	no	—
73	C₁₆	C₄×D₅₃	1	2	yes	—
74	C₂³	—	53	1	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	53	1	no	—
77	C₂×C₄	—	53	2	no	—
78	C₂×C₄	—	53	2	no	—
79	C₂×C₄	—	212	2	no	—
80	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
81	C₈	—	53	2	no	—
82	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
83	C₈	Q₈×D₅₃	1	1	yes	characteristic
84	C₂²	—	53	2	no	—
85	C₂²	—	106	1	no	—
86	C₂²	—	106	1	no	—
87	C₂²	D₈×D₅₃	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	53	2	no	—
90	C₄	—	424	2	no	—
91	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
92	C₄	Q₁₆×D₅₃	1	1	yes	characteristic
93	C₂	—	106	2	no	—
94	C₂	#107×D₅₃	1	1	yes	characteristic
95	C₂	#16811052	1	1	yes	characteristic
96	C₂	#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#366	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#366	D₅₃
hypocenter	C₅₃	#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
3. #16810450	C₂	semidirect
4. #16810449	C₂	semidirect
5. C₅₃×#366	C₂	semidirect
9. C₁₆×D₅₃	C₄	semidirect
53. #366	D₅₃	semidirect
54. C₅₃	#2158	semidirect