Subgroups — #20273019

Order 6784.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719912	C₂	1	2	yes	maximal
3	#16810345	C₂	1	2	yes	maximal, complemented
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719912	C₄	1	2	yes	—
7	C₂×#11719912	C₄	1	4	yes	—
8	C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#13133127	C₄	1	2	yes	—
10	#13133127	C₄	1	4	yes	—
11	C₄×#88322	C₂²	1	2	yes	—
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄²×C₅₃	C₂³	1	1	yes	characteristic
20	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	2	yes	—
23	C₂×#88322	C₂×C₄	1	4	yes	—
24	#11719912	—	2	4	no	—
25	#11719912	—	2	8	no	—
26	C₂²×D₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₅₃	C₄²	1	1	yes	characteristic
30	C₂×Dic₅₃	C₄²	1	2	yes	—
31	C₄×D₅₃	—	2	4	no	—
32	C₄×D₅₃	—	2	2	no	—
33	C₈×C₅₃	—	2	4	no	—
34	C₈×C₅₃	M₁₆	1	4	yes	complemented
35	#88322	—	2	2	no	—
36	#88322	M₁₆	1	8	yes	complemented
37	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
38	C₂×D₅₃	—	2	1	no	—
39	C₂×D₅₃	—	2	2	no	—
40	C₄×C₅₃	C₂×M₁₆	1	4	yes	—
41	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
42	Dic₅₃	—	2	1	no	—
43	Dic₅₃	—	2	2	no	—
44	#1455	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
46	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
47	D₅₃	—	4	1	no	—
48	C₄×M₁₆	—	53	2	no	—
49	C₄×M₁₆	—	53	1	no	—
50	#322	—	53	2	no	—
51	#322	D₅₃	1	1	yes	characteristic, complemented
52	C₈²	—	53	1	no	—
53	C₅₃	#1455	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₁₆	—	53	4	no	—
55	C₂×M₁₆	—	53	2	no	—
56	C₂×C₄²	—	53	1	no	—
57	#97	—	53	4	no	—
58	#97	—	53	2	no	—
59	C₄×C₈	—	53	2	no	—
60	C₄×C₈	—	53	1	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₄×C₈	C₂×D₅₃	1	2	yes	—
63	C₂²×C₄	—	53	2	no	—
64	C₂²×C₄	—	53	1	no	—
65	M₁₆	—	106	8	no	—
66	M₁₆	—	106	4	no	—
67	C₂×C₈	—	53	4	no	—
68	C₂×C₈	—	53	2	no	—
69	C₂×C₈	C₄×D₅₃	1	4	yes	—
70	C₂×C₈	C₄×D₅₃	1	2	yes	—
71	C₄²	—	53	2	no	—
72	C₄²	—	53	1	no	—
73	C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₂³	—	53	1	no	—
75	C₂×C₄	—	53	1	no	—
76	C₂×C₄	—	53	2	no	—
77	C₂×C₄	—	106	4	no	—
78	C₂×C₄	—	106	2	no	—
79	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
80	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	53	8	no	—
83	C₈	—	106	2	no	—
84	C₈	#11719912	1	4	yes	complemented
85	C₂²	—	106	1	no	—
86	C₂²	—	106	2	no	—
87	C₂²	C₄²×D₅₃	1	1	yes	characteristic
88	C₄	—	106	1	no	—
89	C₄	—	106	2	no	—
90	C₄	M₁₆×D₅₃	1	4	yes	—
91	C₄	C₂×#11719912	1	2	yes	—
92	C₂	—	212	1	no	—
93	C₂	C₄×#11719912	1	1	yes	characteristic
94	C₂	#16810575	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#322	C₂
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1455
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #16810345	C₂	semidirect
4. C₈×#88322	C₂	semidirect
5. C₅₃×#322	C₂	semidirect
34. C₈×C₅₃	M₁₆	semidirect
36. #88322	M₁₆	semidirect
51. #322	D₅₃	semidirect
53. C₅₃	#1455	semidirect
84. C₈	#11719912	semidirect