Subgroups — #20274463

Order 6784.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810377	C₂	1	4	yes	maximal
4	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	C₄	1	4	yes	—
7	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
8	#13133128	C₄	1	4	yes	complemented
9	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#97×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
15	C₂×C₈×C₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×Dic₅₃	—	2	4	no	—
20	C₄×Dic₅₃	—	2	8	no	—
21	C₂×#88322	—	2	4	no	—
22	C₂×#88322	C₂×C₄	1	4	yes	—
23	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	C₄²	1	4	yes	—
26	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	#41	1	4	yes	—
29	C₂×Dic₅₃	—	2	4	no	—
30	C₂×Dic₅₃	—	2	4	no	—
31	C₂×Dic₅₃	M₁₆	1	8	yes	—
32	C₈×C₅₃	—	2	4	no	—
33	#88322	—	2	4	no	—
34	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
35	C₂²×C₅₃	#117	1	2	yes	—
36	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
37	C₄×C₅₃	—	2	4	no	—
38	C₄×C₅₃	#117	1	2	yes	—
39	C₄×C₅₃	#117	1	2	yes	—
40	Dic₅₃	—	2	8	no	—
41	#1745	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	#376	1	1	yes	characteristic
43	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
44	C₂×C₅₃	#432	1	4	yes	—
45	C₂×#97	—	53	1	no	—
46	C₂×#97	D₅₃	1	1	yes	characteristic
47	C₄³	—	53	1	no	—
48	#336	—	53	4	no	—
49	C₅₃	#1745	1	1	yes	characteristic, complemented, Sylow
50	C₂²×C₈	—	53	2	no	—
51	C₂²×C₈	C₂×D₅₃	1	2	yes	—
52	C₂×C₄²	—	53	4	no	—
53	C₂×C₄²	—	53	2	no	—
54	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
55	#97	—	53	4	no	—
56	#97	Dic₅₃	1	4	yes	complemented
57	C₂²×C₄	—	53	4	no	—
58	C₂²×C₄	C₄×D₅₃	1	2	yes	—
59	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	53	4	no	—
62	C₂×C₈	—	106	4	no	—
63	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
64	C₄²	—	106	8	no	—
65	C₄²	—	106	4	no	—
66	C₄²	C₄×D₅₃	1	2	yes	—
67	C₄²	C₂×Dic₅₃	1	2	yes	—
68	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	53	8	no	—
71	C₂×C₄	—	106	4	no	—
72	C₂×C₄	—	106	4	no	—
73	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
74	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
75	C₂×C₄	#11719945	1	4	yes	—
76	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
77	C₈	—	2	4	no	—
78	C₈	—	106	4	no	—
79	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
80	C₂²	#13133174	1	2	yes	—
81	C₂²	M₁₆×D₅₃	1	4	yes	—
82	C₄	—	2	4	no	—
83	C₄	—	106	8	no	—
84	C₄	#13133174	1	2	yes	—
85	C₄	#13133176	1	2	yes	—
86	C₂	#16810469	1	1	yes	characteristic
87	C₂	M₁₆×Dic₅₃	1	2	yes	—
88	C₂	#16810678	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1745
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
8. #13133128	C₄	semidirect
11. #97×C₅₃	C₄	semidirect
49. C₅₃	#1745	semidirect
56. #97	Dic₅₃	semidirect