Subgroups — #20274854

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810646	C₂	1	2	yes	maximal
2	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
3	#16810388	C₂	1	1	yes	maximal, characteristic
4	#16810387	C₂	1	2	yes	maximal
5	C₅₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133146	—	2	4	no	—
10	#13133127	C₄	1	4	yes	—
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₃	Q₈	1	2	yes	—
15	M₁₆×C₅₃	—	2	2	no	—
16	M₁₆×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	2	no	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₄×Dic₅₃	D₄	1	2	yes	—
22	#11719917	—	2	2	no	—
23	#11719917	—	2	2	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	C₂×C₄	1	2	yes	—
27	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₄×C₅₃	#41	1	2	yes	—
32	C₂×Dic₅₃	—	4	2	no	—
33	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
34	C₈×C₅₃	—	2	2	no	—
35	C₈×C₅₃	—	4	2	no	—
36	#88322	—	2	2	no	—
37	#88322	—	4	2	no	—
38	C₂²×C₅₃	—	2	2	no	—
39	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₅₃	#125	1	2	yes	—
41	C₄×C₅₃	#117	1	1	yes	characteristic
42	C₄×C₅₃	#117	1	1	yes	characteristic
43	C₄×C₅₃	#124	1	2	yes	—
44	Dic₅₃	—	2	4	no	—
45	#1838	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	2	no	—
47	C₂×C₅₃	#382	1	1	yes	characteristic
48	C₂×C₅₃	#430	1	2	yes	—
49	#423	—	53	2	no	—
50	C₂×#97	—	53	1	no	—
51	#341	—	53	2	no	—
52	#341	—	53	1	no	—
53	#341	D₅₃	1	1	yes	characteristic
54	C₅₃	#1838	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	—	53	2	no	—
56	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
57	C₂²×C₈	—	53	1	no	—
58	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
59	C₂×C₄²	—	53	1	no	—
60	C₄⋊C₈	—	106	4	no	—
61	#97	—	53	4	no	—
62	C₂²×C₄	—	53	2	no	—
63	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
64	M₁₆	—	2	2	no	—
65	M₁₆	—	2	2	no	—
66	M₁₆	—	106	2	no	—
67	M₁₆	—	106	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	53	2	no	—
71	C₂×C₈	—	106	2	no	—
72	C₂×C₈	—	106	2	no	—
73	C₂×C₈	C₄×D₅₃	1	2	yes	—
74	C₄²	—	53	2	no	—
75	C₄²	—	53	2	no	—
76	C₂³	Q₈×D₅₃	1	1	yes	characteristic
77	C₂×C₄	—	53	4	no	—
78	C₂×C₄	—	212	2	no	—
79	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₄	#11719945	1	2	yes	—
81	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
82	C₂×C₄	#11719947	1	2	yes	—
83	C₈	—	2	2	no	—
84	C₈	—	4	2	no	—
85	C₈	—	106	2	no	—
86	C₈	—	212	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
89	C₂²	#13133207	1	2	yes	—
90	C₄	—	106	4	no	—
91	C₄	#13133173	1	1	yes	characteristic
92	C₄	#13133174	1	1	yes	characteristic
93	C₄	#13133203	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	#16810487	1	1	yes	characteristic
96	C₂	#16810670	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#341	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#341	D₅₃
hypocenter	C₅₃	#1838
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

Complements

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

N	H ≅ G/N	type
54. C₅₃	#1838	semidirect