Subgroups — #20272590

Order 6784.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719917	—	2	2	no	—
4	C₄⋊#88322	—	2	2	no	—
5	C₄⋊#88322	C₄	1	4	yes	—
6	C₂²×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	2	no	—
13	C₂×C₈×C₅₃	—	4	2	no	—
14	C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₄²×C₅₃	D₄	1	2	yes	—
16	#11719917	—	4	2	no	—
17	#11719917	—	4	2	no	—
18	C₂×#88322	—	2	4	no	—
19	C₂³×C₅₃	C₄⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	2	1	no	—
22	C₂×C₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
29	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
30	C₈×C₅₃	—	4	2	no	—
31	#88322	—	4	4	no	—
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	—	2	1	no	—
34	C₂²×C₅₃	#95	1	1	yes	characteristic
35	C₂²×C₅₃	#106	1	1	yes	characteristic
36	C₂²×C₅₃	#107	1	1	yes	characteristic
37	C₄×C₅₃	—	2	2	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	#100	1	1	yes	characteristic
40	C₄×C₅₃	#101	1	1	yes	characteristic
41	C₄×C₅₃	#102	1	2	yes	—
42	C₄×C₅₃	#103	1	2	yes	—
43	#1292	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#338	1	1	yes	characteristic
47	C₂×C₅₃	#340	1	1	yes	characteristic
48	C₂×C₅₃	#343	1	1	yes	characteristic
49	#423	—	53	2	no	—
50	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
51	C₅₃	#1292	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₁₆	—	106	2	no	—
53	C₂²×C₈	Dic₅₃	1	2	yes	—
54	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	53	4	no	—
57	C₄⋊C₈	—	106	2	no	—
58	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
59	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
60	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
61	M₁₆	—	212	2	no	—
62	M₁₆	—	212	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	106	4	no	—
66	C₄²	C₄×D₅₃	1	2	yes	—
67	C₄²	C₂²⋊D₅₃	1	2	yes	—
68	C₂³	C₄⋊Dic₅₃	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	#11719913	1	2	yes	—
73	C₂×C₄	D₄₂₄	1	1	yes	characteristic
74	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
75	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
76	C₂×C₄	#11719919	1	2	yes	—
77	C₂×C₄	#11719921	1	2	yes	—
78	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
79	C₈	—	4	2	no	—
80	C₈	—	212	4	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#13133124	1	1	yes	characteristic
84	C₂²	#13133149	1	1	yes	characteristic
85	C₂²	#13133151	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#13133134	1	1	yes	characteristic
89	C₄	#13133136	1	1	yes	characteristic
90	C₄	#13133139	1	2	yes	—
91	C₄	#13133142	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	1	no	—
94	C₂	#16810381	1	1	yes	characteristic
95	C₂	#16810386	1	1	yes	characteristic
96	C₂	#16810392	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133124
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1292
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
51. C₅₃	#1292	semidirect