Subgroups — #27271803

Order 13568.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276499	C₂	1	2	yes	maximal
2	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#16810661	—	2	2	no	—
4	#16810451	C₄	1	4	yes	complemented
5	#16810449	—	2	2	no	—
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×M₆₄	—	2	2	no	—
8	C₂×C₃₂×C₅₃	C₄	1	2	yes	complemented
9	#13133176	—	4	2	no	—
10	#13133151	—	2	4	no	—
11	#13133149	—	4	2	no	—
12	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
16	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
17	C₃₂×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
22	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂²⋊Dic₅₃	—	8	2	no	—
24	C₄⋊Dic₅₃	—	4	4	no	—
25	C₄×Dic₅₃	—	8	2	no	—
26	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
27	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#102	1	2	yes	—
31	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
32	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
33	C₂×C₄×C₅₃	Q₃₂	1	1	yes	characteristic
34	C₂×Dic₅₃	—	8	4	no	—
35	C₈×C₅₃	#95	1	1	yes	characteristic
36	C₈×C₅₃	#103	1	2	yes	—
37	C₈×C₅₃	#106	1	1	yes	characteristic
38	#7475	—	53	1	no	maximal, Sylow
39	C₂²×C₅₃	—	4	1	no	—
40	C₂²×C₅₃	#357	1	2	yes	—
41	C₂²×C₅₃	#366	1	1	yes	characteristic
42	C₄×C₅₃	#340	1	1	yes	characteristic
43	C₄×C₅₃	#358	1	2	yes	—
44	C₄×C₅₃	#367	1	1	yes	characteristic
45	Dic₅₃	—	16	4	no	—
46	C₂×M₆₄	D₅₃	1	1	yes	characteristic
47	#2158	—	53	2	no	—
48	C₂×C₅₃	—	2	2	no	—
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	#1380	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
52	#427	—	106	2	no	—
53	M₆₄	—	2	2	no	—
54	C₂×C₃₂	Dic₅₃	1	2	yes	complemented
55	#367	—	53	4	no	—
56	#366	—	106	2	no	—
57	C₅₃	#7475	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
59	#117	—	212	2	no	—
60	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
61	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
62	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
63	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
64	#107	—	106	4	no	—
65	#106	—	212	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
68	C₂×C₈	#11719913	1	2	yes	—
69	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
70	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
71	C₂×C₈	#11719921	1	2	yes	—
72	C₄⋊C₄	—	212	4	no	—
73	C₂²⋊C₄	—	424	2	no	—
74	C₄²	—	424	2	no	—
75	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
76	C₁₆	#11719919	1	2	yes	—
77	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	C₂³	D₈₄₈	1	1	yes	characteristic
79	C₂×C₄	—	424	4	no	—
80	C₂×C₄	#13133139	1	2	yes	—
81	C₂×C₄	#13133151	1	1	yes	characteristic
82	C₂×C₄	#13133162	1	2	yes	—
83	C₂×C₄	Dic₄₂₄	1	1	yes	characteristic
84	C₈	#13133124	1	1	yes	characteristic
85	C₈	#13133142	1	2	yes	—
86	C₈	#13133149	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	#16810425	1	2	yes	—
89	C₂²	#16810449	1	1	yes	characteristic
90	C₄	—	848	4	no	—
91	C₄	#16810386	1	1	yes	characteristic
92	C₄	#16810428	1	2	yes	—
93	C₄	#16810451	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	#20272807	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810451
commutator	C₁₆×C₅₃	C₄²
Frattini	C₂²×C₁₆	C₂×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂×C₅₃	#1380
prime radical	C₂³×C₅₃	D₁₆
cosocle	C₂²×C₁₆×C₅₃	C₂²
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	#7475
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
4. #16810451	C₄	semidirect
8. C₂×C₃₂×C₅₃	C₄	semidirect
54. C₂×C₃₂	Dic₅₃	semidirect
57. C₅₃	#7475	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276499	2
C₂×C₅₃×M₆₄	2
#7475	53