Subgroups — #27273467

Order 13568.

582 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133127	C₂	1	3	yes	maximal
2	C₂×#16810344	C₂	1	3	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133127	C₄	1	12	yes	—
5	C₂×C₄×#88322	C₂²	1	3	yes	—
6	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
7	#16810344	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
9	C₈²×C₅₃	C₄	1	4	yes	complemented
10	C₂²×#88322	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
12	#13133127	—	2	24	no	—
13	C₄×#88322	C₂×C₄	1	12	yes	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
15	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
17	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
19	C₂²×Dic₅₃	C₄²	1	4	yes	—
20	C₂×C₈×C₅₃	C₄²	1	12	yes	—
21	C₂×C₈×C₅₃	M₁₆	1	24	yes	—
22	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
23	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	—	2	12	no	—
25	C₂×#88322	—	2	12	no	—
26	C₂×#88322	C₄²	1	12	yes	—
27	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₅₃	C₂×M₁₆	1	12	yes	—
31	C₂×Dic₅₃	—	2	12	no	—
32	C₈×C₅₃	#97	1	24	yes	complemented
33	#88322	—	2	12	no	—
34	#7846	—	53	1	no	maximal, Sylow
35	C₂²×C₅₃	C₄³	1	1	yes	characteristic
36	C₂²×C₅₃	C₄×M₁₆	1	6	yes	—
37	C₄×C₅₃	C₂×#97	1	6	yes	—
38	C₄×C₅₃	C₄×M₁₆	1	6	yes	—
39	Dic₅₃	—	4	4	no	—
40	C₄×#97	—	53	3	no	—
41	C₂×#322	—	53	3	no	—
42	C₂×C₈²	D₅₃	1	1	yes	characteristic
43	C₂×C₅₃	#1455	1	4	yes	—
44	C₂×C₅₃	C₄×#97	1	3	yes	—
45	C₂×#97	—	53	12	no	—
46	C₂×C₄×C₈	—	53	3	no	—
47	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
48	C₄³	—	53	1	no	—
49	#322	—	53	12	no	—
50	C₈²	Dic₅₃	1	4	yes	complemented
51	C₅₃	#7846	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₈	—	53	6	no	—
53	C₂²×C₈	C₄×D₅₃	1	6	yes	—
54	C₂×C₄²	—	53	6	no	—
55	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
56	#97	—	106	24	no	—
57	C₄×C₈	—	53	12	no	—
58	C₄×C₈	C₄×D₅₃	1	6	yes	—
59	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
60	C₂²×C₄	—	53	4	no	—
61	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
62	C₂×C₈	—	53	12	no	—
63	C₂×C₈	—	106	12	no	—
64	C₂×C₈	#11719912	1	24	yes	—
65	C₂×C₈	C₄×Dic₅₃	1	12	yes	—
66	C₄²	—	106	12	no	—
67	C₄²	C₂×C₄×D₅₃	1	3	yes	—
68	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
69	C₂³	C₄²×D₅₃	1	1	yes	characteristic
70	C₂×C₄	—	106	12	no	—
71	C₂×C₄	C₄²×D₅₃	1	3	yes	—
72	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
73	C₂×C₄	C₂×#11719912	1	12	yes	—
74	C₈	—	106	12	no	—
75	C₈	#13133127	1	24	yes	complemented
76	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
77	C₂²	C₄×#11719912	1	6	yes	—
78	C₄	—	212	4	no	—
79	C₄	C₂×#13133127	1	6	yes	—
80	C₄	C₄×#11719912	1	6	yes	—
81	C₂	#20273017	1	4	yes	—
82	C₂	C₄×#13133127	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#7846
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. #16810344	C₄	semidirect
9. C₈²×C₅₃	C₄	semidirect
32. C₈×C₅₃	#97	semidirect
50. C₈²	Dic₅₃	semidirect
51. C₅₃	#7846	semidirect
75. C₈	#13133127	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133127	2
C₂×#16810344	2
C₂×C₈²×C₅₃	2
#7846	53