Subgroups — #28300055

Order 13568.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290428	C₂	1	3	yes	maximal
2	#20289911	C₂	1	3	yes	maximal
3	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×D₅₃⋊C₈	C₂²	1	3	yes	—
5	C₄²×#6157	C₂²	1	1	yes	characteristic
6	(C₂×#6157)⋊C₈	C₄	1	12	yes	—
7	#88324⋊C₈	C₄	1	12	yes	complemented
8	C₄×C₈×D₅₃	C₂²	1	3	yes	—
9	C₈×#88322	C₄	1	3	yes	complemented
10	C₈²×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×D₅₃⋊C₈	C₂×C₄	1	6	yes	—
12	C₂×C₄×#6157	C₂×C₄	1	6	yes	—
13	C₄×#88324	C₂×C₄	1	12	yes	—
14	C₂×C₈×D₅₃	C₂×C₄	1	6	yes	—
15	C₄²×D₅₃	C₂³	1	1	yes	characteristic
16	C₄×#88322	C₂×C₄	1	3	yes	—
17	C₈×Dic₅₃	C₂×C₄	1	6	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	3	yes	—
19	C₂²×#6157	C₄²	1	4	yes	—
20	C₂×C₄×D₅₃	C₂²×C₄	1	3	yes	—
21	C₂×#88324	C₄²	1	12	yes	—
22	C₄×#6157	—	2	12	no	—
23	D₅₃⋊C₈	—	2	12	no	—
24	C₂×C₈×C₅₃	C₄²	1	6	yes	—
25	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₄×Dic₅₃	C₂²×C₄	1	3	yes	—
27	C₂×#88322	C₄²	1	6	yes	—
28	C₈×D₅₃	—	2	12	no	—
29	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
30	C₂×#6157	—	2	12	no	—
31	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
32	C₂×Dic₅₃	C₂×C₄²	1	3	yes	—
33	C₄×D₅₃	#131	1	12	yes	—
34	#88324	—	2	12	no	—
35	C₈×C₅₃	—	2	6	no	—
36	#88322	—	2	6	no	—
37	#7845	—	53	1	no	maximal, Sylow
38	C₂²×C₅₃	C₄³	1	1	yes	characteristic
39	C₂×D₅₃	#405	1	6	yes	—
40	#6157	—	4	4	no	—
41	C₄×C₅₃	#405	1	6	yes	—
42	Dic₅₃	#405	1	6	yes	—
43	#1744	—	53	3	no	—
44	C₂×#322	—	53	3	no	—
45	C₂×C₈²	—	53	1	no	—
46	C₂×C₅₃	#1726	1	3	yes	—
47	D₅₃	#1454	1	4	yes	—
48	C₂×C₄×C₈	—	53	3	no	—
49	C₂×C₄×C₈	—	53	3	no	—
50	C₄³	—	53	1	no	—
51	#336	—	53	12	no	—
52	#322	—	53	12	no	—
53	C₈²	—	53	3	no	—
54	C₈²	#6157	1	1	yes	characteristic, complemented
55	C₅₃	#7845	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	53	6	no	—
57	C₂²×C₈	—	53	6	no	—
58	C₂×C₄²	—	53	1	no	—
59	C₂×C₄²	—	53	6	no	—
60	C₄×C₈	—	53	12	no	—
61	C₄×C₈	—	53	6	no	—
62	C₄×C₈	—	53	3	no	—
63	C₄×C₈	C₂×#6157	1	3	yes	—
64	C₂²×C₄	—	53	3	no	—
65	C₂²×C₄	—	53	4	no	—
66	C₂×C₈	—	53	12	no	—
67	C₂×C₈	—	53	6	no	—
68	C₂×C₈	—	106	12	no	—
69	C₂×C₈	—	106	12	no	—
70	C₂×C₈	C₄×#6157	1	6	yes	—
71	C₄²	—	53	3	no	—
72	C₄²	—	106	12	no	—
73	C₄²	C₂²×#6157	1	1	yes	characteristic
74	C₂³	—	53	1	no	—
75	C₂×C₄	—	53	3	no	—
76	C₂×C₄	—	53	12	no	—
77	C₂×C₄	—	106	12	no	—
78	C₂×C₄	C₂×C₄×#6157	1	3	yes	—
79	C₈	—	2	6	no	—
80	C₈	—	106	12	no	—
81	C₈	—	106	6	no	—
82	C₂²	—	53	6	no	—
83	C₂²	C₄²×#6157	1	1	yes	characteristic
84	C₄	—	53	6	no	—
85	C₄	—	212	4	no	—
86	C₄	#16811586	1	6	yes	—
87	C₂	—	53	4	no	—
88	C₂	#20290371	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₄³
Fitting	C₈²×C₅₃	C₄
prime radical	C₂²×D₅₃	C₂×C₄²
cosocle	C₄²×D₅₃	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#7845
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
7. #88324⋊C₈	C₄	semidirect
9. C₈×#88322	C₄	semidirect
10. C₈²×C₅₃	C₄	semidirect
54. C₈²	#6157	semidirect
55. C₅₃	#7845	semidirect

Maximal subgroups

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

subgroup	index
#20290428	2
#20289911	2
C₈²×D₅₃	2
#7845	53