Subgroups — #28313711

Order 13568.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811569	C₂	1	14	yes	maximal
2	C₄³×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#6157	C₂²	1	28	yes	—
4	#16811569	—	2	56	no	—
5	C₂×C₄²×D₅₃	C₂²	1	7	yes	—
6	C₄²×Dic₅₃	C₄	1	7	yes	complemented
7	C₄³×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂³×#6157	C₂³	1	8	yes	—
9	C₂×C₄⋊#6157	—	2	168	no	—
10	C₂²×C₄×D₅₃	C₂³	1	7	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	21	yes	—
12	C₄²×D₅₃	D₄	1	28	yes	—
13	C₄²×D₅₃	Q₈	1	28	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	7	yes	—
15	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
16	C₂²×#6157	—	2	56	no	—
17	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
18	C₂²×Dic₅₃	C₂²×C₄	1	7	yes	—
19	C₂×C₄×D₅₃	C₂×D₄	1	42	yes	—
20	C₂×C₄×D₅₃	C₂×Q₈	1	42	yes	—
21	C₄⋊#6157	—	4	112	no	—
22	C₄²×C₅₃	C₄⋊C₄	1	28	yes	complemented
23	C₄×Dic₅₃	C₄⋊C₄	1	84	yes	complemented
24	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
25	C₂²×D₅₃	C₂²×D₄	1	7	yes	—
26	C₂²×D₅₃	C₂²×Q₈	1	7	yes	—
27	C₂×#6157	—	4	56	no	—
28	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	42	yes	—
29	C₂×Dic₅₃	C₂×C₄⋊C₄	1	42	yes	—
30	C₄×D₅₃	#127	1	28	yes	—
31	C₄×D₅₃	C₄⋊Q₈	1	84	yes	—
32	#14692	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
34	C₂×D₅₃	C₂×#127	1	7	yes	—
35	C₂×D₅₃	C₂×C₄⋊Q₈	1	21	yes	—
36	#6157	—	8	8	no	—
37	C₄×C₅₃	#384	1	28	yes	complemented
38	Dic₅₃	#384	1	28	yes	complemented
39	C₂×#384	—	53	14	no	—
40	C₂×C₄³	—	53	1	no	—
41	C₂×C₅₃	C₂×#384	1	7	yes	—
42	D₅₃	#2867	1	1	yes	characteristic
43	D₅₃	#2868	1	7	yes	—
44	C₂²×C₄⋊C₄	—	53	28	no	—
45	C₂²×C₄²	—	53	7	no	—
46	#384	—	106	56	no	—
47	C₄³	—	53	7	no	—
48	C₄³	#6157	1	1	yes	characteristic, complemented
49	C₅₃	#14692	1	1	yes	characteristic, complemented, Sylow
50	C₂³×C₄	—	53	7	no	—
51	C₂³×C₄	—	53	8	no	—
52	C₂×C₄⋊C₄	—	106	168	no	—
53	C₂×C₄²	—	53	21	no	—
54	C₂×C₄²	—	53	28	no	—
55	C₂×C₄²	—	53	28	no	—
56	C₂×C₄²	C₂×#6157	1	7	yes	—
57	C₂⁴	—	53	1	no	—
58	C₂²×C₄	—	53	7	no	—
59	C₂²×C₄	—	53	42	no	—
60	C₂²×C₄	—	53	42	no	—
61	C₂²×C₄	—	106	56	no	—
62	C₂²×C₄	C₂²×#6157	1	7	yes	—
63	C₄⋊C₄	—	212	112	no	—
64	C₄²	—	53	84	no	—
65	C₄²	C₄⋊#6157	1	28	yes	complemented
66	C₂³	—	53	7	no	—
67	C₂³	—	53	7	no	—
68	C₂³	C₂³×#6157	1	1	yes	characteristic
69	C₂×C₄	—	53	42	no	—
70	C₂×C₄	—	53	28	no	—
71	C₂×C₄	—	53	84	no	—
72	C₂×C₄	—	212	56	no	—
73	C₂×C₄	C₂×C₄⋊#6157	1	42	yes	—
74	C₂²	—	53	7	no	—
75	C₂²	—	53	21	no	—
76	C₂²	C₂²×C₄⋊#6157	1	7	yes	—
77	C₄	—	53	28	no	—
78	C₄	—	424	8	no	—
79	C₄	#16811569	1	28	yes	complemented
80	C₂	—	53	1	no	—
81	C₂	—	53	7	no	—
82	C₂	C₂×#16811569	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×#6157
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₄³×C₅₃	C₄
prime radical, cosocle	C₂³×D₅₃	C₂⁴
hypercenter	C₄³	#6157
hypocenter	C₅₃	#14692
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
6. C₄²×Dic₅₃	C₄	semidirect
7. C₄³×C₅₃	C₄	semidirect
22. C₄²×C₅₃	C₄⋊C₄	semidirect
23. C₄×Dic₅₃	C₄⋊C₄	semidirect
37. C₄×C₅₃	#384	semidirect
38. Dic₅₃	#384	semidirect
48. C₄³	#6157	semidirect
49. C₅₃	#14692	semidirect
65. C₄²	C₄⋊#6157	semidirect
79. C₄	#16811569	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811569	2
C₄³×D₅₃	2
#14692	53