Subgroups — #27315298

Order 13568.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810493	C₂	1	14	yes	maximal
2	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	—
4	#16810493	—	2	56	no	—
5	C₂²×C₄²×C₅₃	C₂²	1	7	yes	—
6	C₄³×C₅₃	C₄	1	8	yes	complemented
7	C₂³×Dic₅₃	C₂³	1	8	yes	—
8	C₂×C₄⋊Dic₅₃	—	2	168	no	—
9	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
10	C₂×C₄²×C₅₃	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₅₃	D₄	1	28	yes	—
12	C₂×C₄²×C₅₃	Q₈	1	28	yes	—
13	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂²×C₄	1	14	yes	—
15	C₂²×C₄×C₅₃	C₂×D₄	1	42	yes	—
16	C₂²×C₄×C₅₃	C₂×Q₈	1	42	yes	—
17	C₂²×Dic₅₃	—	2	56	no	—
18	C₄²×C₅₃	C₄⋊C₄	1	112	yes	complemented
19	C₄⋊Dic₅₃	—	4	112	no	—
20	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
21	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
22	C₂³×C₅₃	C₂²×Q₈	1	7	yes	—
23	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	84	yes	—
24	C₂×C₄×C₅₃	#127	1	28	yes	—
25	C₂×C₄×C₅₃	C₄⋊Q₈	1	84	yes	—
26	C₂×Dic₅₃	—	4	56	no	—
27	#14692	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₂²×C₄⋊C₄	1	7	yes	—
29	C₂²×C₅₃	C₂×#127	1	7	yes	—
30	C₂²×C₅₃	C₂×C₄⋊Q₈	1	21	yes	—
31	C₄×C₅₃	#384	1	56	yes	complemented
32	Dic₅₃	—	8	8	no	—
33	C₂×#384	—	53	14	no	—
34	C₂×C₄³	D₅₃	1	1	yes	characteristic
35	C₂×C₅₃	C₂×#384	1	7	yes	—
36	C₂×C₅₃	#2867	1	1	yes	characteristic
37	C₂×C₅₃	#2868	1	7	yes	—
38	C₂²×C₄⋊C₄	—	53	28	no	—
39	C₂²×C₄²	C₂×D₅₃	1	7	yes	—
40	#384	—	106	56	no	—
41	C₄³	Dic₅₃	1	8	yes	complemented
42	C₅₃	#14692	1	1	yes	characteristic, complemented, Sylow
43	C₂³×C₄	—	53	8	no	—
44	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
45	C₂×C₄⋊C₄	—	106	168	no	—
46	C₂×C₄²	Dic₁₀₆	1	28	yes	—
47	C₂×C₄²	D₂₁₂	1	28	yes	—
48	C₂×C₄²	C₂×Dic₅₃	1	28	yes	—
49	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
50	C₂²×C₄	—	106	56	no	—
51	C₂²×C₄	C₂×Dic₁₀₆	1	42	yes	—
52	C₂²×C₄	C₂×D₂₁₂	1	42	yes	—
53	C₂²×C₄	C₂²×Dic₅₃	1	14	yes	—
54	C₄⋊C₄	—	212	112	no	—
55	C₄²	C₄⋊Dic₅₃	1	112	yes	complemented
56	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
57	C₂³	C₂²×D₂₁₂	1	7	yes	—
58	C₂³	C₂²×Dic₁₀₆	1	7	yes	—
59	C₂×C₄	—	212	56	no	—
60	C₂×C₄	C₂×C₄⋊Dic₅₃	1	84	yes	—
61	C₂×C₄	#13133214	1	28	yes	—
62	C₂×C₄	#13133217	1	84	yes	—
63	C₂²	C₂²×C₄⋊Dic₅₃	1	7	yes	—
64	C₂²	C₂×#13133214	1	7	yes	—
65	C₂²	C₂×#13133217	1	21	yes	—
66	C₄	—	424	8	no	—
67	C₄	#16810493	1	56	yes	complemented
68	C₂	C₂×#16810493	1	7	yes	—
69	C₂	#20283110	1	1	yes	characteristic
70	C₂	#20283113	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄³×C₅₃	C₂
socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#14692
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
6. C₄³×C₅₃	C₄	semidirect
18. C₄²×C₅₃	C₄⋊C₄	semidirect
31. C₄×C₅₃	#384	semidirect
41. C₄³	Dic₅₃	semidirect
42. C₅₃	#14692	semidirect
55. C₄²	C₄⋊Dic₅₃	semidirect
67. C₄	#16810493	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810493	2
C₂×C₄³×C₅₃	2
#14692	53