Subgroups — #27277871

Order 13568.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274843	C₂	1	6	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810493	C₂²	1	4	yes	—
4	#16810386	—	2	12	no	—
5	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
6	C₈²×C₅₃	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₅₃	—	2	12	no	—
8	C₂²×C₈×C₅₃	D₄	1	6	yes	—
9	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₅₃	Q₈	1	6	yes	—
12	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
13	C₂²×Dic₅₃	—	4	4	no	—
14	C₂×C₈×C₅₃	—	2	12	no	—
15	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
16	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
17	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
18	C₄⋊Dic₅₃	—	4	24	no	—
19	C₂³×C₅₃	#127	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
21	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
22	C₂×C₄×C₅₃	#135	1	12	yes	—
23	C₂×Dic₅₃	—	8	12	no	—
24	C₈×C₅₃	—	2	12	no	—
25	#8817	—	53	1	no	maximal, Sylow
26	C₂²×C₅₃	#384	1	1	yes	characteristic
27	C₂²×C₅₃	#495	1	6	yes	—
28	C₄×C₅₃	#427	1	6	yes	—
29	C₄×C₅₃	#499	1	6	yes	—
30	Dic₅₃	—	16	4	no	—
31	#1834	—	53	6	no	—
32	C₂×C₈²	D₅₃	1	1	yes	characteristic
33	C₂×C₅₃	#1711	1	4	yes	—
34	C₂×C₅₃	#1846	1	3	yes	—
35	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
36	#384	—	53	4	no	—
37	#340	—	106	12	no	—
38	C₈²	Dic₅₃	1	4	yes	complemented
39	C₅₃	#8817	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₈	D₂₁₂	1	6	yes	—
41	C₂×C₄⋊C₄	—	106	12	no	—
42	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
43	C₄×C₈	Dic₁₀₆	1	6	yes	—
44	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
45	C₂²×C₄	—	212	4	no	—
46	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊Dic₅₃	1	12	yes	—
49	C₄⋊C₄	—	212	24	no	—
50	C₄²	C₂×Dic₁₀₆	1	3	yes	—
51	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
52	C₂³	#13133214	1	1	yes	characteristic
53	C₂×C₄	—	424	12	no	—
54	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
55	C₂×C₄	#13133217	1	3	yes	—
56	C₂×C₄	#13133239	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#16810493	1	1	yes	characteristic
59	C₂²	#16811007	1	6	yes	—
60	C₄	—	848	4	no	—
61	C₄	#16810661	1	6	yes	—
62	C₄	#16811025	1	6	yes	—
63	C₂	#20274327	1	4	yes	—
64	C₂	#20274879	1	3	yes	—

	subgroup	G/H
center	C₂³	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8817
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₅₃	C₄	semidirect
38. C₈²	Dic₅₃	semidirect
39. C₅₃	#8817	semidirect

Maximal subgroups

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

subgroup	index
#20274843	2
C₂×C₈²×C₅₃	2
#8817	53