Subgroups — #27277873

Order 13568.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274875	C₂	1	6	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133151	—	2	12	no	—
4	#16810493	C₂²	1	4	yes	—
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	#13133151	—	4	24	no	—
9	C₂²×C₈×C₅₃	D₄	1	6	yes	—
10	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
12	C₄×C₈×C₅₃	Q₈	1	6	yes	—
13	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
14	C₂²×Dic₅₃	—	4	4	no	—
15	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
16	C₂×C₈×C₅₃	D₈	1	12	yes	—
17	C₂×C₈×C₅₃	Q₁₆	1	12	yes	—
18	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
20	C₄⋊Dic₅₃	—	4	24	no	—
21	C₂³×C₅₃	#127	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
23	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
24	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
25	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
26	C₂×Dic₅₃	—	8	12	no	—
27	C₈×C₅₃	#107	1	24	yes	complemented
28	#8818	—	53	1	no	maximal, Sylow
29	C₂²×C₅₃	#384	1	1	yes	characteristic
30	C₂²×C₅₃	#493	1	3	yes	—
31	C₂²×C₅₃	#494	1	3	yes	—
32	C₄×C₅₃	C₂×#107	1	6	yes	—
33	C₄×C₅₃	#500	1	6	yes	—
34	Dic₅₃	—	16	4	no	—
35	#1845	—	53	6	no	—
36	C₂×C₈²	D₅₃	1	1	yes	characteristic
37	C₂×C₅₃	#1712	1	1	yes	characteristic
38	C₂×C₅₃	#1713	1	3	yes	—
39	C₂×C₅₃	#1845	1	3	yes	—
40	C₂×#107	—	106	12	no	—
41	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
42	#384	—	53	4	no	—
43	C₈²	Dic₅₃	1	4	yes	complemented
44	C₅₃	#8818	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₈	D₂₁₂	1	6	yes	—
46	C₂×C₄⋊C₄	—	106	12	no	—
47	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
48	#107	—	212	24	no	—
49	C₄×C₈	Dic₁₀₆	1	6	yes	—
50	C₄×C₈	C₂×Dic₅₃	1	6	yes	—
51	C₂²×C₄	—	212	4	no	—
52	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
53	C₂×C₈	D₄₂₄	1	12	yes	—
54	C₂×C₈	Dic₂₁₂	1	12	yes	—
55	C₂×C₈	C₄⋊Dic₅₃	1	12	yes	—
56	C₄⋊C₄	—	212	24	no	—
57	C₄²	C₂×Dic₁₀₆	1	3	yes	—
58	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
59	C₂³	#13133214	1	1	yes	characteristic
60	C₂×C₄	—	424	12	no	—
61	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
62	C₂×C₄	#13133217	1	3	yes	—
63	C₂×C₄	C₂×D₄₂₄	1	6	yes	—
64	C₂×C₄	C₂×Dic₂₁₂	1	6	yes	—
65	C₈	#13133151	1	24	yes	complemented
66	C₂²	#16810493	1	1	yes	characteristic
67	C₂²	#16811000	1	3	yes	—
68	C₂²	#16811003	1	3	yes	—
69	C₄	—	848	4	no	—
70	C₄	C₂×#13133151	1	6	yes	—
71	C₄	#16811028	1	6	yes	—
72	C₂	#20274330	1	1	yes	characteristic
73	C₂	#20274333	1	3	yes	—
74	C₂	#20274875	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	#8818
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
27. C₈×C₅₃	#107	semidirect
43. C₈²	Dic₅₃	semidirect
44. C₅₃	#8818	semidirect
65. C₈	#13133151	semidirect

Maximal subgroups

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

subgroup	index
#20274875	2
C₂×C₈²×C₅₃	2
#8818	53