Subgroups — #27270706

Order 13568.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719908	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#11719908	C₄	1	4	yes	—
4	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
5	C₈²×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×#11719908	C₈	1	8	yes	—
7	C₂×C₁₆×C₅₃	C₈	1	4	yes	—
8	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
9	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
11	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
12	C₄²×C₅₃	C₄²	1	1	yes	characteristic
13	C₁₆×C₅₃	C₁₆	1	8	yes	complemented
14	#11719908	—	2	8	no	—
15	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
16	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
17	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
18	C₈×C₅₃	M₃₂	1	8	yes	—
19	#7052	—	53	1	no	maximal, Sylow
20	C₂²×C₅₃	C₈²	1	1	yes	characteristic
21	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
22	C₄×C₅₃	#346	1	4	yes	—
23	C₈×C₁₆	—	53	2	no	—
24	C₈×C₁₆	D₅₃	1	1	yes	characteristic
25	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
26	C₂×C₅₃	#1311	1	2	yes	—
27	C₄×C₁₆	—	53	4	no	—
28	C₄×C₁₆	Dic₅₃	1	2	yes	—
29	C₈²	C₂×D₅₃	1	1	yes	characteristic
30	C₅₃	#7052	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₁₆	—	53	8	no	—
32	C₂×C₁₆	#88322	1	4	yes	—
33	C₄×C₈	C₄×D₅₃	1	2	yes	—
34	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
35	C₂×C₈	C₈×D₅₃	1	4	yes	—
36	C₂×C₈	C₂×#88322	1	2	yes	—
37	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
38	C₁₆	—	106	8	no	—
39	C₁₆	#11719908	1	8	yes	complemented
40	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
41	C₂×C₄	C₄×#88322	1	1	yes	characteristic
42	C₈	C₂×#11719908	1	4	yes	—
43	C₈	#13133156	1	8	yes	—
44	C₂²	C₈×#88322	1	1	yes	characteristic
45	C₄	C₄×#11719908	1	2	yes	—
46	C₄	#16810398	1	4	yes	—
47	C₂	C₈×#11719908	1	1	yes	characteristic
48	C₂	#20272631	1	2	yes	—

	subgroup	G/H
center, Frattini	C₈²	C₂×D₅₃
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₈²
cosocle	C₈²×C₅₃	C₂²
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#7052
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
13. C₁₆×C₅₃	C₁₆	semidirect
30. C₅₃	#7052	semidirect
39. C₁₆	#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719908	2
C₈×C₁₆×C₅₃	2
#7052	53