Subgroups — #27270707

Order 13568.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₄×#11719908	C₄	1	2	yes	—
5	C₄×#11719908	C₄	1	2	yes	—
6	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
7	C₈²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#11719908	C₈	1	4	yes	—
9	C₂×#11719908	C₈	1	4	yes	—
10	C₂×C₁₆×C₅₃	C₈	1	4	yes	—
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
15	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
16	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
17	C₄²×C₅₃	C₄²	1	1	yes	characteristic
18	C₁₆×C₅₃	—	2	4	no	—
19	#11719908	—	2	4	no	—
20	#11719908	C₁₆	1	8	yes	complemented
21	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
22	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
24	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
25	C₈×C₅₃	M₃₂	1	4	yes	—
26	C₈×C₅₃	M₃₂	1	4	yes	—
27	#7052	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	C₈²	1	1	yes	characteristic
29	C₄×C₅₃	C₄×C₁₆	1	2	yes	—
30	C₄×C₅₃	#346	1	2	yes	—
31	C₄×C₅₃	#346	1	2	yes	—
32	C₈×C₁₆	—	53	1	no	—
33	C₈×C₁₆	—	53	1	no	—
34	C₈×C₁₆	D₅₃	1	1	yes	characteristic
35	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
36	C₂×C₅₃	#1311	1	1	yes	characteristic
37	C₂×C₅₃	#1311	1	1	yes	characteristic
38	C₄×C₁₆	—	53	2	no	—
39	C₄×C₁₆	—	53	2	no	—
40	C₄×C₁₆	Dic₅₃	1	2	yes	—
41	C₈²	C₂×D₅₃	1	1	yes	characteristic
42	C₅₃	#7052	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	—	53	4	no	—
44	C₂×C₁₆	—	53	4	no	—
45	C₂×C₁₆	#88322	1	4	yes	—
46	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
47	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
48	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
49	C₂×C₈	C₈×D₅₃	1	2	yes	—
50	C₂×C₈	C₈×D₅₃	1	2	yes	—
51	C₂×C₈	C₂×#88322	1	2	yes	—
52	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
53	C₁₆	—	2	4	no	—
54	C₁₆	—	53	8	no	—
55	C₁₆	—	106	4	no	—
56	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
57	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
58	C₂×C₄	C₄×#88322	1	1	yes	characteristic
59	C₈	C₁₆×D₅₃	1	4	yes	—
60	C₈	#13133156	1	4	yes	—
61	C₈	#13133157	1	4	yes	—
62	C₂²	C₈×#88322	1	1	yes	characteristic
63	C₄	C₁₆×Dic₅₃	1	2	yes	—
64	C₄	#16810398	1	2	yes	—
65	C₄	#16810399	1	2	yes	—
66	C₂	C₁₆×#88322	1	1	yes	characteristic
67	C₂	#20272631	1	1	yes	characteristic
68	C₂	#20272632	1	1	yes	characteristic

	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
20. #11719908	C₁₆	semidirect
42. C₅₃	#7052	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