Subgroups — #27271419

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133123	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	2	yes	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₄	1	2	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	2	yes	—
10	C₄×#88322	C₈	1	2	yes	—
11	#13133123	C₈	1	4	yes	complemented
12	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₈	1	4	yes	complemented
16	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂×#88322	C₁₆	1	4	yes	—
20	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
21	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
22	#11719908	C₁₆	1	4	yes	—
23	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
25	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
26	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
27	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
28	#88322	—	2	4	no	—
29	#7329	—	53	1	no	maximal, Sylow
30	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
31	C₄×C₅₃	C₈²	1	1	yes	characteristic
32	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
33	C₄×C₅₃	M₆₄	1	4	yes	—
34	C₄×C₃₂	—	53	1	no	—
35	C₄×C₃₂	D₅₃	1	1	yes	characteristic
36	C₈×C₁₆	—	53	1	no	—
37	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
38	C₂×C₅₃	#1397	1	2	yes	—
39	C₂×C₃₂	—	53	2	no	—
40	C₂×C₃₂	Dic₅₃	1	2	yes	—
41	C₄×C₁₆	—	53	1	no	—
42	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
43	C₈²	—	53	1	no	—
44	C₅₃	#7329	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	53	2	no	—
46	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
48	C₄×C₈	—	53	2	no	—
49	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
50	C₃₂	—	53	4	no	—
51	C₃₂	#88322	1	4	yes	complemented
52	C₂×C₈	—	53	4	no	—
53	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
54	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
55	C₄²	C₈×D₅₃	1	1	yes	characteristic
56	C₁₆	—	53	4	no	—
57	C₁₆	C₈×D₅₃	1	2	yes	—
58	C₁₆	C₂×#88322	1	2	yes	—
59	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
60	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
61	C₈	—	106	4	no	—
62	C₈	C₈×Dic₅₃	1	1	yes	characteristic
63	C₈	C₄×#88322	1	1	yes	characteristic
64	C₈	C₁₆×D₅₃	1	2	yes	—
65	C₂²	C₁₆×Dic₅₃	1	1	yes	characteristic
66	C₄	C₈×#88322	1	1	yes	characteristic
67	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
68	C₄	#16810456	1	4	yes	—
69	C₂	C₁₆×#88322	1	1	yes	characteristic
70	C₂	#20272848	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#7329
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
11. #13133123	C₈	semidirect
15. C₃₂×C₅₃	C₈	semidirect
44. C₅₃	#7329	semidirect
51. C₃₂	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133123	2
C₁₆×#88322	2
C₄×C₃₂×C₅₃	2
#7329	53