Subgroups — #28299385

Order 13568.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811462	C₂	1	2	yes	maximal
2	C₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16811462	—	2	2	no	—
4	C₂×#13133123	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#11719908	C₈	1	1	yes	characteristic
8	C₈×Dic₅₃	C₈	1	1	yes	characteristic
9	#13133123	D₄	1	1	yes	characteristic
10	#13133123	Q₈	1	1	yes	characteristic
11	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
15	C₂×#88322	C₁₆	1	1	yes	characteristic
16	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
18	#11719908	—	2	1	no	—
19	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
20	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
21	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
22	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₅₃	M₃₂	1	1	yes	characteristic
24	#88322	—	2	1	no	—
25	C₄⋊C₆₄	—	53	1	no	maximal, Sylow
26	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
27	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₅₃	M₆₄	1	1	yes	characteristic
29	Dic₅₃	—	2	1	no	—
30	Dic₅₃	C₆₄	1	2	yes	complemented
31	C₂×C₆₄	—	53	2	no	—
32	C₄×C₃₂	—	53	1	no	—
33	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₅₃	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₅₃	M₁₂₈	1	1	yes	characteristic
36	C₂×C₃₂	—	53	1	no	—
37	C₂×C₃₂	#6157	1	1	yes	characteristic
38	C₄×C₁₆	—	53	1	no	—
39	C₆₄	—	106	2	no	—
40	C₅₃	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₁₆	—	53	1	no	—
42	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
43	C₄×C₈	—	53	1	no	—
44	C₃₂	—	2	1	no	—
45	C₃₂	—	53	1	no	—
46	C₃₂	—	53	1	no	—
47	C₂×C₈	—	53	1	no	—
48	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
49	C₄²	—	53	1	no	—
50	C₁₆	—	106	1	no	—
51	C₁₆	C₄⋊#6157	1	1	yes	characteristic
52	C₁₆	#11719940	1	1	yes	characteristic
53	C₂×C₄	—	53	1	no	—
54	C₂×C₄	—	53	1	no	—
55	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
56	C₈	—	106	1	no	—
57	C₈	#13133284	1	1	yes	characteristic
58	C₈	#13133292	1	1	yes	characteristic
59	C₂²	D₅₃⋊C₃₂	1	1	yes	characteristic
60	C₄	—	53	2	no	—
61	C₄	—	106	1	no	—
62	C₄	#16811544	1	1	yes	characteristic
63	C₄	#16811556	1	1	yes	characteristic
64	C₂	#20289873	1	1	yes	characteristic
65	C₂	D₅₃⋊C₆₄	1	1	yes	characteristic
66	C₂	#20289885	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₂×C₆₄
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₂×C₃₂
cosocle	C₂×#13133123	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	C₄⋊C₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
30. Dic₅₃	C₆₄	semidirect
40. C₅₃	C₄⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811462	2
C₃₂×Dic₅₃	2
C₄⋊C₆₄	53