Subgroups — #27271899

Order 13568.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	2	yes	maximal
2	C₅₃×#1397	C₂	1	1	yes	maximal, characteristic
3	#16810342	C₄	1	4	yes	complemented
4	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
10	C₃₂×C₅₃	C₈	1	2	yes	—
11	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
13	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
14	C₄²×C₅₃	C₁₆	1	1	yes	characteristic
15	C₁₆×C₅₃	C₁₆	1	2	yes	—
16	C₁₆×C₅₃	C₄²	1	1	yes	characteristic
17	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	1	no	—
19	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
20	C₈×C₅₃	—	2	1	no	—
21	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
22	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
23	#7511	—	53	1	no	maximal, Sylow
24	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
25	C₄×C₅₃	—	4	1	no	—
26	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₅₃	M₆₄	1	1	yes	characteristic
28	M₁₂₈	—	53	2	no	—
29	#1397	D₅₃	1	1	yes	characteristic
30	C₂×C₅₃	—	2	1	no	—
31	C₂×C₅₃	#1397	1	1	yes	characteristic
32	C₂×C₃₂	Dic₅₃	1	1	yes	characteristic
33	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
34	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
35	C₆₄	—	53	4	no	—
36	C₅₃	#7511	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	#88322	1	1	yes	characteristic
38	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
39	C₄×C₈	#88322	1	1	yes	characteristic
40	C₃₂	#88322	1	2	yes	—
41	C₃₂	C₄×D₅₃	1	2	yes	—
42	C₂×C₈	#11719908	1	1	yes	characteristic
43	C₂×C₈	C₂×#88322	1	1	yes	characteristic
44	C₄²	#11719908	1	1	yes	characteristic
45	C₁₆	#11719908	1	2	yes	—
46	C₁₆	C₂×#88322	1	1	yes	characteristic
47	C₁₆	C₄×Dic₅₃	1	1	yes	characteristic
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	C₄×#88322	1	1	yes	characteristic
52	C₈	C₂×#11719908	1	1	yes	characteristic
53	C₂²	#16810457	1	1	yes	characteristic
54	C₄	—	4	1	no	—
55	C₄	C₄×#11719908	1	1	yes	characteristic
56	C₄	#16810457	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	#20272849	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×#1397	C₂
socle	C₂×C₅₃	#1397
prime radical	C₂²×C₅₃	M₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	#1397	D₅₃
hypocenter	C₅₃	#7511
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
3. #16810342	C₄	semidirect
36. C₅₃	#7511	semidirect

Maximal subgroups

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

subgroup	index
#20272926	2
C₅₃×#1397	2
#7511	53