Subgroups — #27298925

Order 13568.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276320	C₂	1	3	yes	maximal
2	#20276312	C₂	1	3	yes	maximal
3	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₂²	1	3	yes	—
5	C₂²×#11719908	C₂²	1	1	yes	characteristic
6	#16810401	C₄	1	12	yes	complemented
7	#16810399	C₄	1	6	yes	complemented
8	C₄×#11719908	C₄	1	6	yes	complemented
9	C₅₃×#405	C₂²	1	3	yes	—
10	C₅₃×#338	C₄	1	1	yes	characteristic
11	C₅₃×H(3,ℤ/4)	C₄	1	3	yes	—
12	#13133157	C₂×C₄	1	12	yes	—
13	C₂×#11719908	C₂×C₄	1	3	yes	—
14	C₂×#11719908	C₂×C₄	1	3	yes	—
15	C₂×#11719908	C₂×C₄	1	6	yes	—
16	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	#117×C₅₃	C₂×C₄	1	3	yes	—
19	#97×C₅₃	C₂×C₄	1	6	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	6	no	—
23	C₂×C₈×C₅₃	C₄²	1	6	yes	—
24	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
26	C₄⋊C₄×C₅₃	—	2	3	no	—
27	C₂²⋊C₄×C₅₃	—	2	3	no	—
28	C₄²×C₅₃	C₄²	1	6	yes	—
29	#11719908	C₄²	1	1	yes	characteristic
30	#11719908	C₄²	1	3	yes	—
31	#11719908	C₄²	1	12	yes	—
32	C₂³×C₅₃	#131	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	6	no	—
34	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₅₃	#131	1	3	yes	—
36	C₈×C₅₃	—	2	6	no	—
37	C₈×C₅₃	C₂×C₄²	1	3	yes	—
38	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	#12542	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	—	4	1	no	—
41	C₂²×C₅₃	#405	1	3	yes	—
42	C₄×C₅₃	—	4	6	no	—
43	C₄×C₅₃	C₄³	1	1	yes	characteristic
44	C₄×C₅₃	#405	1	3	yes	—
45	#2110	—	53	3	no	—
46	#2108	—	53	3	no	—
47	#1747	D₅₃	1	1	yes	characteristic
48	C₂×C₅₃	—	2	3	no	—
49	C₂×C₅₃	#1726	1	1	yes	characteristic
50	C₂×M₃₂	—	53	3	no	—
51	C₂²×C₁₆	—	53	1	no	—
52	#405	C₂×D₅₃	1	3	yes	—
53	#347	—	53	12	no	—
54	#346	—	53	6	no	—
55	C₄×C₁₆	—	53	6	no	—
56	#338	Dic₅₃	1	1	yes	characteristic
57	H(3,ℤ/4)	Dic₅₃	1	3	yes	—
58	C₅₃	#12542	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
60	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
61	#117	C₂×Dic₅₃	1	3	yes	—
62	M₃₂	—	53	12	no	—
63	C₂×C₁₆	—	53	3	no	—
64	C₂×C₁₆	—	53	6	no	—
65	C₂×C₁₆	—	53	3	no	—
66	#97	C₄×D₅₃	1	6	yes	—
67	C₄×C₈	C₄×D₅₃	1	6	yes	—
68	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	C₄×Dic₅₃	1	6	yes	—
71	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
72	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
73	C₄⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₄²	C₄×Dic₅₃	1	6	yes	—
76	C₁₆	—	53	12	no	—
77	C₁₆	—	53	1	no	—
78	C₁₆	—	53	3	no	—
79	C₂³	#13133224	1	1	yes	characteristic
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₄×Dic₅₃	1	3	yes	—
82	C₂×C₄	#13133224	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	C₄²×D₅₃	1	3	yes	—
85	C₈	C₄²×D₅₃	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	#16810579	1	3	yes	—
88	C₄	—	4	6	no	—
89	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
90	C₄	#16810579	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	#20274402	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1747	C₂
socle	C₂×C₅₃	#1726
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1747	D₅₃
hypocenter	C₅₃	#12542
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
6. #16810401	C₄	semidirect
7. #16810399	C₄	semidirect
8. C₄×#11719908	C₄	semidirect
58. C₅₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#20276320	2
#20276312	2
C₅₃×#1747	2
#12542	53