Subgroups — #27270815

Order 13568.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272779	C₂	1	2	yes	maximal
2	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₅₃×#500	C₄	1	2	yes	—
5	C₅₃×#487	—	2	2	no	—
6	C₈²×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719908	—	4	2	no	—
8	C₄⋊Q₈×C₅₃	—	2	2	no	—
9	#107×C₅₃	—	4	2	no	—
10	#103×C₅₃	—	4	2	no	—
11	#103×C₅₃	—	4	2	no	—
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×Q₈×C₅₃	—	4	2	no	—
15	C₂×C₈×C₅₃	—	2	1	no	—
16	C₂×C₈×C₅₃	D₈	1	2	yes	—
17	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
18	C₄⋊C₄×C₅₃	—	4	2	no	—
19	C₄⋊C₄×C₅₃	—	4	2	no	—
20	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	#11719908	—	8	2	no	—
22	Q₈×C₅₃	—	8	2	no	—
23	Q₈×C₅₃	—	8	2	no	—
24	C₂×C₄×C₅₃	—	8	2	no	—
25	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
26	C₂×C₄×C₅₃	#102	1	2	yes	—
27	C₈×C₅₃	—	2	2	no	—
28	C₈×C₅₃	—	2	2	no	—
29	C₈×C₅₃	—	2	2	no	—
30	#7093	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#331	1	1	yes	characteristic
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	—	16	2	no	—
34	C₄×C₅₃	#359	1	2	yes	—
35	C₄×C₅₃	#360	1	2	yes	—
36	#1710	D₅₃	1	1	yes	characteristic
37	#1367	—	53	2	no	—
38	C₂×C₅₃	#1358	1	1	yes	characteristic
39	C₂×C₅₃	#1360	1	2	yes	—
40	#500	Dic₅₃	1	2	yes	—
41	#487	—	2	2	no	—
42	C₄⋊C₁₆	—	106	2	no	—
43	C₈²	C₂×D₅₃	1	1	yes	characteristic
44	C₅₃	#7093	1	1	yes	characteristic, complemented, Sylow
45	C₄⋊Q₈	—	2	2	no	—
46	C₂×C₁₆	—	212	2	no	—
47	#107	—	4	2	no	—
48	#103	—	4	2	no	—
49	#103	—	4	2	no	—
50	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×C₈	—	2	1	no	—
54	C₂×C₈	#11719922	1	2	yes	—
55	C₂×C₈	#11719923	1	2	yes	—
56	C₄⋊C₄	—	4	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
59	C₁₆	—	424	2	no	—
60	Q₈	—	8	2	no	—
61	Q₈	—	8	2	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	#13133134	1	1	yes	characteristic
64	C₂×C₄	#13133138	1	2	yes	—
65	C₈	—	2	2	no	—
66	C₈	—	2	2	no	—
67	C₈	—	2	2	no	—
68	C₂²	#16810367	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	16	2	no	—
71	C₄	#16810430	1	2	yes	—
72	C₄	#16810433	1	2	yes	—
73	C₂	#20272754	1	1	yes	characteristic
74	C₂	#20272758	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1710	C₂
socle, prime radical	C₂²×C₅₃	#331
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1710	D₅₃
hypocenter	C₅₃	#7093
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
44. C₅₃	#7093	semidirect

Maximal subgroups

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

subgroup	index
#20272779	2
C₅₃×#1710	2
#7093	53