Finite Group Databasebeta

← C53×#63103 #27270615

Subgroups — C53×#63103 #27270615

Order 13568.

59452 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×2−1+6C2128yesmaximal, complemented
2C53×H(7,𝔽2)C2136yesmaximal, complemented
3C2×C53×#585C2163yesmaximal, complemented
4C53×#585C221336yescomplemented
5C53×#585C2211008yescomplemented
6C2×2−1+4×C53C221378yescomplemented
7C2×H(5,𝔽2)×C53C221630yescomplemented
8C22×#41×C53C221315yescomplemented
92−1+4×C53C2312016yescomplemented
10H(5,𝔽2)×C53C2313360yescomplemented
11C2×#41×C53C2311260yescomplemented
12C2×#41×C53C2313780yescomplemented
13C22×Q8×C53C231315yescomplemented
14C22×D4×C53C231945yescomplemented
15C23×C4×C53C231135yescomplemented
16C24×C53C241135yes—
17#41×C53C241336yes—
18#41×C53C2415040yes—
19C2×Q8×C53C2411260yes—
20C2×D4×C53C2413780yes—
21C22×C4×C53C241315yes—
22C22×C4×C53C241945yes—
23C23×C53—8135no—
24C23×C53C251315yes—
25Q8×C53C251336yes—
26D4×C53C2511008yes—
27C2×C4×C53C251945yes—
28C2×C4×C53C25163yes—
29#63103C5311yesmaximal, characteristic, direct product factor, Sylow
30C22×C53—4315no—
31C22×C53C26163yes—
32C4×C53C26163yes—
33C4×C53C2611yescharacteristic
342−1+6C2×C53128yescomplemented
35H(7,𝔽2)C2×C53136yescomplemented
36C2×#585C2×C53163yescomplemented
37C2×C53—263no—
38C2×C53C2711yescharacteristic
39#585C22×C531336yescomplemented
40#585C22×C5311008yescomplemented
41C2×2−1+4C22×C531378yescomplemented
42C2×H(5,𝔽2)C22×C531630yescomplemented
43C22×#41C22×C531315yescomplemented
44C53#6310311yescharacteristic, direct product factor, Sylow
452−1+4C23×C5312016yescomplemented
46H(5,𝔽2)C23×C5313360yescomplemented
47C2×#41C23×C5311260yescomplemented
48C2×#41C23×C5313780yescomplemented
49C22×Q8C23×C531315yescomplemented
50C22×D4C23×C531945yescomplemented
51C23×C4C23×C531135yescomplemented
52C24C24×C531135yes—
53#41C24×C531336yes—
54#41C24×C5315040yes—
55C2×Q8C24×C5311260yes—
56C2×D4C24×C5313780yes—
57C22×C4C24×C531315yes—
58C22×C4C24×C531945yes—
59C23—8135no—
60C23C25×C531315yes—
61Q8C25×C531336yes—
62D4C25×C5311008yes—
63C2×C4C25×C53163yes—
64C2×C4C25×C531945yes—
65C22—4315no—
66C22C26×C53163yes—
67C4C26×C5311yescharacteristic
68C4C26×C53163yes—
69C2—263no—
70C2C27×C5311yescharacteristic
subgroupG/H
centerC4×C53C26
commutator, Frattini, cosocleC2C27×C53
socleC2×C53C27
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×2−1+6C2semidirect
2. C53×H(7,𝔽2)C2semidirect
3. C2×C53×#585C2semidirect
4. C53×#585C22semidirect
5. C53×#585C22semidirect
6. C2×2−1+4×C53C22semidirect
7. C2×H(5,𝔽2)×C53C22semidirect
8. C22×#41×C53C22semidirect
9. 2−1+4×C53C23semidirect
10. H(5,𝔽2)×C53C23semidirect
11. C2×#41×C53C23semidirect
12. C2×#41×C53C23semidirect
13. C22×Q8×C53C23semidirect
14. C22×D4×C53C23semidirect
15. C23×C4×C53C23semidirect
29. #63103C53direct (N × H)
34. 2−1+6C2×C53semidirect
35. H(7,𝔽2)C2×C53semidirect
36. C2×#585C2×C53semidirect
39. #585C22×C53semidirect
40. #585C22×C53semidirect
41. C2×2−1+4C22×C53semidirect
42. C2×H(5,𝔽2)C22×C53semidirect
43. C22×#41C22×C53semidirect
44. C53#63103direct (N × H)
45. 2−1+4C23×C53semidirect
46. H(5,𝔽2)C23×C53semidirect
47. C2×#41C23×C53semidirect
48. C2×#41C23×C53semidirect
49. C22×Q8C23×C53semidirect
50. C22×D4C23×C53semidirect
51. C23×C4C23×C53semidirect

Maximal subgroups

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

subgroupindex
C53×2−1+62
C53×H(7,𝔽2)2
C2×C53×#5852
#6310353