Finite Group Databasebeta

← C2×C61×2−1+6 #29566535

Subgroups — C2×C61×2−1+6 #29566535

Order 15616.

59186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×2−1+6C2164yesmaximal, direct product factor
2C2×C61×#585C2136yesmaximal, complemented
3C22×2−1+4×C61C2127yesmaximal, complemented
4C61×#585C2211152yescomplemented
5C2×2−1+4×C61C221216yescomplemented
6C2×2−1+4×C61C221864yescomplemented
7C2×H(5,𝔽2)×C61C221120yescomplemented
8C22×#41×C61C221270yescomplemented
9C23×Q8×C61C22145yescomplemented
102−1+4×C61C2313456yescomplemented
11H(5,𝔽2)×C61C2311920yescomplemented
12C2×#41×C61C231720yes—
13C2×#41×C61C2314320yescomplemented
14C22×Q8×C61C231270yes—
15C22×Q8×C61C231720yescomplemented
16C22×D4×C61C231270yes—
17C23×C4×C61C231135yes—
18C24×C61C24145yes—
19#41×C61C2415760yes—
20C2×Q8×C61C241120yes—
21C2×Q8×C61C2412160yes—
22C2×D4×C61C241216yes—
23C2×D4×C61C2412160yes—
24C22×C4×C61C241270yes—
25C22×C4×C61C2411080yes—
26C23×C61—490no—
27C23×C61C25127yes—
28C23×C61C251180yes—
29Q8×C61C251480yes—
30D4×C61C251864yes—
31C2×C4×C61C2511080yes—
32C2×C4×C61C25136yes—
33C2×2−1+6C6111yesmaximal, characteristic, direct product factor, Sylow
34C22×C61—254no—
35C22×C61—4180no—
36C22×C61C26154yes—
37C22×C61C2611yescharacteristic
38C4×C61C26172yes—
392−1+6C2×C61164yesdirect product factor
40C2×#585C2×C61136yescomplemented
41C22×2−1+4C2×C61127yescomplemented
42C2×C61—254no—
43C2×C612−1+612yesdirect product factor
44C2×C61C2711yescharacteristic
45#585C22×C6111152yescomplemented
46C2×2−1+4C22×C611216yescomplemented
47C2×2−1+4C22×C611864yescomplemented
48C2×H(5,𝔽2)C22×C611120yescomplemented
49C22×#41C22×C611270yescomplemented
50C23×Q8C22×C61145yescomplemented
51C61C2×2−1+611yescharacteristic, direct product factor, Sylow
522−1+4C23×C6113456yescomplemented
53H(5,𝔽2)C23×C6111920yescomplemented
54C2×#41C23×C611720yes—
55C2×#41C23×C6114320yescomplemented
56C22×Q8C23×C611270yes—
57C22×Q8C23×C611720yescomplemented
58C22×D4C23×C611270yes—
59C23×C4C23×C611135yes—
60C24C24×C61145yes—
61#41C24×C6115760yes—
62C2×Q8C24×C611120yes—
63C2×Q8C24×C6112160yes—
64C2×D4C24×C611216yes—
65C2×D4C24×C6112160yes—
66C22×C4C24×C611270yes—
67C22×C4C24×C6111080yes—
68C23—490no—
69C23C25×C61127yes—
70C23C25×C611180yes—
71Q8C25×C611480yes—
72D4C25×C611864yes—
73C2×C4C25×C61136yes—
74C2×C4C25×C6111080yes—
75C22—254no—
76C22—4180no—
77C22C26×C61154yes—
78C22C26×C6111yescharacteristic
79C4C26×C61172yes—
80C2—254no—
81C2C61×2−1+612yesdirect product factor
82C2C27×C6111yescharacteristic
subgroupG/H
center, socleC22×C61C26
commutator, Frattini, cosocleC2C27×C61
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×2−1+6C2direct (N × H)
2. C2×C61×#585C2semidirect
3. C22×2−1+4×C61C2semidirect
4. C61×#585C22semidirect
5. C2×2−1+4×C61C22semidirect
6. C2×2−1+4×C61C22semidirect
7. C2×H(5,𝔽2)×C61C22semidirect
8. C22×#41×C61C22semidirect
9. C23×Q8×C61C22semidirect
10. 2−1+4×C61C23semidirect
11. H(5,𝔽2)×C61C23semidirect
13. C2×#41×C61C23semidirect
15. C22×Q8×C61C23semidirect
33. C2×2−1+6C61direct (N × H)
39. 2−1+6C2×C61direct (N × H)
40. C2×#585C2×C61semidirect
41. C22×2−1+4C2×C61semidirect
43. C2×C612−1+6direct (N × H)
45. #585C22×C61semidirect
46. C2×2−1+4C22×C61semidirect
47. C2×2−1+4C22×C61semidirect
48. C2×H(5,𝔽2)C22×C61semidirect
49. C22×#41C22×C61semidirect
50. C23×Q8C22×C61semidirect
51. C61C2×2−1+6direct (N × H)
52. 2−1+4C23×C61semidirect
53. H(5,𝔽2)C23×C61semidirect
55. C2×#41C23×C61semidirect
57. C22×Q8C23×C61semidirect
81. C2C61×2−1+6direct (N × H)

Maximal subgroups

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

subgroupindex
C61×2−1+62
C2×C61×#5852
C22×2−1+4×C612
C2×2−1+661