Finite Group Databasebeta

← C2×#20297350 #28384411

Subgroups — C2×#20297350 #28384411

Order 14272.

898 conjugacy classes of subgroups; only the 447 normal subgroups are shown.

normal subgroups only

This group has 898 conjugacy classes of subgroups — too many to draw. Only its 447 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20297350C2116yesmaximal, direct product factor
2C2×#16813171C212yesmaximal, complemented
3C2×#16813170C213yesmaximal, complemented
4C22×D892C213yesmaximal, complemented
5C2×D4×D223C216yesmaximal, complemented
6C2×#41×C223C211yesmaximal, characteristic, complemented
7C23×D223C2216yescomplemented
8#16813171C22116yescomplemented
9#16813170C22124yescomplemented
10C2×Dic446C2211yescharacteristic, complemented
11C2×D892C2216yescomplemented
12C2×D892C2213yescomplemented
13C2×D892C22124yescomplemented
14C2×C22⋊D223C2216yescomplemented
15D4×D223C22148yescomplemented
16C2×C4×D223C2216yescomplemented
17#41×C223C2218yescomplemented
18C2×Q8×C223C2211yescharacteristic, complemented
19C2×D4×C223C2213yescomplemented
20C22×C4×C223C2213yescomplemented
21C22×D223—412no—
22C22×D223C2316yes—
23C22×D223C23124yescomplemented
24Dic446C2314yescomplemented
25D892C23124yescomplemented
26D892C23112yescomplemented
27C22⋊D223C23124yescomplemented
28C2×Dic223C2312yes—
29C4×D223C23124yescomplemented
30C23×C223C2313yes—
31Q8×C223C2314yescomplemented
32D4×C223C23112yescomplemented
33C2×C4×C223C2311yescharacteristic
34C2×C4×C223C2313yes—
35C2×C4×C223C23112yescomplemented
36C22×C223—26no—
37C22×C223C2416yes—
38C22×C223C2411yescharacteristic
39C2×D223—212no—
40C2×D223—424no—
41C2×D223C24112yes—
42C4×C223C2412yes—
43C4×C223C2416yes—
44Dic223C2414yes—
45C2×C223—26no—
46C2×C223H(5,𝔽2)12yescomplemented
47C2×C223C2511yescharacteristic
48D223—212no—
49C223C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—2231nomaximal, Sylow
51H(5,𝔽2)—22316no—
52C2×#41—2232no—
53C2×#41—2233no—
54C2×#41D22311yescharacteristic, complemented
55C22×D4—2236no—
56C22×D4—2233no—
57C24—2236no—
58#41—22316no—
59#41—22324no—
60#41C2×D22318yescomplemented
61C2×Q8—2231no—
62C2×Q8C2×D22311yescharacteristic, complemented
63C2×D4—2236no—
64C2×D4—22348no—
65C2×D4—2236no—
66C2×D4—2233no—
67C2×D4—22324no—
68C2×D4C2×D22313yescomplemented
69C22×C4—2236no—
70C22×C4C2×D22313yescomplemented
71C23—89212no—
72C23—22324no—
73C23—2236no—
74C23C22×D22313yes—
75Q8—2234no—
76Q8C22×D22314yescomplemented
77D4—22324no—
78D4—22324no—
79D4—22312no—
80D4C22×D223112yescomplemented
81C2×C4—2232no—
82C2×C4—22324no—
83C2×C4C22×D22313yes—
84C2×C4C22×D22311yescharacteristic
85C2×C4C22×D223112yescomplemented
86C22—26no—
87C22—89224no—
88C22—44612no—
89C22—22312no—
90C22C23×D22316yes—
91C22C23×D22311yescharacteristic
92C4—2234no—
93C4C23×D22316yes—
94C4C23×D22312yes—
95C2—26no—
96C2—44612no—
97C2#2029735012yesdirect product factor
98C2C24×D22311yescharacteristic
subgroupG/H
centerC22C23×D223
commutator, cosocleC2×C223C25
FrattiniC2C24×D223
FittingC2×#41×C223C2
socleC22×C223C24
hypercenterC2×#41D223
hypocenterC223C2×H(5,𝔽2)
LCSG ▷ C2×C223 ▷ C223
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20297350C2direct (N × H)
2. C2×#16813171C2semidirect
3. C2×#16813170C2semidirect
4. C22×D892C2semidirect
5. C2×D4×D223C2semidirect
6. C2×#41×C223C2semidirect
7. C23×D223C22semidirect
8. #16813171C22semidirect
9. #16813170C22semidirect
10. C2×Dic446C22semidirect
11. C2×D892C22semidirect
12. C2×D892C22semidirect
13. C2×D892C22semidirect
14. C2×C22⋊D223C22semidirect
15. D4×D223C22semidirect
16. C2×C4×D223C22semidirect
17. #41×C223C22semidirect
18. C2×Q8×C223C22semidirect
19. C2×D4×C223C22semidirect
20. C22×C4×C223C22semidirect
23. C22×D223C23semidirect
24. Dic446C23semidirect
25. D892C23semidirect
26. D892C23semidirect
27. C22⋊D223C23semidirect
29. C4×D223C23semidirect
31. Q8×C223C23semidirect
32. D4×C223C23semidirect
35. C2×C4×C223C23semidirect
46. C2×C223H(5,𝔽2)semidirect
49. C223C2×H(5,𝔽2)semidirect
54. C2×#41D223semidirect
60. #41C2×D223semidirect
62. C2×Q8C2×D223semidirect
68. C2×D4C2×D223semidirect
70. C22×C4C2×D223semidirect
76. Q8C22×D223semidirect
80. D4C22×D223semidirect
85. C2×C4C22×D223semidirect
97. C2#20297350direct (N × H)

Maximal subgroups

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

subgroupindex
#202973502
C2×#168131712
C2×#168131702
C22×D8922
C2×D4×D2232
C2×#41×C2232
C2×H(5,𝔽2)223