Finite Group Databasebeta

← C2×#26013685 #36374975

Subgroups — C2×#26013685 #36374975

Order 22336.

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#26013685C2116yesmaximal, direct product factor
2C2×#19150234C212yesmaximal, complemented
3C2×#19150233C213yesmaximal, complemented
4C22×D1396C213yesmaximal, complemented
5C2×D4×D349C216yesmaximal, complemented
6C2×#41×C349C211yesmaximal, characteristic, complemented
7C23×D349C2216yescomplemented
8#19150234C22116yescomplemented
9#19150233C22124yescomplemented
10C2×Dic698C2211yescharacteristic, complemented
11C2×D1396C2216yescomplemented
12C2×D1396C2213yescomplemented
13C2×D1396C22124yescomplemented
14C2×C22⋊D349C2216yescomplemented
15D4×D349C22148yescomplemented
16C2×C4×D349C2216yescomplemented
17#41×C349C2218yescomplemented
18C2×Q8×C349C2211yescharacteristic, complemented
19C2×D4×C349C2213yescomplemented
20C22×C4×C349C2213yescomplemented
21C22×D349—412no—
22C22×D349C2316yes—
23C22×D349C23124yescomplemented
24Dic698C2314yescomplemented
25D1396C23124yescomplemented
26D1396C23112yescomplemented
27C22⋊D349C23124yescomplemented
28C2×Dic349C2312yes—
29C4×D349C23124yescomplemented
30C23×C349C2313yes—
31Q8×C349C2314yescomplemented
32D4×C349C23112yescomplemented
33C2×C4×C349C2311yescharacteristic
34C2×C4×C349C2313yes—
35C2×C4×C349C23112yescomplemented
36C22×C349—26no—
37C22×C349C2416yes—
38C22×C349C2411yescharacteristic
39C2×D349—212no—
40C2×D349—424no—
41C2×D349C24112yes—
42C4×C349C2412yes—
43C4×C349C2416yes—
44Dic349C2414yes—
45C2×C349—26no—
46C2×C349H(5,𝔽2)12yescomplemented
47C2×C349C2511yescharacteristic
48D349—212no—
49C349C2×H(5,𝔽2)11yescharacteristic, complemented, Sylow
50C2×H(5,𝔽2)—3491nomaximal, Sylow
51H(5,𝔽2)—34916no—
52C2×#41—3492no—
53C2×#41—3493no—
54C2×#41D34911yescharacteristic, complemented
55C22×D4—3496no—
56C22×D4—3493no—
57C24—3496no—
58#41—34916no—
59#41—34924no—
60#41C2×D34918yescomplemented
61C2×Q8—3491no—
62C2×Q8C2×D34911yescharacteristic, complemented
63C2×D4—3496no—
64C2×D4—34948no—
65C2×D4—3496no—
66C2×D4—3493no—
67C2×D4—34924no—
68C2×D4C2×D34913yescomplemented
69C22×C4—3496no—
70C22×C4C2×D34913yescomplemented
71C23—34924no—
72C23—3496no—
73C23—139612no—
74C23C22×D34913yes—
75Q8—3494no—
76Q8C22×D34914yescomplemented
77D4—34924no—
78D4—34924no—
79D4—34912no—
80D4C22×D349112yescomplemented
81C2×C4—3492no—
82C2×C4—34924no—
83C2×C4C22×D34913yes—
84C2×C4C22×D34911yescharacteristic
85C2×C4C22×D349112yescomplemented
86C22—26no—
87C22—34912no—
88C22—139624no—
89C22—69812no—
90C22C23×D34916yes—
91C22C23×D34911yescharacteristic
92C4—3494no—
93C4C23×D34916yes—
94C4C23×D34912yes—
95C2—26no—
96C2—69812no—
97C2#2601368512yesdirect product factor
98C2C24×D34911yescharacteristic
subgroupG/H
centerC22C23×D349
commutator, cosocleC2×C349C25
FrattiniC2C24×D349
FittingC2×#41×C349C2
socleC22×C349C24
hypercenterC2×#41D349
hypocenterC349C2×H(5,𝔽2)
LCSG ▷ C2×C349 ▷ C349
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26013685C2direct (N × H)
2. C2×#19150234C2semidirect
3. C2×#19150233C2semidirect
4. C22×D1396C2semidirect
5. C2×D4×D349C2semidirect
6. C2×#41×C349C2semidirect
7. C23×D349C22semidirect
8. #19150234C22semidirect
9. #19150233C22semidirect
10. C2×Dic698C22semidirect
11. C2×D1396C22semidirect
12. C2×D1396C22semidirect
13. C2×D1396C22semidirect
14. C2×C22⋊D349C22semidirect
15. D4×D349C22semidirect
16. C2×C4×D349C22semidirect
17. #41×C349C22semidirect
18. C2×Q8×C349C22semidirect
19. C2×D4×C349C22semidirect
20. C22×C4×C349C22semidirect
23. C22×D349C23semidirect
24. Dic698C23semidirect
25. D1396C23semidirect
26. D1396C23semidirect
27. C22⋊D349C23semidirect
29. C4×D349C23semidirect
31. Q8×C349C23semidirect
32. D4×C349C23semidirect
35. C2×C4×C349C23semidirect
46. C2×C349H(5,𝔽2)semidirect
49. C349C2×H(5,𝔽2)semidirect
54. C2×#41D349semidirect
60. #41C2×D349semidirect
62. C2×Q8C2×D349semidirect
68. C2×D4C2×D349semidirect
70. C22×C4C2×D349semidirect
76. Q8C22×D349semidirect
80. D4C22×D349semidirect
85. C2×C4C22×D349semidirect
97. C2#26013685direct (N × H)

Maximal subgroups

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

subgroupindex
#260136852
C2×#191502342
C2×#191502332
C22×D13962
C2×D4×D3492
C2×#41×C3492
C2×H(5,𝔽2)349