Finite Group Databasebeta

← C2×#41×D223 #28384405

Subgroups — C2×#41×D223 #28384405

Order 14272.

890 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 455 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
1C2×#16813171C211yesmaximal, characteristic, complemented
2C2×#16813170C213yesmaximal, complemented
3C2×#16813169C213yesmaximal, complemented
4#41×D223C2116yesmaximal, direct product factor
5C2×Q8×D223C211yesmaximal, characteristic, complemented
6C2×D4×D223C213yesmaximal, complemented
7C22×C4×D223C213yesmaximal, complemented
8C2×#41×C223C211yesmaximal, characteristic, complemented
9C23×D223C2213yes—
10#16813171C2218yescomplemented
11#16813170C22124yescomplemented
12#16813169C22124yescomplemented
13C2×Dic446C2213yescomplemented
14Q8×D223C2218yescomplemented
15C2×D892C2213yescomplemented
16C2×C22⋊D223C2216yescomplemented
17D4×D223C22124yescomplemented
18C22×Dic223C2213yescomplemented
19C2×C4×D223C2211yescharacteristic
20C2×C4×D223C2213yes—
21C2×C4×D223C2213yescomplemented
22C2×C4×D223C2213yescomplemented
23C2×C4×D223C22124yescomplemented
24#41×C223C2218yescomplemented
25C2×Q8×C223C2211yescharacteristic, complemented
26C2×D4×C223C2213yescomplemented
27C22×C4×C223C2213yescomplemented
28C22×D223—212no—
29C22×D223C23112yes—
30C22×D223C2313yes—
31C22×D223C2311yescharacteristic
32Dic446C23112yescomplemented
33D892C23112yescomplemented
34C22⋊D223C23124yescomplemented
35C2×Dic223C2311yescharacteristic
36C2×Dic223C2313yes—
37C2×Dic223C23112yescomplemented
38C4×D223C2314yes—
39C4×D223C23112yes—
40C4×D223C23112yescomplemented
41C4×D223C23112yescomplemented
42C23×C223C2313yes—
43Q8×C223C2314yescomplemented
44D4×C223C23112yescomplemented
45C2×C4×C223C2313yes—
46C2×C4×C223C2311yescharacteristic
47C2×C4×C223C23112yescomplemented
48C22×C223—26no—
49C22×C223C2416yes—
50C22×C223C2411yescharacteristic
51C2×D223—26no—
52C2×D223—224no—
53C2×D223#4114yesdirect product factor
54C2×D223C2416yes—
55C2×D223C2412yes—
56C4×C223C2412yes—
57C4×C223C2416yes—
58Dic223C2412yes—
59Dic223C2416yes—
60C2×C223—26no—
61C2×C223C2×#4112yescomplemented
62C2×C223C2511yescharacteristic
63D223—26no—
64D223C2×#4114yesdirect product factor
65C223C22×#4111yescharacteristic, complemented, Sylow
66C22×#41—2231nomaximal, Sylow
67C2×#41—22316no—
68C2×#41—2231no—
69C2×#41—2233no—
70C2×#41—2233no—
71C2×#41D22311yescharacteristic, direct product factor
72C22×Q8—2231no—
73C22×D4—2233no—
74C23×C4—2233no—
75C24—2233no—
76#41—2238no—
77#41—22324no—
78#41—22324no—
79#41C2×D22318yesdirect product factor
80C2×Q8—2238no—
81C2×Q8—2233no—
82C2×Q8C2×D22311yescharacteristic, complemented
83C2×D4—2236no—
84C2×D4—22324no—
85C2×D4—2233no—
86C2×D4C2×D22313yescomplemented
87C22×C4—2233no—
88C22×C4—2233no—
89C22×C4—2233no—
90C22×C4—2231no—
91C22×C4—2233no—
92C22×C4—22324no—
93C22×C4C2×D22313yescomplemented
94C23—44612no—
95C23—22312no—
96C23—2233no—
97C23—2231no—
98C23C22×D22313yes—
99Q8—22312no—
100Q8C22×D22314yescomplemented
101D4—22324no—
102D4—22312no—
103D4C22×D223112yescomplemented
104C2×C4—2231no—
105C2×C4—2233no—
106C2×C4—22312no—
107C2×C4—2234no—
108C2×C4—22312no—
109C2×C4—22312no—
110C2×C4—22312no—
111C2×C4C22×D22313yes—
112C2×C4C22×D22311yescharacteristic
113C2×C4C22×D223112yescomplemented
114C22—26no—
115C22—44624no—
116C22—4466no—
117C22—2236no—
118C22—2234no—
119C22—2232no—
120C22C23×D22316yes—
121C22C23×D22311yescharacteristic
122C4—2232no—
123C4—2236no—
124C4C23×D22312yes—
125C4C23×D22316yes—
126C2—26no—
127C2—4466no—
128C2—2234no—
129C2#41×D22312yesdirect product factor
130C2C24×D22311yescharacteristic
subgroupG/H
centerC2×C4C22×D223
commutator, cosocleC2×C223C25
FrattiniC2C24×D223
FittingC2×#41×C223C2
socleC22×C223C24
hypercenterC2×#41D223
hypocenterC223C22×#41
LCSG ▷ C2×C223 ▷ C223
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16813171C2semidirect
2. C2×#16813170C2semidirect
3. C2×#16813169C2semidirect
4. #41×D223C2direct (N × H)
5. C2×Q8×D223C2semidirect
6. C2×D4×D223C2semidirect
7. C22×C4×D223C2semidirect
8. C2×#41×C223C2semidirect
10. #16813171C22semidirect
11. #16813170C22semidirect
12. #16813169C22semidirect
13. C2×Dic446C22semidirect
14. Q8×D223C22semidirect
15. C2×D892C22semidirect
16. C2×C22⋊D223C22semidirect
17. D4×D223C22semidirect
18. C22×Dic223C22semidirect
21. C2×C4×D223C22semidirect
22. C2×C4×D223C22semidirect
23. C2×C4×D223C22semidirect
24. #41×C223C22semidirect
25. C2×Q8×C223C22semidirect
26. C2×D4×C223C22semidirect
27. C22×C4×C223C22semidirect
32. Dic446C23semidirect
33. D892C23semidirect
34. C22⋊D223C23semidirect
37. C2×Dic223C23semidirect
40. C4×D223C23semidirect
41. C4×D223C23semidirect
43. Q8×C223C23semidirect
44. D4×C223C23semidirect
47. C2×C4×C223C23semidirect
53. C2×D223#41direct (N × H)
61. C2×C223C2×#41semidirect
64. D223C2×#41direct (N × H)
65. C223C22×#41semidirect
71. C2×#41D223direct (N × H)
79. #41C2×D223direct (N × H)
82. C2×Q8C2×D223semidirect
86. C2×D4C2×D223semidirect
93. C22×C4C2×D223semidirect
100. Q8C22×D223semidirect
103. D4C22×D223semidirect
113. C2×C4C22×D223semidirect
129. C2#41×D223direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168131712
C2×#168131702
C2×#168131692
#41×D2232
C2×Q8×D2232
C2×D4×D2232
C22×C4×D2232
C2×#41×C2232
C22×#41223