Finite Group Databasebeta

← C22×#16812745 #28377834

Subgroups — C22×#16812745 #28377834

Order 13952.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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×#16812745C2148yesmaximal, direct product factor
2C22×#13182030C212yesmaximal, complemented
3C22×#13182029C213yesmaximal, complemented
4C23×D436C213yesmaximal, complemented
5C22×D4×D109C216yesmaximal, complemented
6C22×#41×C109C211yesmaximal, characteristic, complemented
7C24×D109C2216yescomplemented
8#16812745C221256yesdirect product factor
9C2×#13182030C22148yescomplemented
10C2×#13182029C22172yescomplemented
11C22×Dic218C2211yescharacteristic, complemented
12C22×D436C2216yescomplemented
13C22×D436C2213yescomplemented
14C22×D436C22172yescomplemented
15C22×C22⋊D109C2216yescomplemented
16C2×D4×D109C221144yescomplemented
17C22×C4×D109C2216yescomplemented
18C2×#41×C109C22124yescomplemented
19C22×Q8×C109C2211yescharacteristic, complemented
20C22×D4×C109C2213yescomplemented
21C23×C4×C109C2213yescomplemented
22C23×D109—424no—
23C23×D109C2316yes—
24C23×D109C23172yescomplemented
25#13182030C231128yescomplemented
26#13182029C231192yescomplemented
27C2×Dic218C23112yescomplemented
28C2×D436C23172yescomplemented
29C2×D436C23136yescomplemented
30C2×D436C231192yescomplemented
31C2×C22⋊D109C23172yescomplemented
32D4×D109C231384yescomplemented
33C22×Dic109C2312yes—
34C2×C4×D109C23172yescomplemented
35C24×C109C2313yes—
36#41×C109C23164yescomplemented
37C2×Q8×C109C23112yescomplemented
38C2×D4×C109C23136yescomplemented
39C22×C4×C109C2311yescharacteristic
40C22×C4×C109C2313yes—
41C22×C4×C109C23136yescomplemented
42C23×C109—212no—
43C23×C109C24118yes—
44C23×C109C2411yescharacteristic
45C22×D109—224no—
46C22×D109—4144no—
47C22×D109C24136yes—
48C22×D109C24196yescomplemented
49Q8×C109C24116yescomplemented
50D4×C109C24148yescomplemented
51C2×C4×C109C2416yes—
52C2×C4×C109C24118yes—
53C2×C4×C109C24148yescomplemented
54C22⋊D109C24196yescomplemented
55C2×Dic109C24112yes—
56D436C24196yescomplemented
57D436C24148yescomplemented
58C4×D109C24196yescomplemented
59Dic218C24116yescomplemented
60C22×C109—236no—
61C22×C109H(5,𝔽2)14yescomplemented
62C22×C109C25112yes—
63C22×C109C2513yes—
64C2×D109—272no—
65C2×D109—496no—
66C2×D109C25124yes—
67C4×C109C2514yes—
68C4×C109C25112yes—
69Dic109C2518yes—
70C2×C109—212no—
71C2×C109C2×H(5,𝔽2)16yescomplemented
72C2×C109C2611yescharacteristic
73D109—224no—
74C22×H(5,𝔽2)—1091nomaximal, Sylow
75C109C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—10948no—
77C22×#41—1092no—
78C22×#41—1093no—
79C22×#41D10911yescharacteristic, complemented
80C23×D4—1096no—
81C23×D4—1093no—
82C25—1096no—
83H(5,𝔽2)—109256no—
84C2×#41—10948no—
85C2×#41—10972no—
86C2×#41C2×D109124yescomplemented
87C22×Q8—1091no—
88C22×Q8C2×D10911yescharacteristic, complemented
89C22×D4—1096no—
90C22×D4—109144no—
91C22×D4—1096no—
92C22×D4—1093no—
93C22×D4—10972no—
94C22×D4C2×D10913yescomplemented
95C23×C4—1096no—
96C23×C4C2×D10913yescomplemented
97C24—10972no—
98C24—1096no—
99C24—43624no—
100C24C22×D10913yes—
101#41—109128no—
102#41—109192no—
103#41C22×D109164yescomplemented
104C2×Q8—10912no—
105C2×Q8C22×D109112yescomplemented
106C2×D4—10972no—
107C2×D4—109384no—
108C2×D4—10972no—
109C2×D4—10936no—
110C2×D4—109192no—
111C2×D4C22×D109136yescomplemented
112C22×C4—1092no—
113C22×C4—10972no—
114C22×C4C22×D10913yes—
115C22×C4C22×D10911yescharacteristic
116C22×C4C22×D109136yescomplemented
117C23—212no—
118C23—10996no—
119C23—10936no—
120C23—436144no—
121C23—21824no—
122C23C23×D109118yes—
123C23C23×D10911yescharacteristic
124Q8—10916no—
125Q8C23×D109116yescomplemented
126D4—10996no—
127D4—10996no—
128D4—10948no—
129D4C23×D109148yescomplemented
130C2×C4—10912no—
131C2×C4—10996no—
132C2×C4C23×D109118yes—
133C2×C4C23×D10916yes—
134C2×C4C23×D109148yescomplemented
135C22—236no—
136C22—10924no—
137C22—43696no—
138C22—21872no—
139C22#1681274514yesdirect product factor
140C22C24×D109112yes—
141C22C24×D10913yes—
142C4—1098no—
143C4C24×D109112yes—
144C4C24×D10914yes—
145C2—212no—
146C2—21824no—
147C2C2×#1681274516yesdirect product factor
148C2C25×D10911yescharacteristic
subgroupG/H
centerC23C23×D109
commutator, cosocleC2×C109C26
FrattiniC2C25×D109
FittingC22×#41×C109C2
socleC23×C109C24
hypercenterC22×#41D109
hypocenterC109C22×H(5,𝔽2)
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812745C2direct (N × H)
2. C22×#13182030C2semidirect
3. C22×#13182029C2semidirect
4. C23×D436C2semidirect
5. C22×D4×D109C2semidirect
6. C22×#41×C109C2semidirect
7. C24×D109C22semidirect
8. #16812745C22direct (N × H)
9. C2×#13182030C22semidirect
10. C2×#13182029C22semidirect
11. C22×Dic218C22semidirect
12. C22×D436C22semidirect
13. C22×D436C22semidirect
14. C22×D436C22semidirect
15. C22×C22⋊D109C22semidirect
16. C2×D4×D109C22semidirect
17. C22×C4×D109C22semidirect
18. C2×#41×C109C22semidirect
19. C22×Q8×C109C22semidirect
20. C22×D4×C109C22semidirect
21. C23×C4×C109C22semidirect
24. C23×D109C23semidirect
25. #13182030C23semidirect
26. #13182029C23semidirect
27. C2×Dic218C23semidirect
28. C2×D436C23semidirect
29. C2×D436C23semidirect
30. C2×D436C23semidirect
31. C2×C22⋊D109C23semidirect
32. D4×D109C23semidirect
34. C2×C4×D109C23semidirect
36. #41×C109C23semidirect
37. C2×Q8×C109C23semidirect
38. C2×D4×C109C23semidirect
41. C22×C4×C109C23semidirect
48. C22×D109C24semidirect
49. Q8×C109C24semidirect
50. D4×C109C24semidirect
53. C2×C4×C109C24semidirect
54. C22⋊D109C24semidirect
56. D436C24semidirect
57. D436C24semidirect
58. C4×D109C24semidirect
59. Dic218C24semidirect
61. C22×C109H(5,𝔽2)semidirect
71. C2×C109C2×H(5,𝔽2)semidirect
75. C109C22×H(5,𝔽2)semidirect
79. C22×#41D109semidirect
86. C2×#41C2×D109semidirect
88. C22×Q8C2×D109semidirect
94. C22×D4C2×D109semidirect
96. C23×C4C2×D109semidirect
103. #41C22×D109semidirect
105. C2×Q8C22×D109semidirect
111. C2×D4C22×D109semidirect
116. C22×C4C22×D109semidirect
125. Q8C23×D109semidirect
129. D4C23×D109semidirect
134. C2×C4C23×D109semidirect
139. C22#16812745direct (N × H)
147. C2C2×#16812745direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168127452
C22×#131820302
C22×#131820292
C23×D4362
C22×D4×D1092
C22×#41×C1092
C22×H(5,𝔽2)109