Finite Group Databasebeta

← C2×#22554117 #30680714

Subgroups — C2×#22554117 #30680714

Order 16768.

1512 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 549 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×#16862292C212yesmaximal, complemented
2#22554117C2116yesmaximal, direct product factor
3C2×#16862274C216yesmaximal, complemented
4C2×#16862268C213yesmaximal, complemented
5C22×D1048C213yesmaximal, complemented
6C2×#131×C131C211yesmaximal, characteristic, complemented
7#16862292—28no—
8#16862292C22116yescomplemented
9C2×#14523514—22no—
10C2×#14523513C2216yescomplemented
11C22×D524C2216yescomplemented
12C2×D4×D131—26no—
13#16862274C22148yescomplemented
14#16862268C22124yescomplemented
15C2×Dic524C2211yescharacteristic, complemented
16C2×#14523502C2216yescomplemented
17C2×D1048C2213yescomplemented
18C2×D1048C2216yescomplemented
19C2×D1048C22124yescomplemented
20C2×#41×C131C2211yescharacteristic
21#131×C131C2218yescomplemented
22C2×M16×C131C2213yescomplemented
23C22×C8×C131C2213yescomplemented
24C23×D131—26no—
25#14523514—216no—
26#14523513—212no—
27#14523513C23124yescomplemented
28C2×Dic262C2312yes—
29C2×D524—26no—
30C2×D524—212no—
31C2×D524C2316yes—
32C2×D524C23124yescomplemented
33C2×C22⋊D131—26no—
34D4×D131—248no—
35C2×C4×D131—26no—
36Dic524C2314yescomplemented
37#14523502C23124yescomplemented
38D1048C23112yescomplemented
39D1048C23124yescomplemented
40#41×C131D414yes—
41#41×C131C2314yes—
42C2×Q8×C131D411yescharacteristic
43C2×D4×C131D413yes—
44C22×C4×C131C2313yes—
45M16×C131C23112yescomplemented
46C2×C8×C131C2311yescharacteristic
47C2×C8×C131C2313yes—
48C2×C8×C131C23112yescomplemented
49C22×D131—26no—
50C22×D131—224no—
51C22×D131—812no—
52Dic262—22no—
53Dic262C2414yes—
54D524—224no—
55D524—26no—
56D524C24112yes—
57C22⋊D131—224no—
58C2×Dic131—22no—
59C4×D131—224no—
60C23×C131C2×D413yes—
61Q8×C131C2×D414yes—
62D4×C131C2×D4112yes—
63C2×C4×C131C2×D413yes—
64C2×C4×C131C2×D416yes—
65C2×C4×C131C2411yescharacteristic
66C2×C4×C131C2416yes—
67C8×C131C2412yes—
68C8×C131C2416yes—
69C22×C131—26no—
70C22×C131C22×D416yes—
71C22×C131C22×D411yescharacteristic
72C2×D131—212no—
73C2×D131—412no—
74C2×D131—824no—
75C4×C131C22×D411yescharacteristic
76C4×C131C22×D416yes—
77C4×C131C2511yescharacteristic
78Dic131—24no—
79C2×C131—26no—
80C2×C131#57612yescomplemented
81C2×C131C23×D411yescharacteristic
82D131—412no—
83C131C2×#57611yescharacteristic, complemented, Sylow
84C2×#576—1311nomaximal, Sylow
85C2×H(5,𝔽2)—1312no—
86#576—13116no—
87C2×AGL(1,ℤ/8)—1316no—
88C2×#135—1313no—
89C22×D8—1313no—
90C2×#131D13111yescharacteristic, complemented
91H(5,𝔽2)—2628no—
92H(5,𝔽2)—13116no—
93C2×#41—2622no—
94C2×#41—1316no—
95C2×#41C2×D13111yescharacteristic
96C22×D4—2626no—
97C22×D4—1316no—
98AGL(1,ℤ/8)—13148no—
99#135—13124no—
100C2×Q16—1311no—
101C2×SD16—1316no—
102C2×D8—1313no—
103C2×D8—1316no—
104C2×D8—13124no—
105#131C2×D13118yescomplemented
106C2×M16C2×D13113yescomplemented
107C22×C8C2×D13113yescomplemented
108C24—2626no—
109#41—26216no—
110#41—26212no—
111#41—13124no—
112#41D52414yes—
113#41C22×D13114yes—
114C2×Q8—1312no—
115C2×Q8D52411yescharacteristic
116C2×D4—2626no—
117C2×D4—26248no—
118C2×D4—2626no—
119C2×D4—26212no—
120C2×D4—1316no—
121C2×D4—13124no—
122C2×D4D52413yes—
123C22×C4—2626no—
124C22×C4C22×D13113yes—
125Q16—1314no—
126SD16—13124no—
127D8—13112no—
128D8—13124no—
129M16C22×D131112yescomplemented
130C2×C8C22×D13111yescharacteristic
131C2×C8C22×D13113yes—
132C2×C8C22×D131112yescomplemented
133C23—2626no—
134C23—26224no—
135C23—104812no—
136C23C2×D52413yes—
137Q8—2622no—
138Q8—1314no—
139Q8C2×D52414yes—
140D4—26224no—
141D4—26224no—
142D4—2626no—
143D4—13112no—
144D4C2×D524112yes—
145C2×C4—2622no—
146C2×C4—26224no—
147C2×C4C2×D52416yes—
148C2×C4C2×D52413yes—
149C2×C4C23×D13116yes—
150C2×C4C23×D13111yescharacteristic
151C8C23×D13112yes—
152C8C23×D13116yes—
153C22—26no—
154C22—26212no—
155C22—52412no—
156C22—104824no—
157C22C22×D52416yes—
158C22C22×D52411yescharacteristic
159C4—2624no—
160C4C22×D52416yes—
161C4C22×D52411yescharacteristic
162C4C24×D13111yescharacteristic
163C2—26no—
164C2—52412no—
165C2#2255411712yesdirect product factor
166C2C23×D52411yescharacteristic
subgroupG/H
centerC22C22×D524
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC2×#131×C131C2
socleC22×C131C22×D4
hypercenterC2×#131D131
hypocenterC131C2×#576
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#16862292C2semidirect
2. #22554117C2direct (N × H)
3. C2×#16862274C2semidirect
4. C2×#16862268C2semidirect
5. C22×D1048C2semidirect
6. C2×#131×C131C2semidirect
8. #16862292C22semidirect
10. C2×#14523513C22semidirect
11. C22×D524C22semidirect
13. #16862274C22semidirect
14. #16862268C22semidirect
15. C2×Dic524C22semidirect
16. C2×#14523502C22semidirect
17. C2×D1048C22semidirect
18. C2×D1048C22semidirect
19. C2×D1048C22semidirect
21. #131×C131C22semidirect
22. C2×M16×C131C22semidirect
23. C22×C8×C131C22semidirect
27. #14523513C23semidirect
32. C2×D524C23semidirect
36. Dic524C23semidirect
37. #14523502C23semidirect
38. D1048C23semidirect
39. D1048C23semidirect
45. M16×C131C23semidirect
48. C2×C8×C131C23semidirect
80. C2×C131#576semidirect
83. C131C2×#576semidirect
90. C2×#131D131semidirect
105. #131C2×D131semidirect
106. C2×M16C2×D131semidirect
107. C22×C8C2×D131semidirect
129. M16C22×D131semidirect
132. C2×C8C22×D131semidirect
165. C2#22554117direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622922
#225541172
C2×#168622742
C2×#168622682
C22×D10482
C2×#131×C1312
C2×#576131