Finite Group Databasebeta

← C2×#27189741 #39910032

Subgroups — C2×#27189741 #39910032

Order 26176.

298 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 103 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×#20266987C211yesmaximal, characteristic, complemented
2C2×D4×D409C211yesmaximal, characteristic, complemented
3#27189741C218yesmaximal, direct product factor
4C2×#20266977C211yesmaximal, characteristic, complemented
5C2×#20266976C211yesmaximal, characteristic, complemented
6C2×#20266973C211yesmaximal, characteristic, complemented
7C2×#20266971C211yesmaximal, characteristic, complemented
8C2×D8×C409C211yesmaximal, characteristic, complemented
9C23×D409—21no—
10#20266987—22no—
11#20266987C2214yescomplemented
12C2×Dic818C2211yescharacteristic, complemented
13C2×D1636C2211yescharacteristic
14C2×C22⋊D409—21no—
15C2×C22⋊D409—21no—
16D4×D409—22no—
17D4×D409C2214yescomplemented
18C22×Dic409—21no—
19C2×C4×D409C2211yescharacteristic
20#20266977C2214yescomplemented
21#20266976C2214yescomplemented
22#20266973C2214yescomplemented
23#20266971C2214yescomplemented
24C2×#15692543C2211yescharacteristic, complemented
25C2×D4×C409C2211yescharacteristic
26C2×D4×C409C2211yescharacteristic, complemented
27D8×C409C2214yescomplemented
28C2×C8×C409C2211yescharacteristic, complemented
29C22×D409—21no—
30C22×D409—24no—
31C22×D409—44no—
32C22×D409D411yescharacteristic
33Dic818—21no—
34Dic818C2312yescomplemented
35D1636—21no—
36D1636C2312yes—
37C22⋊D409—24no—
38C22⋊D409—24no—
39C2×Dic409—21no—
40C2×Dic409—24no—
41C2×Dic409D411yescharacteristic
42C4×D409D412yes—
43C4×D409C2312yes—
44#15692543C2312yescomplemented
45C23×C409—21no—
46C23×C409—21no—
47D4×C409—21no—
48D4×C409—21no—
49D4×C409C2312yes—
50D4×C409C2312yescomplemented
51C2×C4×C409C2311yescharacteristic
52C8×C409C2312yescomplemented
53C22×C409—22no—
54C22×C409—22no—
55C22×C409—42no—
56C22×C409—42no—
57C22×C409C2×D411yescharacteristic
58C2×D409—22no—
59C2×D409—22no—
60C2×D409—48no—
61C2×D409—42no—
62C2×D409C2×D412yes—
63C4×C409C2×D411yescharacteristic
64C4×C409C2411yescharacteristic
65Dic409—22no—
66Dic409C2×D412yes—
67C2×C409—42no—
68C2×C409—42no—
69C2×C409AGL(1,ℤ/8)12yescomplemented
70C2×C409C22×D411yescharacteristic
71D409—22no—
72D409—42no—
73C409C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
74C2×AGL(1,ℤ/8)—4091nomaximal, Sylow
75C2×#41—4091no—
76C22×D4—4091no—
77AGL(1,ℤ/8)—4098no—
78C2×SD16—4091no—
79C2×SD16—4091no—
80C2×D8—4091no—
81C2×D8D40911yescharacteristic, complemented
82C2×M16—4091no—
83C24—8181no—
84#41—8182no—
85#41—4094no—
86C2×Q8—4091no—
87C2×D4—8181no—
88C2×D4—8182no—
89C2×D4—8181no—
90C2×D4—4094no—
91C2×D4—4091no—
92C2×D4C2×D40911yescharacteristic
93C2×D4C2×D40911yescharacteristic, complemented
94C22×C4—8181no—
95C22×C4—4091no—
96SD16—4094no—
97SD16—4094no—
98D8—4094no—
99D8C2×D40914yescomplemented
100M16—4094no—
101C2×C8—4091no—
102C2×C8C2×D40911yescharacteristic, complemented
103C23—21no—
104C23—21no—
105C23—8184no—
106C23—8181no—
107C23—16364no—
108C23—4091no—
109Q8—8181no—
110Q8—4092no—
111D4—21no—
112D4—21no—
113D4—8184no—
114D4—8184no—
115D4—8181no—
116D4—4092no—
117D4C22×D40912yes—
118D4C22×D40912yescomplemented
119C2×C4—8181no—
120C2×C4—8184no—
121C2×C4—4091no—
122C2×C4—4092no—
123C2×C4—4092no—
124C2×C4C22×D40911yescharacteristic
125C8—4092no—
126C8C22×D40912yescomplemented
127C22—22no—
128C22—22no—
129C22—42no—
130C22—42no—
131C22—8182no—
132C22—8182no—
133C22—16368no—
134C22—16362no—
135C22—4092no—
136C22D4×D40911yescharacteristic
137C4—8182no—
138C4—4092no—
139C4D4×D40911yescharacteristic
140C4C23×D40911yescharacteristic
141C2—42no—
142C2—42no—
143C2—8182no—
144C2—16362no—
145C2#2718974112yesdirect product factor
146C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC22D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
FittingC2×D8×C409C2
socleC22×C409C2×D4
hypercenterC2×D8D409
hypocenterC409C2×AGL(1,ℤ/8)
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20266987C2semidirect
2. C2×D4×D409C2semidirect
3. #27189741C2direct (N × H)
4. C2×#20266977C2semidirect
5. C2×#20266976C2semidirect
6. C2×#20266973C2semidirect
7. C2×#20266971C2semidirect
8. C2×D8×C409C2semidirect
11. #20266987C22semidirect
12. C2×Dic818C22semidirect
17. D4×D409C22semidirect
20. #20266977C22semidirect
21. #20266976C22semidirect
22. #20266973C22semidirect
23. #20266971C22semidirect
24. C2×#15692543C22semidirect
26. C2×D4×C409C22semidirect
27. D8×C409C22semidirect
28. C2×C8×C409C22semidirect
34. Dic818C23semidirect
44. #15692543C23semidirect
50. D4×C409C23semidirect
52. C8×C409C23semidirect
69. C2×C409AGL(1,ℤ/8)semidirect
73. C409C2×AGL(1,ℤ/8)semidirect
81. C2×D8D409semidirect
93. C2×D4C2×D409semidirect
99. D8C2×D409semidirect
102. C2×C8C2×D409semidirect
118. D4C22×D409semidirect
126. C8C22×D409semidirect
145. C2#27189741direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202669872
C2×D4×D4092
#271897412
C2×#202669772
C2×#202669762
C2×#202669732
C2×#202669712
C2×D8×C4092
C2×AGL(1,ℤ/8)409