Finite Group Databasebeta

← #135×D409 #39910025

Subgroups — #135×D409 #39910025

Order 26176.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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#41×D409C212yesmaximal, complemented
2#27189740C211yesmaximal, characteristic, complemented
3#27189739C211yesmaximal, characteristic, complemented
4#27189738C212yesmaximal, complemented
5#27189737C212yesmaximal, complemented
6#27189736C211yesmaximal, characteristic, complemented
7Q16×D409C211yesmaximal, characteristic, complemented
8SD16×D409C212yesmaximal, complemented
9D8×D409C211yesmaximal, characteristic, complemented
10C2×C8×D409C211yesmaximal, characteristic, complemented
11#135×C409C211yesmaximal, characteristic, complemented
12#20266989C2212yescomplemented
13#20266988—22no—
14#20266988C2212yescomplemented
15#20266987—22no—
16#20266987C2212yescomplemented
17Q8×D409C2212yes—
18D4×D409—22no—
19D4×D409C2212yes—
20C2×C4×D409—22no—
21C2×C4×D409C2211yescharacteristic
22Dic1636C2211yescharacteristic, complemented
23#20266978C2212yescomplemented
24#20266977C2212yescomplemented
25#20266976C2212yescomplemented
26#20266975C2212yescomplemented
27D3272C2211yescharacteristic, complemented
28#20266973C2212yescomplemented
29C2×#15692543C2211yescharacteristic, complemented
30C8×D409C2211yescharacteristic
31C8×D409C2211yescharacteristic
32C8×D409C2211yescharacteristic, complemented
33C8×D409C2211yescharacteristic, complemented
34#41×C409C2212yescomplemented
35Q16×C409C2211yescharacteristic, complemented
36SD16×C409C2212yescomplemented
37D8×C409C2211yescharacteristic, complemented
38C2×C8×C409C2211yescharacteristic, complemented
39C22×D409—22no—
40C22×D409D411yescharacteristic
41Dic818—22no—
42Dic818C2312yes—
43D1636—22no—
44D1636C2312yes—
45C22⋊D409—22no—
46C22⋊D409—22no—
47C22⋊D409—22no—
48C2×Dic409—22no—
49C2×Dic409D411yescharacteristic
50C4×D409—22no—
51C4×D409—22no—
52C4×D409—22no—
53C4×D409D411yescharacteristic
54C4×D409D411yescharacteristic
55C4×D409C2311yescharacteristic
56C4×D409C2311yescharacteristic
57#15692543C2311yescharacteristic
58#15692543C2311yescharacteristic
59Q8×C409C2312yes—
60D4×C409—22no—
61D4×C409C2312yes—
62C2×C4×C409—22no—
63C2×C4×C409C2311yescharacteristic
64C8×C409C2311yescharacteristic
65C8×C409C2311yescharacteristic
66C22×C409—22no—
67C22×C409C2×D411yescharacteristic
68C2×D409—22no—
69C2×D409—22no—
70C2×D409—44no—
71C2×D409C2×D411yescharacteristic
72C2×D409C2×D411yescharacteristic
73C4×C409—22no—
74C4×C409C2×D411yescharacteristic
75C4×C409C2411yescharacteristic
76Dic409—22no—
77Dic409C2×D411yescharacteristic
78Dic409C2×D411yescharacteristic
79C2×C409—21no—
80C2×C409—42no—
81C2×C409C22×D411yescharacteristic
82D409—21no—
83D409—42no—
84D409#13512yesdirect product factor
85C409C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—4091nomaximal, Sylow
87C2×#41—4092no—
88#135—4092no—
89#135—4091no—
90#135—4092no—
91#135—4091no—
92#135—4091no—
93#135D40911yescharacteristic, direct product factor
94C2×Q16—4091no—
95C2×SD16—4092no—
96C2×D8—4091no—
97C22×C8—4091no—
98#41—8182no—
99#41—8182no—
100#41—4092no—
101#41—4092no—
102#41—4092no—
103#41C2×D40912yescomplemented
104C2×Q8—4092no—
105C2×D4—8182no—
106C2×D4—4092no—
107C22×C4—8182no—
108C22×C4—4091no—
109Q16—4092no—
110Q16—4091no—
111Q16C2×D40911yescharacteristic, complemented
112SD16—4092no—
113SD16—4092no—
114SD16—4092no—
115SD16C2×D40912yescomplemented
116D8—4092no—
117D8—4091no—
118D8C2×D40911yescharacteristic, complemented
119C2×C8—4091no—
120C2×C8—4091no—
121C2×C8—4091no—
122C2×C8—4091no—
123C2×C8—4091no—
124C2×C8C2×D40911yescharacteristic, complemented
125C23—8182no—
126C23—4091no—
127Q8—8182no—
128Q8—4092no—
129Q8C22×D40912yes—
130D4—22no—
131D4—8182no—
132D4—8182no—
133D4—8182no—
134D4—8182no—
135D4—4092no—
136D4C22×D40912yes—
137C2×C4—22no—
138C2×C4—8182no—
139C2×C4—8182no—
140C2×C4—8182no—
141C2×C4—8182no—
142C2×C4—4091no—
143C2×C4—4091no—
144C2×C4—4091no—
145C2×C4—4091no—
146C2×C4—4091no—
147C2×C4C22×D40911yescharacteristic
148C8—4091no—
149C8—4091no—
150C8C22×D40911yescharacteristic
151C8C22×D40911yescharacteristic
152C22—22no—
153C22—8182no—
154C22—8182no—
155C22—16364no—
156C22—4091no—
157C22—4091no—
158C22D4×D40911yescharacteristic
159C4—22no—
160C4—8182no—
161C4—4091no—
162C4—4091no—
163C4D4×D40911yescharacteristic
164C4C23×D40911yescharacteristic
165C2—21no—
166C2—42no—
167C2—8181no—
168C2—16362no—
169C2—4092no—
170C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC4D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
Fitting#135×C409C2
socleC2×C409C22×D4
hypercenter#135D409
hypocenterC409C2×#135
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D409C2semidirect
2. #27189740C2semidirect
3. #27189739C2semidirect
4. #27189738C2semidirect
5. #27189737C2semidirect
6. #27189736C2semidirect
7. Q16×D409C2semidirect
8. SD16×D409C2semidirect
9. D8×D409C2semidirect
10. C2×C8×D409C2semidirect
11. #135×C409C2semidirect
12. #20266989C22semidirect
14. #20266988C22semidirect
16. #20266987C22semidirect
22. Dic1636C22semidirect
23. #20266978C22semidirect
24. #20266977C22semidirect
25. #20266976C22semidirect
26. #20266975C22semidirect
27. D3272C22semidirect
28. #20266973C22semidirect
29. C2×#15692543C22semidirect
32. C8×D409C22semidirect
33. C8×D409C22semidirect
34. #41×C409C22semidirect
35. Q16×C409C22semidirect
36. SD16×C409C22semidirect
37. D8×C409C22semidirect
38. C2×C8×C409C22semidirect
84. D409#135direct (N × H)
85. C409C2×#135semidirect
93. #135D409direct (N × H)
103. #41C2×D409semidirect
111. Q16C2×D409semidirect
115. SD16C2×D409semidirect
118. D8C2×D409semidirect
124. C2×C8C2×D409semidirect

Maximal subgroups

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

subgroupindex
#41×D4092
#271897402
#271897392
#271897382
#271897372
#271897362
Q16×D4092
SD16×D4092
D8×D4092
C2×C8×D4092
#135×C4092
C2×#135409