Finite Group Databasebeta

← #39910053

Subgroups — #39910053

Order 26176.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#27189765C211yesmaximal, characteristic, complemented
2#27189762C211yesmaximal, characteristic, complemented
3#27189748C211yesmaximal, characteristic, complemented
4#27189746C212yesmaximal, complemented
5#27189744C212yesmaximal, complemented
6#27189743C211yesmaximal, characteristic, complemented
7#27189739C211yesmaximal, characteristic, complemented
8#27189738C212yesmaximal, complemented
9SD16×D409C212yesmaximal, complemented
10#27189723C211yesmaximal, characteristic, complemented
11C2×SD16×C409C211yesmaximal, characteristic, complemented
12#20266989—21no—
13#20266989C2212yescomplemented
14#20266988—22no—
15#20266988C2211yescharacteristic
16#20266988C2211yescharacteristic
17#20266988C2211yescharacteristic, complemented
18#20266987—21no—
19#20266987C2212yes—
20Q8×D409—21no—
21Q8×D409C2212yescomplemented
22C2×C22⋊D409—22no—
23D4×D409—21no—
24D4×D409C2212yes—
25Dic1636C2211yescharacteristic, complemented
26#20266978C2212yescomplemented
27#20266977C2211yescharacteristic, complemented
28#20266977C2211yescharacteristic, complemented
29#20266976C2212yescomplemented
30#20266975C2212yescomplemented
31D3272C2211yescharacteristic, complemented
32#20266973C2212yescomplemented
33#20266972C2211yescharacteristic, complemented
34#20266971C2212yescomplemented
35C8×D409C2212yescomplemented
36C2×Q8×C409C2211yescharacteristic, complemented
37C2×D4×C409C2211yescharacteristic
38SD16×C409C2212yescomplemented
39SD16×C409C2212yescomplemented
40C2×C8×C409C2211yescharacteristic, complemented
41C22×D409—22no—
42Dic818—22no—
43Dic818D411yescharacteristic
44Dic818C2311yescharacteristic
45Dic818C2311yescharacteristic
46D1636—22no—
47D1636D411yescharacteristic
48D1636C2311yescharacteristic
49D1636C2311yescharacteristic
50C22⋊D409—21no—
51C22⋊D409—21no—
52C22⋊D409—22no—
53C22⋊D409—22no—
54C22⋊D409D412yes—
55C2×Dic409—22no—
56C4×D409—22no—
57C4×D409—21no—
58C4×D409—21no—
59C4×D409—22no—
60C4×D409C2312yes—
61#15692543C2312yes—
62C23×C409—21no—
63Q8×C409—21no—
64Q8×C409C2312yes—
65D4×C409—21no—
66D4×C409C2312yes—
67C2×C4×C409—21no—
68C2×C4×C409C2311yescharacteristic
69C8×C409C2311yescharacteristic
70C8×C409C2311yescharacteristic
71C22×C409—22no—
72C22×C409—81no—
73C22×C409C2×D411yescharacteristic
74C2×D409—21no—
75C2×D409—21no—
76C2×D409—82no—
77C2×D409C2×D412yes—
78C4×C409—22no—
79C4×C409C2×D411yescharacteristic
80C4×C409C2411yescharacteristic
81Dic409—21no—
82Dic409—21no—
83Dic409C2×D412yes—
84C2×C409—21no—
85C2×C409—42no—
86C2×C409C22×D411yescharacteristic
87D409—22no—
88D409—41no—
89D409—41no—
90C409#57711yescharacteristic, complemented, Sylow
91#577—4091nomaximal, Sylow
922−1+4—4091no—
93H(5,𝔽2)—4091no—
94#137—4091no—
95#137—4092no—
96AGL(1,ℤ/8)—4091no—
97AGL(1,ℤ/8)—4092no—
98#135—4092no—
99#135—4091no—
100C2×SD16—4092no—
101C2×SD16D40911yescharacteristic, complemented
102#131—4091no—
103#41—8182no—
104#41—8181no—
105#41—8181no—
106#41—4092no—
107#41—4092no—
108#41—4091no—
109#41—4091no—
110#41—4091no—
111C2×Q8—8181no—
112C2×Q8—4092no—
113C2×Q8C2×D40911yescharacteristic, complemented
114C2×D4—8181no—
115C2×D4—8182no—
116C2×D4—4092no—
117C2×D4C2×D40911yescharacteristic
118Q16—4092no—
119Q16—4091no—
120SD16—4092no—
121SD16—4092no—
122SD16—4091no—
123SD16—4091no—
124SD16C2×D40912yescomplemented
125SD16C2×D40912yescomplemented
126D8—4092no—
127D8—4091no—
128M16—4091no—
129M16—4092no—
130C2×C8—4092no—
131C2×C8C2×D40911yescharacteristic, complemented
132C23—21no—
133C23—8182no—
134Q8—21no—
135Q8—8182no—
136Q8—4091no—
137Q8—4091no—
138Q8—4091no—
139Q8C22×D40912yes—
140D4—21no—
141D4—8182no—
142D4—8182no—
143D4—8181no—
144D4—8181no—
145D4—8182no—
146D4—4092no—
147D4—4091no—
148D4—4091no—
149D4—4091no—
150D4C22×D40912yes—
151C2×C4—21no—
152C2×C4—8182no—
153C2×C4—8182no—
154C2×C4—8182no—
155C2×C4—8181no—
156C2×C4—8181no—
157C2×C4—4092no—
158C2×C4C22×D40911yescharacteristic
159C8—4092no—
160C8C22×D40911yescharacteristic
161C8C22×D40911yescharacteristic
162C22—22no—
163C22—81no—
164C22—8181no—
165C22—8181no—
166C22—4092no—
167C22—32722no—
168C22D4×D40911yescharacteristic
169C4—22no—
170C4—8181no—
171C4—8181no—
172C4—4092no—
173C4D4×D40911yescharacteristic
174C4C23×D40911yescharacteristic
175C2—21no—
176C2—42no—
177C2—8182no—
178C2—16361no—
179C2—16361no—
180C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC2C2×D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
FittingC2×SD16×C409C2
socleC2×C409C22×D4
hypercenterC2×SD16D409
hypocenterC409#577
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #27189765C2semidirect
2. #27189762C2semidirect
3. #27189748C2semidirect
4. #27189746C2semidirect
5. #27189744C2semidirect
6. #27189743C2semidirect
7. #27189739C2semidirect
8. #27189738C2semidirect
9. SD16×D409C2semidirect
10. #27189723C2semidirect
11. C2×SD16×C409C2semidirect
13. #20266989C22semidirect
17. #20266988C22semidirect
21. Q8×D409C22semidirect
25. Dic1636C22semidirect
26. #20266978C22semidirect
27. #20266977C22semidirect
28. #20266977C22semidirect
29. #20266976C22semidirect
30. #20266975C22semidirect
31. D3272C22semidirect
32. #20266973C22semidirect
33. #20266972C22semidirect
34. #20266971C22semidirect
35. C8×D409C22semidirect
36. C2×Q8×C409C22semidirect
38. SD16×C409C22semidirect
39. SD16×C409C22semidirect
40. C2×C8×C409C22semidirect
90. C409#577semidirect
101. C2×SD16D409semidirect
113. C2×Q8C2×D409semidirect
124. SD16C2×D409semidirect
125. SD16C2×D409semidirect
131. C2×C8C2×D409semidirect

Maximal subgroups

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

subgroupindex
#271897652
#271897622
#271897482
#271897462
#271897442
#271897432
#271897392
#271897382
SD16×D4092
#271897232
C2×SD16×C4092
#577409