Finite Group Databasebeta

← #39910056

Subgroups — #39910056

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#27189763C211yesmaximal, characteristic, complemented
3#27189749C212yesmaximal, complemented
4#27189745C211yesmaximal, characteristic, complemented
5#27189744C212yesmaximal, complemented
6#27189740C212yesmaximal, complemented
7#27189737C211yesmaximal, characteristic, complemented
8C2×#20266975C211yesmaximal, characteristic, complemented
9SD16×D409C212yesmaximal, complemented
10#27189725C211yesmaximal, characteristic, complemented
11#137×C409C211yesmaximal, characteristic, complemented
12#20266989—21no—
13#20266989C2212yes—
14#20266989C2212yescomplemented
15#20266988—21no—
16#20266988—22no—
17#20266988C2211yescharacteristic
18#20266988C2211yescharacteristic, complemented
19Q8×D409—21no—
20Q8×D409C2212yescomplemented
21C2×D1636—21no—
22C2×D1636C2211yescharacteristic
23D4×D409—22no—
24D4×D409C2212yes—
25#20266978C2211yescharacteristic, complemented
26#20266977C2212yescomplemented
27#20266976C2211yescharacteristic, complemented
28#20266975C2212yescomplemented
29#20266975C2212yescomplemented
30#20266975C2211yescharacteristic, complemented
31D3272C2212yescomplemented
32#20266973C2211yescharacteristic, complemented
33#20266971C2212yescomplemented
34C2×#15692543C2211yescharacteristic, complemented
35C8×D409C2212yescomplemented
36#41×C409C2211yescharacteristic
37C2×Q8×C409C2211yescharacteristic, complemented
38Q16×C409C2212yescomplemented
39SD16×C409C2212yescomplemented
40M16×C409C2211yescharacteristic, complemented
41C22×D409—21no—
42C22×D409—22no—
43Dic818—22no—
44Dic818D411yescharacteristic
45Dic818C2311yescharacteristic
46D1636—22no—
47D1636—22no—
48D1636—21no—
49D1636D411yescharacteristic
50D1636C2311yescharacteristic
51D1636C2312yes—
52C22⋊D409—21no—
53C22⋊D409—22no—
54C22⋊D409D412yes—
55C4×D409—22no—
56C4×D409—21no—
57C4×D409—22no—
58C4×D409—22no—
59C4×D409C2312yes—
60#15692543C2311yescharacteristic
61#15692543C2311yescharacteristic
62Q8×C409—21no—
63Q8×C409C2312yes—
64Q8×C409C2311yescharacteristic
65D4×C409—21no—
66D4×C409C2311yescharacteristic
67C2×C4×C409—21no—
68C2×C4×C409—21no—
69C2×C4×C409C2311yescharacteristic
70C8×C409C2312yes—
71C22×C409—21no—
72C22×C409C2×D411yescharacteristic
73C2×D409—21no—
74C2×D409—22no—
75C2×D409—82no—
76C2×D409—81no—
77C2×D409C2×D412yes—
78C4×C409—21no—
79C4×C409—22no—
80C4×C409C2×D411yescharacteristic
81C4×C409C2411yescharacteristic
82Dic409—21no—
83Dic409C2×D412yes—
84C2×C409—21no—
85C2×C409—41no—
86C2×C409C22×D411yescharacteristic
87D409—22no—
88D409—41no—
89D409—42no—
90C409#57711yescharacteristic, complemented, Sylow
91#577—4091nomaximal, Sylow
922−1+4—4091no—
93H(5,𝔽2)—4091no—
94#137—4092no—
95#137D40911yescharacteristic, complemented
96AGL(1,ℤ/8)—4092no—
97AGL(1,ℤ/8)—4091no—
98#135—4091no—
99#135—4092no—
100C2×SD16—4091no—
101C2×SD16—4092no—
102#131—4091no—
103#41—8181no—
104#41—8182no—
105#41—8181no—
106#41—4092no—
107#41—4092no—
108#41—4091no—
109#41—4091no—
110#41C2×D40911yescharacteristic
111C2×Q8—8181no—
112C2×Q8—4092no—
113C2×Q8C2×D40911yescharacteristic, complemented
114C2×D4—8181no—
115C2×D4—8182no—
116C2×D4—4092no—
117C2×D4—4091no—
118Q16—4091no—
119Q16C2×D40912yescomplemented
120SD16—4092no—
121SD16—4092no—
122SD16—4091no—
123SD16—4091no—
124SD16—4092no—
125SD16C2×D40912yescomplemented
126D8—4091no—
127D8—4092no—
128M16—4092no—
129M16C2×D40911yescharacteristic, complemented
130C2×C8—4091no—
131C2×C8—4092no—
132C23—8182no—
133C23—8181no—
134Q8—21no—
135Q8—8182no—
136Q8—4091no—
137Q8—4091no—
138Q8C22×D40912yes—
139Q8C22×D40911yescharacteristic
140D4—21no—
141D4—8182no—
142D4—8182no—
143D4—8181no—
144D4—8182no—
145D4—8181no—
146D4—4092no—
147D4—4091no—
148D4—4091no—
149D4—4092no—
150D4C22×D40911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8182no—
154C2×C4—8182no—
155C2×C4—8182no—
156C2×C4—8181no—
157C2×C4—4092no—
158C2×C4C22×D40911yescharacteristic
159C8—4091no—
160C8—4091no—
161C8C22×D40912yes—
162C22—21no—
163C22—8181no—
164C22—8182no—
165C22—4092no—
166C22—32721no—
167C22—32722no—
168C22D4×D40911yescharacteristic
169C4—21no—
170C4—22no—
171C4—8181no—
172C4—4092no—
173C4D4×D40911yescharacteristic
174C4C23×D40911yescharacteristic
175C2—21no—
176C2—41no—
177C2—8182no—
178C2—16362no—
179C2—16361no—
180C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC2C2×D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
Fitting#137×C409C2
socleC2×C409C22×D4
hypercenter#137D409
hypocenterC409#577
LCSG ▷ C4×C409 ▷ C2×C409 ▷ C409
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #27189765C2semidirect
2. #27189763C2semidirect
3. #27189749C2semidirect
4. #27189745C2semidirect
5. #27189744C2semidirect
6. #27189740C2semidirect
7. #27189737C2semidirect
8. C2×#20266975C2semidirect
9. SD16×D409C2semidirect
10. #27189725C2semidirect
11. #137×C409C2semidirect
14. #20266989C22semidirect
18. #20266988C22semidirect
20. Q8×D409C22semidirect
25. #20266978C22semidirect
26. #20266977C22semidirect
27. #20266976C22semidirect
28. #20266975C22semidirect
29. #20266975C22semidirect
30. #20266975C22semidirect
31. D3272C22semidirect
32. #20266973C22semidirect
33. #20266971C22semidirect
34. C2×#15692543C22semidirect
35. C8×D409C22semidirect
37. C2×Q8×C409C22semidirect
38. Q16×C409C22semidirect
39. SD16×C409C22semidirect
40. M16×C409C22semidirect
90. C409#577semidirect
95. #137D409semidirect
113. C2×Q8C2×D409semidirect
119. Q16C2×D409semidirect
125. SD16C2×D409semidirect
129. M16C2×D409semidirect

Maximal subgroups

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

subgroupindex
#271897652
#271897632
#271897492
#271897452
#271897442
#271897402
#271897372
C2×#202669752
SD16×D4092
#271897252
#137×C4092
#577409