Finite Group Databasebeta

← #39910046

Subgroups — #39910046

Order 26176.

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

normal subgroups only

This group has 262 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
1C2×#20266989C211yesmaximal, characteristic, complemented
2#41×D409C211yesmaximal, characteristic, complemented
3#27189749C212yesmaximal, complemented
4#27189748C211yesmaximal, characteristic, complemented
5#27189745C211yesmaximal, characteristic, complemented
6#27189744C212yesmaximal, complemented
7#27189742C211yesmaximal, characteristic, complemented
8#27189740C212yesmaximal, complemented
9#27189738C212yesmaximal, complemented
10M16×D409C211yesmaximal, characteristic, complemented
11#137×C409C211yesmaximal, characteristic, complemented
12#20266989—21no—
13#20266989—21no—
14#20266989C2212yes—
15#20266989C2212yescomplemented
16#20266989C2211yescharacteristic, complemented
17#20266988—21no—
18#20266988C2211yescharacteristic
19#20266987—21no—
20#20266987C2211yescharacteristic, complemented
21Q8×D409C2211yescharacteristic, complemented
22C2×D1636—21no—
23C2×D1636C2211yescharacteristic, complemented
24D4×D409—21no—
25D4×D409C2211yescharacteristic, complemented
26C2×C4×D409—21no—
27C2×C4×D409—21no—
28C2×C4×D409C2211yescharacteristic
29#20266978C2212yescomplemented
30#20266977C2212yescomplemented
31#20266975C2212yescomplemented
32#20266975C2212yescomplemented
33D3272C2212yescomplemented
34#20266973C2212yescomplemented
35#20266972C2211yescharacteristic, complemented
36#20266971C2212yescomplemented
37C8×D409C2212yescomplemented
38#41×C409C2211yescharacteristic
39C2×Q8×C409C2211yescharacteristic, complemented
40Q16×C409C2212yescomplemented
41SD16×C409C2212yescomplemented
42M16×C409C2211yescharacteristic, complemented
43C22×D409—21no—
44C22×D409—21no—
45C22×D409D411yescharacteristic
46Dic818—21no—
47Dic818C2311yescharacteristic
48D1636—21no—
49D1636—22no—
50D1636—22no—
51D1636—21no—
52D1636C2312yes—
53D1636C2311yescharacteristic
54C22⋊D409—21no—
55C22⋊D409—21no—
56C22⋊D409—21no—
57C2×Dic409—21no—
58C2×Dic409D411yescharacteristic
59C4×D409—21no—
60C4×D409—21no—
61C4×D409—21no—
62C4×D409—22no—
63C4×D409—22no—
64C4×D409D411yescharacteristic
65C4×D409D411yescharacteristic
66C4×D409C2311yescharacteristic
67C4×D409C2311yescharacteristic
68#15692543C2312yes—
69Q8×C409—21no—
70Q8×C409C2312yes—
71Q8×C409C2311yescharacteristic
72D4×C409—21no—
73D4×C409C2311yescharacteristic
74C2×C4×C409—21no—
75C2×C4×C409—21no—
76C2×C4×C409C2311yescharacteristic
77C8×C409C2312yes—
78C22×C409—21no—
79C22×C409C2×D411yescharacteristic
80C2×D409—21no—
81C2×D409—22no—
82C2×D409—41no—
83C2×D409—42no—
84C2×D409—42no—
85C2×D409C2×D411yescharacteristic
86C2×D409C2×D411yescharacteristic
87C4×C409—22no—
88C4×C409—21no—
89C4×C409C2×D411yescharacteristic
90C4×C409C2411yescharacteristic
91Dic409—21no—
92Dic409C2×D411yescharacteristic
93Dic409C2×D411yescharacteristic
94C2×C409—21no—
95C2×C409—41no—
96C2×C409C22×D411yescharacteristic
97D409—21no—
98D409—21no—
99D409—41no—
100D409—42no—
101C409#57511yescharacteristic, complemented, Sylow
102#575—4091nomaximal, Sylow
103C2×#41—4091no—
104C2×#41—4091no—
105#137—4091no—
106#137—4092no—
107#137D40911yescharacteristic, complemented
108AGL(1,ℤ/8)—4091no—
109AGL(1,ℤ/8)—4092no—
110AGL(1,ℤ/8)—4091no—
111#135—4092no—
112#135—4092no—
113C2×M16—4091no—
114#41—8181no—
115#41—8181no—
116#41—8181no—
117#41—8181no—
118#41—4091no—
119#41—4092no—
120#41—4092no—
121#41—4091no—
122#41—4091no—
123#41C2×D40911yescharacteristic
124C2×Q8—4091no—
125C2×Q8C2×D40911yescharacteristic, complemented
126C2×D4—8181no—
127C2×D4—8181no—
128C2×D4—4091no—
129C2×D4—4091no—
130C22×C4—8181no—
131C22×C4—8181no—
132C22×C4—4091no—
133Q16—4092no—
134Q16C2×D40912yescomplemented
135SD16—4092no—
136SD16—4092no—
137SD16—4092no—
138SD16C2×D40912yescomplemented
139D8—4092no—
140D8—4092no—
141M16—4091no—
142M16—4092no—
143M16C2×D40911yescharacteristic, complemented
144C2×C8—4092no—
145C23—8181no—
146C23—8181no—
147C23—4091no—
148Q8—21no—
149Q8—8181no—
150Q8—4091no—
151Q8C22×D40911yescharacteristic
152Q8C22×D40912yes—
153D4—21no—
154D4—8181no—
155D4—8182no—
156D4—8182no—
157D4—8181no—
158D4—8181no—
159D4—8181no—
160D4—8181no—
161D4—4092no—
162D4—4091no—
163D4C22×D40911yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—8181no—
167C2×C4—8181no—
168C2×C4—8182no—
169C2×C4—8182no—
170C2×C4—8181no—
171C2×C4—8181no—
172C2×C4—4091no—
173C2×C4—4091no—
174C2×C4—4091no—
175C2×C4—4091no—
176C2×C4—4091no—
177C2×C4C22×D40911yescharacteristic
178C8—4092no—
179C8C22×D40912yes—
180C22—21no—
181C22—8182no—
182C22—8181no—
183C22—16361no—
184C22—16362no—
185C22—16362no—
186C22—4091no—
187C22—4091no—
188C22D4×D40911yescharacteristic
189C4—21no—
190C4—22no—
191C4—8181no—
192C4—4091no—
193C4—4091no—
194C4D4×D40911yescharacteristic
195C4C23×D40911yescharacteristic
196C2—21no—
197C2—41no—
198C2—8181no—
199C2—8181no—
200C2—16362no—
201C2—16361no—
202C2C2×D4×D40911yescharacteristic
subgroupG/H
centerC2C2×D4×D409
commutator, cosocleC4×C409C24
FrattiniC4C23×D409
Fitting#137×C409C2
socleC2×C409C22×D4
hypercenter#137D409
hypocenterC409#575
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. C2×#20266989C2semidirect
2. #41×D409C2semidirect
3. #27189749C2semidirect
4. #27189748C2semidirect
5. #27189745C2semidirect
6. #27189744C2semidirect
7. #27189742C2semidirect
8. #27189740C2semidirect
9. #27189738C2semidirect
10. M16×D409C2semidirect
11. #137×C409C2semidirect
15. #20266989C22semidirect
16. #20266989C22semidirect
20. #20266987C22semidirect
21. Q8×D409C22semidirect
23. C2×D1636C22semidirect
25. D4×D409C22semidirect
29. #20266978C22semidirect
30. #20266977C22semidirect
31. #20266975C22semidirect
32. #20266975C22semidirect
33. D3272C22semidirect
34. #20266973C22semidirect
35. #20266972C22semidirect
36. #20266971C22semidirect
37. C8×D409C22semidirect
39. C2×Q8×C409C22semidirect
40. Q16×C409C22semidirect
41. SD16×C409C22semidirect
42. M16×C409C22semidirect
101. C409#575semidirect
107. #137D409semidirect
125. C2×Q8C2×D409semidirect
134. Q16C2×D409semidirect
138. SD16C2×D409semidirect
143. M16C2×D409semidirect

Maximal subgroups

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

subgroupindex
C2×#202669892
#41×D4092
#271897492
#271897482
#271897452
#271897442
#271897422
#271897402
#271897382
M16×D4092
#137×C4092
#575409