Finite Group Databasebeta

← #135×D397 #38763300

Subgroups — #135×D397 #38763300

Order 25408.

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×D397C212yesmaximal, complemented
2#27163119C211yesmaximal, characteristic, complemented
3#27163118C211yesmaximal, characteristic, complemented
4#27163117C212yesmaximal, complemented
5#27163116C212yesmaximal, complemented
6#27163115C211yesmaximal, characteristic, complemented
7Q16×D397C211yesmaximal, characteristic, complemented
8SD16×D397C212yesmaximal, complemented
9D8×D397C211yesmaximal, characteristic, complemented
10C2×C8×D397C211yesmaximal, characteristic, complemented
11#135×C397C211yesmaximal, characteristic, complemented
12#20263732C2212yescomplemented
13#20263731—22no—
14#20263731C2212yescomplemented
15#20263730—22no—
16#20263730C2212yescomplemented
17Q8×D397C2212yes—
18D4×D397—22no—
19D4×D397C2212yes—
20C2×C4×D397—22no—
21C2×C4×D397C2211yescharacteristic
22Dic1588C2211yescharacteristic, complemented
23#20263721C2212yescomplemented
24#20263720C2212yescomplemented
25#20263719C2212yescomplemented
26#20263718C2212yescomplemented
27D3176C2211yescharacteristic, complemented
28#20263716C2212yescomplemented
29C2×#15691787C2211yescharacteristic, complemented
30C8×D397C2211yescharacteristic
31C8×D397C2211yescharacteristic
32C8×D397C2211yescharacteristic, complemented
33C8×D397C2211yescharacteristic, complemented
34#41×C397C2212yescomplemented
35Q16×C397C2211yescharacteristic, complemented
36SD16×C397C2212yescomplemented
37D8×C397C2211yescharacteristic, complemented
38C2×C8×C397C2211yescharacteristic, complemented
39C22×D397—22no—
40C22×D397D411yescharacteristic
41Dic794—22no—
42Dic794C2312yes—
43D1588—22no—
44D1588C2312yes—
45C22⋊D397—22no—
46C22⋊D397—22no—
47C22⋊D397—22no—
48C2×Dic397—22no—
49C2×Dic397D411yescharacteristic
50C4×D397—22no—
51C4×D397—22no—
52C4×D397—22no—
53C4×D397D411yescharacteristic
54C4×D397D411yescharacteristic
55C4×D397C2311yescharacteristic
56C4×D397C2311yescharacteristic
57#15691787C2311yescharacteristic
58#15691787C2311yescharacteristic
59Q8×C397C2312yes—
60D4×C397—22no—
61D4×C397C2312yes—
62C2×C4×C397—22no—
63C2×C4×C397C2311yescharacteristic
64C8×C397C2311yescharacteristic
65C8×C397C2311yescharacteristic
66C22×C397—22no—
67C22×C397C2×D411yescharacteristic
68C2×D397—22no—
69C2×D397—22no—
70C2×D397—44no—
71C2×D397C2×D411yescharacteristic
72C2×D397C2×D411yescharacteristic
73C4×C397—22no—
74C4×C397C2×D411yescharacteristic
75C4×C397C2411yescharacteristic
76Dic397—22no—
77Dic397C2×D411yescharacteristic
78Dic397C2×D411yescharacteristic
79C2×C397—21no—
80C2×C397—42no—
81C2×C397C22×D411yescharacteristic
82D397—21no—
83D397—42no—
84D397#13512yesdirect product factor
85C397C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—3971nomaximal, Sylow
87C2×#41—3972no—
88#135—3972no—
89#135—3971no—
90#135—3972no—
91#135—3971no—
92#135—3971no—
93#135D39711yescharacteristic, direct product factor
94C2×Q16—3971no—
95C2×SD16—3972no—
96C2×D8—3971no—
97C22×C8—3971no—
98#41—7942no—
99#41—7942no—
100#41—3972no—
101#41—3972no—
102#41—3972no—
103#41C2×D39712yescomplemented
104C2×Q8—3972no—
105C2×D4—7942no—
106C2×D4—3972no—
107C22×C4—7942no—
108C22×C4—3971no—
109Q16—3972no—
110Q16—3971no—
111Q16C2×D39711yescharacteristic, complemented
112SD16—3972no—
113SD16—3972no—
114SD16—3972no—
115SD16C2×D39712yescomplemented
116D8—3972no—
117D8—3971no—
118D8C2×D39711yescharacteristic, complemented
119C2×C8—3971no—
120C2×C8—3971no—
121C2×C8—3971no—
122C2×C8—3971no—
123C2×C8—3971no—
124C2×C8C2×D39711yescharacteristic, complemented
125C23—7942no—
126C23—3971no—
127Q8—7942no—
128Q8—3972no—
129Q8C22×D39712yes—
130D4—22no—
131D4—7942no—
132D4—7942no—
133D4—7942no—
134D4—7942no—
135D4—3972no—
136D4C22×D39712yes—
137C2×C4—22no—
138C2×C4—7942no—
139C2×C4—7942no—
140C2×C4—7942no—
141C2×C4—7942no—
142C2×C4—3971no—
143C2×C4—3971no—
144C2×C4—3971no—
145C2×C4—3971no—
146C2×C4—3971no—
147C2×C4C22×D39711yescharacteristic
148C8—3971no—
149C8—3971no—
150C8C22×D39711yescharacteristic
151C8C22×D39711yescharacteristic
152C22—22no—
153C22—7942no—
154C22—7942no—
155C22—15884no—
156C22—3971no—
157C22—3971no—
158C22D4×D39711yescharacteristic
159C4—22no—
160C4—7942no—
161C4—3971no—
162C4—3971no—
163C4D4×D39711yescharacteristic
164C4C23×D39711yescharacteristic
165C2—21no—
166C2—42no—
167C2—7941no—
168C2—15882no—
169C2—3972no—
170C2C2×D4×D39711yescharacteristic
subgroupG/H
centerC4D4×D397
commutator, cosocleC4×C397C24
FrattiniC4C23×D397
Fitting#135×C397C2
socleC2×C397C22×D4
hypercenter#135D397
hypocenterC397C2×#135
LCSG ▷ C4×C397 ▷ C2×C397 ▷ C397
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D397C2semidirect
2. #27163119C2semidirect
3. #27163118C2semidirect
4. #27163117C2semidirect
5. #27163116C2semidirect
6. #27163115C2semidirect
7. Q16×D397C2semidirect
8. SD16×D397C2semidirect
9. D8×D397C2semidirect
10. C2×C8×D397C2semidirect
11. #135×C397C2semidirect
12. #20263732C22semidirect
14. #20263731C22semidirect
16. #20263730C22semidirect
22. Dic1588C22semidirect
23. #20263721C22semidirect
24. #20263720C22semidirect
25. #20263719C22semidirect
26. #20263718C22semidirect
27. D3176C22semidirect
28. #20263716C22semidirect
29. C2×#15691787C22semidirect
32. C8×D397C22semidirect
33. C8×D397C22semidirect
34. #41×C397C22semidirect
35. Q16×C397C22semidirect
36. SD16×C397C22semidirect
37. D8×C397C22semidirect
38. C2×C8×C397C22semidirect
84. D397#135direct (N × H)
85. C397C2×#135semidirect
93. #135D397direct (N × H)
103. #41C2×D397semidirect
111. Q16C2×D397semidirect
115. SD16C2×D397semidirect
118. D8C2×D397semidirect
124. C2×C8C2×D397semidirect

Maximal subgroups

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

subgroupindex
#41×D3972
#271631192
#271631182
#271631172
#271631162
#271631152
Q16×D3972
SD16×D3972
D8×D3972
C2×C8×D3972
#135×C3972
C2×#135397