Finite Group Databasebeta

← #135×D373 #37560200

Subgroups — #135×D373 #37560200

Order 23872.

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×D373C212yesmaximal, complemented
2#26045815C211yesmaximal, characteristic, complemented
3#26045814C211yesmaximal, characteristic, complemented
4#26045813C212yesmaximal, complemented
5#26045812C212yesmaximal, complemented
6#26045811C211yesmaximal, characteristic, complemented
7Q16×D373C211yesmaximal, characteristic, complemented
8SD16×D373C212yesmaximal, complemented
9D8×D373C211yesmaximal, characteristic, complemented
10C2×C8×D373C211yesmaximal, characteristic, complemented
11#135×C373C211yesmaximal, characteristic, complemented
12#20238611C2212yescomplemented
13#20238610—22no—
14#20238610C2212yescomplemented
15#20238609—22no—
16#20238609C2212yescomplemented
17Q8×D373C2212yes—
18D4×D373—22no—
19D4×D373C2212yes—
20C2×C4×D373—22no—
21C2×C4×D373C2211yescharacteristic
22Dic1492C2211yescharacteristic, complemented
23#20238600C2212yescomplemented
24#20238599C2212yescomplemented
25#20238598C2212yescomplemented
26#20238597C2212yescomplemented
27D2984C2211yescharacteristic, complemented
28#20238595C2212yescomplemented
29C2×#15688412C2211yescharacteristic, complemented
30C8×D373C2211yescharacteristic
31C8×D373C2211yescharacteristic
32C8×D373C2211yescharacteristic, complemented
33C8×D373C2211yescharacteristic, complemented
34#41×C373C2212yescomplemented
35Q16×C373C2211yescharacteristic, complemented
36SD16×C373C2212yescomplemented
37D8×C373C2211yescharacteristic, complemented
38C2×C8×C373C2211yescharacteristic, complemented
39C22×D373—22no—
40C22×D373D411yescharacteristic
41Dic746—22no—
42Dic746C2312yes—
43D1492—22no—
44D1492C2312yes—
45C22⋊D373—22no—
46C22⋊D373—22no—
47C22⋊D373—22no—
48C2×Dic373—22no—
49C2×Dic373D411yescharacteristic
50C4×D373—22no—
51C4×D373—22no—
52C4×D373—22no—
53C4×D373D411yescharacteristic
54C4×D373D411yescharacteristic
55C4×D373C2311yescharacteristic
56C4×D373C2311yescharacteristic
57#15688412C2311yescharacteristic
58#15688412C2311yescharacteristic
59Q8×C373C2312yes—
60D4×C373—22no—
61D4×C373C2312yes—
62C2×C4×C373—22no—
63C2×C4×C373C2311yescharacteristic
64C8×C373C2311yescharacteristic
65C8×C373C2311yescharacteristic
66C22×C373—22no—
67C22×C373C2×D411yescharacteristic
68C2×D373—22no—
69C2×D373—22no—
70C2×D373—44no—
71C2×D373C2×D411yescharacteristic
72C2×D373C2×D411yescharacteristic
73C4×C373—22no—
74C4×C373C2×D411yescharacteristic
75C4×C373C2411yescharacteristic
76Dic373—22no—
77Dic373C2×D411yescharacteristic
78Dic373C2×D411yescharacteristic
79C2×C373—21no—
80C2×C373—42no—
81C2×C373C22×D411yescharacteristic
82D373—21no—
83D373—42no—
84D373#13512yesdirect product factor
85C373C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—3731nomaximal, Sylow
87C2×#41—3732no—
88#135—3732no—
89#135—3731no—
90#135—3732no—
91#135—3731no—
92#135—3731no—
93#135D37311yescharacteristic, direct product factor
94C2×Q16—3731no—
95C2×SD16—3732no—
96C2×D8—3731no—
97C22×C8—3731no—
98#41—3732no—
99#41—3732no—
100#41—3732no—
101#41—7462no—
102#41—7462no—
103#41C2×D37312yescomplemented
104C2×Q8—3732no—
105C2×D4—3732no—
106C2×D4—7462no—
107C22×C4—3731no—
108C22×C4—7462no—
109Q16—3732no—
110Q16—3731no—
111Q16C2×D37311yescharacteristic, complemented
112SD16—3732no—
113SD16—3732no—
114SD16—3732no—
115SD16C2×D37312yescomplemented
116D8—3732no—
117D8—3731no—
118D8C2×D37311yescharacteristic, complemented
119C2×C8—3731no—
120C2×C8—3731no—
121C2×C8—3731no—
122C2×C8—3731no—
123C2×C8—3731no—
124C2×C8C2×D37311yescharacteristic, complemented
125C23—3731no—
126C23—7462no—
127Q8—3732no—
128Q8—7462no—
129Q8C22×D37312yes—
130D4—22no—
131D4—3732no—
132D4—7462no—
133D4—7462no—
134D4—7462no—
135D4—7462no—
136D4C22×D37312yes—
137C2×C4—22no—
138C2×C4—3731no—
139C2×C4—3731no—
140C2×C4—3731no—
141C2×C4—3731no—
142C2×C4—3731no—
143C2×C4—7462no—
144C2×C4—7462no—
145C2×C4—7462no—
146C2×C4—7462no—
147C2×C4C22×D37311yescharacteristic
148C8—3731no—
149C8—3731no—
150C8C22×D37311yescharacteristic
151C8C22×D37311yescharacteristic
152C22—22no—
153C22—3731no—
154C22—3731no—
155C22—14924no—
156C22—7462no—
157C22—7462no—
158C22D4×D37311yescharacteristic
159C4—22no—
160C4—3731no—
161C4—3731no—
162C4—7462no—
163C4D4×D37311yescharacteristic
164C4C23×D37311yescharacteristic
165C2—21no—
166C2—42no—
167C2—3732no—
168C2—14922no—
169C2—7461no—
170C2C2×D4×D37311yescharacteristic
subgroupG/H
centerC4D4×D373
commutator, cosocleC4×C373C24
FrattiniC4C23×D373
Fitting#135×C373C2
socleC2×C373C22×D4
hypercenter#135D373
hypocenterC373C2×#135
LCSG ▷ C4×C373 ▷ C2×C373 ▷ C373
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D373C2semidirect
2. #26045815C2semidirect
3. #26045814C2semidirect
4. #26045813C2semidirect
5. #26045812C2semidirect
6. #26045811C2semidirect
7. Q16×D373C2semidirect
8. SD16×D373C2semidirect
9. D8×D373C2semidirect
10. C2×C8×D373C2semidirect
11. #135×C373C2semidirect
12. #20238611C22semidirect
14. #20238610C22semidirect
16. #20238609C22semidirect
22. Dic1492C22semidirect
23. #20238600C22semidirect
24. #20238599C22semidirect
25. #20238598C22semidirect
26. #20238597C22semidirect
27. D2984C22semidirect
28. #20238595C22semidirect
29. C2×#15688412C22semidirect
32. C8×D373C22semidirect
33. C8×D373C22semidirect
34. #41×C373C22semidirect
35. Q16×C373C22semidirect
36. SD16×C373C22semidirect
37. D8×C373C22semidirect
38. C2×C8×C373C22semidirect
84. D373#135direct (N × H)
85. C373C2×#135semidirect
93. #135D373direct (N × H)
103. #41C2×D373semidirect
111. Q16C2×D373semidirect
115. SD16C2×D373semidirect
118. D8C2×D373semidirect
124. C2×C8C2×D373semidirect

Maximal subgroups

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

subgroupindex
#41×D3732
#260458152
#260458142
#260458132
#260458122
#260458112
Q16×D3732
SD16×D3732
D8×D3732
C2×C8×D3732
#135×C3732
C2×#135373