Finite Group Databasebeta

← #135×D229 #28412259

Subgroups — #135×D229 #28412259

Order 14656.

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×D229C212yesmaximal, complemented
2#20301011C211yesmaximal, characteristic, complemented
3#20301010C211yesmaximal, characteristic, complemented
4#20301009C212yesmaximal, complemented
5#20301008C212yesmaximal, complemented
6#20301007C211yesmaximal, characteristic, complemented
7Q16×D229C211yesmaximal, characteristic, complemented
8SD16×D229C212yesmaximal, complemented
9D8×D229C211yesmaximal, characteristic, complemented
10C2×C8×D229C211yesmaximal, characteristic, complemented
11#135×C229C211yesmaximal, characteristic, complemented
12#16814667C2212yescomplemented
13#16814666—22no—
14#16814666C2212yescomplemented
15#16814665—22no—
16#16814665C2212yescomplemented
17Q8×D229C2212yes—
18D4×D229—22no—
19D4×D229C2212yes—
20C2×C4×D229—22no—
21C2×C4×D229C2211yescharacteristic
22Dic916C2211yescharacteristic, complemented
23#16814656C2212yescomplemented
24#16814655C2212yescomplemented
25#16814654C2212yescomplemented
26#16814653C2212yescomplemented
27D1832C2211yescharacteristic, complemented
28#16814651C2212yescomplemented
29C2×#14270080C2211yescharacteristic, complemented
30C8×D229C2211yescharacteristic
31C8×D229C2211yescharacteristic
32C8×D229C2211yescharacteristic, complemented
33C8×D229C2211yescharacteristic, complemented
34#41×C229C2212yescomplemented
35Q16×C229C2211yescharacteristic, complemented
36SD16×C229C2212yescomplemented
37D8×C229C2211yescharacteristic, complemented
38C2×C8×C229C2211yescharacteristic, complemented
39C22×D229—22no—
40C22×D229D411yescharacteristic
41Dic458—22no—
42Dic458C2312yes—
43D916—22no—
44D916C2312yes—
45C22⋊D229—22no—
46C22⋊D229—22no—
47C22⋊D229—22no—
48C2×Dic229—22no—
49C2×Dic229D411yescharacteristic
50C4×D229—22no—
51C4×D229—22no—
52C4×D229—22no—
53C4×D229D411yescharacteristic
54C4×D229D411yescharacteristic
55C4×D229C2311yescharacteristic
56C4×D229C2311yescharacteristic
57#14270080C2311yescharacteristic
58#14270080C2311yescharacteristic
59Q8×C229C2312yes—
60D4×C229—22no—
61D4×C229C2312yes—
62C2×C4×C229—22no—
63C2×C4×C229C2311yescharacteristic
64C8×C229C2311yescharacteristic
65C8×C229C2311yescharacteristic
66C22×C229—22no—
67C22×C229C2×D411yescharacteristic
68C2×D229—22no—
69C2×D229—22no—
70C2×D229—44no—
71C2×D229C2×D411yescharacteristic
72C2×D229C2×D411yescharacteristic
73C4×C229—22no—
74C4×C229C2×D411yescharacteristic
75C4×C229C2411yescharacteristic
76Dic229—22no—
77Dic229C2×D411yescharacteristic
78Dic229C2×D411yescharacteristic
79C2×C229—21no—
80C2×C229—42no—
81C2×C229C22×D411yescharacteristic
82D229—21no—
83D229—42no—
84D229#13512yesdirect product factor
85C229C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2291nomaximal, Sylow
87C2×#41—2292no—
88#135—2292no—
89#135—2291no—
90#135—2292no—
91#135—2291no—
92#135—2291no—
93#135D22911yescharacteristic, direct product factor
94C2×Q16—2291no—
95C2×SD16—2292no—
96C2×D8—2291no—
97C22×C8—2291no—
98#41—4582no—
99#41—4582no—
100#41—2292no—
101#41—2292no—
102#41—2292no—
103#41C2×D22912yescomplemented
104C2×Q8—2292no—
105C2×D4—4582no—
106C2×D4—2292no—
107C22×C4—4582no—
108C22×C4—2291no—
109Q16—2292no—
110Q16—2291no—
111Q16C2×D22911yescharacteristic, complemented
112SD16—2292no—
113SD16—2292no—
114SD16—2292no—
115SD16C2×D22912yescomplemented
116D8—2292no—
117D8—2291no—
118D8C2×D22911yescharacteristic, complemented
119C2×C8—2291no—
120C2×C8—2291no—
121C2×C8—2291no—
122C2×C8—2291no—
123C2×C8—2291no—
124C2×C8C2×D22911yescharacteristic, complemented
125C23—4582no—
126C23—2291no—
127Q8—4582no—
128Q8—2292no—
129Q8C22×D22912yes—
130D4—22no—
131D4—4582no—
132D4—4582no—
133D4—4582no—
134D4—4582no—
135D4—2292no—
136D4C22×D22912yes—
137C2×C4—22no—
138C2×C4—4582no—
139C2×C4—4582no—
140C2×C4—4582no—
141C2×C4—4582no—
142C2×C4—2291no—
143C2×C4—2291no—
144C2×C4—2291no—
145C2×C4—2291no—
146C2×C4—2291no—
147C2×C4C22×D22911yescharacteristic
148C8—2291no—
149C8—2291no—
150C8C22×D22911yescharacteristic
151C8C22×D22911yescharacteristic
152C22—22no—
153C22—9164no—
154C22—4582no—
155C22—4582no—
156C22—2291no—
157C22—2291no—
158C22D4×D22911yescharacteristic
159C4—22no—
160C4—4582no—
161C4—2291no—
162C4—2291no—
163C4D4×D22911yescharacteristic
164C4C23×D22911yescharacteristic
165C2—21no—
166C2—42no—
167C2—9162no—
168C2—4581no—
169C2—2292no—
170C2C2×D4×D22911yescharacteristic
subgroupG/H
centerC4D4×D229
commutator, cosocleC4×C229C24
FrattiniC4C23×D229
Fitting#135×C229C2
socleC2×C229C22×D4
hypercenter#135D229
hypocenterC229C2×#135
LCSG ▷ C4×C229 ▷ C2×C229 ▷ C229
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D229C2semidirect
2. #20301011C2semidirect
3. #20301010C2semidirect
4. #20301009C2semidirect
5. #20301008C2semidirect
6. #20301007C2semidirect
7. Q16×D229C2semidirect
8. SD16×D229C2semidirect
9. D8×D229C2semidirect
10. C2×C8×D229C2semidirect
11. #135×C229C2semidirect
12. #16814667C22semidirect
14. #16814666C22semidirect
16. #16814665C22semidirect
22. Dic916C22semidirect
23. #16814656C22semidirect
24. #16814655C22semidirect
25. #16814654C22semidirect
26. #16814653C22semidirect
27. D1832C22semidirect
28. #16814651C22semidirect
29. C2×#14270080C22semidirect
32. C8×D229C22semidirect
33. C8×D229C22semidirect
34. #41×C229C22semidirect
35. Q16×C229C22semidirect
36. SD16×C229C22semidirect
37. D8×C229C22semidirect
38. C2×C8×C229C22semidirect
84. D229#135direct (N × H)
85. C229C2×#135semidirect
93. #135D229direct (N × H)
103. #41C2×D229semidirect
111. Q16C2×D229semidirect
115. SD16C2×D229semidirect
118. D8C2×D229semidirect
124. C2×C8C2×D229semidirect

Maximal subgroups

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

subgroupindex
#41×D2292
#203010112
#203010102
#203010092
#203010082
#203010072
Q16×D2292
SD16×D2292
D8×D2292
C2×C8×D2292
#135×C2292
C2×#135229