Finite Group Databasebeta

← #135×D199 #27164796

Subgroups — #135×D199 #27164796

Order 12736.

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×D199C212yesmaximal, complemented
2#20263978C211yesmaximal, characteristic, complemented
3#20263977C211yesmaximal, characteristic, complemented
4#20263976C212yesmaximal, complemented
5#20263975C212yesmaximal, complemented
6#20263974C211yesmaximal, characteristic, complemented
7Q16×D199C211yesmaximal, characteristic, complemented
8SD16×D199C212yesmaximal, complemented
9D8×D199C211yesmaximal, characteristic, complemented
10C2×C8×D199C211yesmaximal, characteristic, complemented
11#135×C199C211yesmaximal, characteristic, complemented
12#15691887C2212yescomplemented
13#15691886—22no—
14#15691886C2212yescomplemented
15#15691885—22no—
16#15691885C2212yescomplemented
17Q8×D199C2212yes—
18D4×D199—22no—
19D4×D199C2212yes—
20C2×C4×D199—22no—
21C2×C4×D199C2211yescharacteristic
22Dic796C2211yescharacteristic, complemented
23#15691876C2212yescomplemented
24#15691875C2212yescomplemented
25#15691874C2212yescomplemented
26#15691873C2212yescomplemented
27D1592C2211yescharacteristic, complemented
28#15691871C2212yescomplemented
29C2×C199⋊C8C2211yescharacteristic, complemented
30C8×D199C2211yescharacteristic
31C8×D199C2211yescharacteristic
32C8×D199C2211yescharacteristic, complemented
33C8×D199C2211yescharacteristic, complemented
34#41×C199C2212yescomplemented
35Q16×C199C2211yescharacteristic, complemented
36SD16×C199C2212yescomplemented
37D8×C199C2211yescharacteristic, complemented
38C2×C8×C199C2211yescharacteristic, complemented
39C22×D199—22no—
40C22×D199D411yescharacteristic
41Dic398—22no—
42Dic398C2312yes—
43D796—22no—
44D796C2312yes—
45C22⋊D199—22no—
46C22⋊D199—22no—
47C22⋊D199—22no—
48C2×Dic199—22no—
49C2×Dic199D411yescharacteristic
50C4×D199—22no—
51C4×D199—22no—
52C4×D199—22no—
53C4×D199D411yescharacteristic
54C4×D199D411yescharacteristic
55C4×D199C2311yescharacteristic
56C4×D199C2311yescharacteristic
57C199⋊C8C2311yescharacteristic
58C199⋊C8C2311yescharacteristic
59Q8×C199C2312yes—
60D4×C199—22no—
61D4×C199C2312yes—
62C2×C4×C199—22no—
63C2×C4×C199C2311yescharacteristic
64C8×C199C2311yescharacteristic
65C8×C199C2311yescharacteristic
66C22×C199—22no—
67C22×C199C2×D411yescharacteristic
68C2×D199—22no—
69C2×D199—22no—
70C2×D199—44no—
71C2×D199C2×D411yescharacteristic
72C2×D199C2×D411yescharacteristic
73C4×C199—22no—
74C4×C199C2×D411yescharacteristic
75C4×C199C2411yescharacteristic
76Dic199—22no—
77Dic199C2×D411yescharacteristic
78Dic199C2×D411yescharacteristic
79C2×C199—21no—
80C2×C199—42no—
81C2×C199C22×D411yescharacteristic
82D199—21no—
83D199—42no—
84D199#13512yesdirect product factor
85C199C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—1991nomaximal, Sylow
87C2×#41—1992no—
88#135—1992no—
89#135—1991no—
90#135—1992no—
91#135—1991no—
92#135—1991no—
93#135D19911yescharacteristic, direct product factor
94C2×Q16—1991no—
95C2×SD16—1992no—
96C2×D8—1991no—
97C22×C8—1991no—
98#41—3982no—
99#41—3982no—
100#41—1992no—
101#41—1992no—
102#41—1992no—
103#41C2×D19912yescomplemented
104C2×Q8—1992no—
105C2×D4—3982no—
106C2×D4—1992no—
107C22×C4—3982no—
108C22×C4—1991no—
109Q16—1992no—
110Q16—1991no—
111Q16C2×D19911yescharacteristic, complemented
112SD16—1992no—
113SD16—1992no—
114SD16—1992no—
115SD16C2×D19912yescomplemented
116D8—1992no—
117D8—1991no—
118D8C2×D19911yescharacteristic, complemented
119C2×C8—1991no—
120C2×C8—1991no—
121C2×C8—1991no—
122C2×C8—1991no—
123C2×C8—1991no—
124C2×C8C2×D19911yescharacteristic, complemented
125C23—3982no—
126C23—1991no—
127Q8—3982no—
128Q8—1992no—
129Q8C22×D19912yes—
130D4—22no—
131D4—3982no—
132D4—3982no—
133D4—3982no—
134D4—3982no—
135D4—1992no—
136D4C22×D19912yes—
137C2×C4—22no—
138C2×C4—3982no—
139C2×C4—3982no—
140C2×C4—3982no—
141C2×C4—3982no—
142C2×C4—1991no—
143C2×C4—1991no—
144C2×C4—1991no—
145C2×C4—1991no—
146C2×C4—1991no—
147C2×C4C22×D19911yescharacteristic
148C8—1991no—
149C8—1991no—
150C8C22×D19911yescharacteristic
151C8C22×D19911yescharacteristic
152C22—22no—
153C22—7964no—
154C22—3982no—
155C22—3982no—
156C22—1991no—
157C22—1991no—
158C22D4×D19911yescharacteristic
159C4—22no—
160C4—3982no—
161C4—1991no—
162C4—1991no—
163C4D4×D19911yescharacteristic
164C4C23×D19911yescharacteristic
165C2—21no—
166C2—42no—
167C2—7962no—
168C2—3981no—
169C2—1992no—
170C2C2×D4×D19911yescharacteristic
subgroupG/H
centerC4D4×D199
commutator, cosocleC4×C199C24
FrattiniC4C23×D199
Fitting#135×C199C2
socleC2×C199C22×D4
hypercenter#135D199
hypocenterC199C2×#135
LCSG ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D199C2semidirect
2. #20263978C2semidirect
3. #20263977C2semidirect
4. #20263976C2semidirect
5. #20263975C2semidirect
6. #20263974C2semidirect
7. Q16×D199C2semidirect
8. SD16×D199C2semidirect
9. D8×D199C2semidirect
10. C2×C8×D199C2semidirect
11. #135×C199C2semidirect
12. #15691887C22semidirect
14. #15691886C22semidirect
16. #15691885C22semidirect
22. Dic796C22semidirect
23. #15691876C22semidirect
24. #15691875C22semidirect
25. #15691874C22semidirect
26. #15691873C22semidirect
27. D1592C22semidirect
28. #15691871C22semidirect
29. C2×C199⋊C8C22semidirect
32. C8×D199C22semidirect
33. C8×D199C22semidirect
34. #41×C199C22semidirect
35. Q16×C199C22semidirect
36. SD16×C199C22semidirect
37. D8×C199C22semidirect
38. C2×C8×C199C22semidirect
84. D199#135direct (N × H)
85. C199C2×#135semidirect
93. #135D199direct (N × H)
103. #41C2×D199semidirect
111. Q16C2×D199semidirect
115. SD16C2×D199semidirect
118. D8C2×D199semidirect
124. C2×C8C2×D199semidirect

Maximal subgroups

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

subgroupindex
#41×D1992
#202639782
#202639772
#202639762
#202639752
#202639742
Q16×D1992
SD16×D1992
D8×D1992
C2×C8×D1992
#135×C1992
C2×#135199