Finite Group Databasebeta

← #135×D241 #29508495

Subgroups — #135×D241 #29508495

Order 15424.

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×D241C212yesmaximal, complemented
2#21440665C211yesmaximal, characteristic, complemented
3#21440664C211yesmaximal, characteristic, complemented
4#21440663C212yesmaximal, complemented
5#21440662C212yesmaximal, complemented
6#21440661C211yesmaximal, characteristic, complemented
7Q16×D241C211yesmaximal, characteristic, complemented
8SD16×D241C212yesmaximal, complemented
9D8×D241C211yesmaximal, characteristic, complemented
10C2×C8×D241C211yesmaximal, characteristic, complemented
11#135×C241C211yesmaximal, characteristic, complemented
12#16838771C2212yescomplemented
13#16838770—22no—
14#16838770C2212yescomplemented
15#16838769—22no—
16#16838769C2212yescomplemented
17Q8×D241C2212yes—
18D4×D241—22no—
19D4×D241C2212yes—
20C2×C4×D241—22no—
21C2×C4×D241C2211yescharacteristic
22Dic964C2211yescharacteristic, complemented
23#16838760C2212yescomplemented
24#16838759C2212yescomplemented
25#16838758C2212yescomplemented
26#16838757C2212yescomplemented
27D1928C2211yescharacteristic, complemented
28#16838755C2212yescomplemented
29C2×#14515349C2211yescharacteristic, complemented
30C8×D241C2211yescharacteristic
31C8×D241C2211yescharacteristic
32C8×D241C2211yescharacteristic, complemented
33C8×D241C2211yescharacteristic, complemented
34#41×C241C2212yescomplemented
35Q16×C241C2211yescharacteristic, complemented
36SD16×C241C2212yescomplemented
37D8×C241C2211yescharacteristic, complemented
38C2×C8×C241C2211yescharacteristic, complemented
39C22×D241—22no—
40C22×D241D411yescharacteristic
41Dic482—22no—
42Dic482C2312yes—
43D964—22no—
44D964C2312yes—
45C22⋊D241—22no—
46C22⋊D241—22no—
47C22⋊D241—22no—
48C2×Dic241—22no—
49C2×Dic241D411yescharacteristic
50C4×D241—22no—
51C4×D241—22no—
52C4×D241—22no—
53C4×D241D411yescharacteristic
54C4×D241D411yescharacteristic
55C4×D241C2311yescharacteristic
56C4×D241C2311yescharacteristic
57#14515349C2311yescharacteristic
58#14515349C2311yescharacteristic
59Q8×C241C2312yes—
60D4×C241—22no—
61D4×C241C2312yes—
62C2×C4×C241—22no—
63C2×C4×C241C2311yescharacteristic
64C8×C241C2311yescharacteristic
65C8×C241C2311yescharacteristic
66C22×C241—22no—
67C22×C241C2×D411yescharacteristic
68C2×D241—22no—
69C2×D241—22no—
70C2×D241—44no—
71C2×D241C2×D411yescharacteristic
72C2×D241C2×D411yescharacteristic
73C4×C241—22no—
74C4×C241C2×D411yescharacteristic
75C4×C241C2411yescharacteristic
76Dic241—22no—
77Dic241C2×D411yescharacteristic
78Dic241C2×D411yescharacteristic
79C2×C241—21no—
80C2×C241—42no—
81C2×C241C22×D411yescharacteristic
82D241—21no—
83D241—42no—
84D241#13512yesdirect product factor
85C241C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2411nomaximal, Sylow
87C2×#41—2412no—
88#135—2412no—
89#135—2411no—
90#135—2412no—
91#135—2411no—
92#135—2411no—
93#135D24111yescharacteristic, direct product factor
94C2×Q16—2411no—
95C2×SD16—2412no—
96C2×D8—2411no—
97C22×C8—2411no—
98#41—4822no—
99#41—4822no—
100#41—2412no—
101#41—2412no—
102#41—2412no—
103#41C2×D24112yescomplemented
104C2×Q8—2412no—
105C2×D4—4822no—
106C2×D4—2412no—
107C22×C4—4822no—
108C22×C4—2411no—
109Q16—2412no—
110Q16—2411no—
111Q16C2×D24111yescharacteristic, complemented
112SD16—2412no—
113SD16—2412no—
114SD16—2412no—
115SD16C2×D24112yescomplemented
116D8—2412no—
117D8—2411no—
118D8C2×D24111yescharacteristic, complemented
119C2×C8—2411no—
120C2×C8—2411no—
121C2×C8—2411no—
122C2×C8—2411no—
123C2×C8—2411no—
124C2×C8C2×D24111yescharacteristic, complemented
125C23—4822no—
126C23—2411no—
127Q8—4822no—
128Q8—2412no—
129Q8C22×D24112yes—
130D4—22no—
131D4—4822no—
132D4—4822no—
133D4—4822no—
134D4—4822no—
135D4—2412no—
136D4C22×D24112yes—
137C2×C4—22no—
138C2×C4—4822no—
139C2×C4—4822no—
140C2×C4—4822no—
141C2×C4—4822no—
142C2×C4—2411no—
143C2×C4—2411no—
144C2×C4—2411no—
145C2×C4—2411no—
146C2×C4—2411no—
147C2×C4C22×D24111yescharacteristic
148C8—2411no—
149C8—2411no—
150C8C22×D24111yescharacteristic
151C8C22×D24111yescharacteristic
152C22—22no—
153C22—9644no—
154C22—4822no—
155C22—4822no—
156C22—2411no—
157C22—2411no—
158C22D4×D24111yescharacteristic
159C4—22no—
160C4—4822no—
161C4—2411no—
162C4—2411no—
163C4D4×D24111yescharacteristic
164C4C23×D24111yescharacteristic
165C2—21no—
166C2—42no—
167C2—9642no—
168C2—4821no—
169C2—2412no—
170C2C2×D4×D24111yescharacteristic
subgroupG/H
centerC4D4×D241
commutator, cosocleC4×C241C24
FrattiniC4C23×D241
Fitting#135×C241C2
socleC2×C241C22×D4
hypercenter#135D241
hypocenterC241C2×#135
LCSG ▷ C4×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D241C2semidirect
2. #21440665C2semidirect
3. #21440664C2semidirect
4. #21440663C2semidirect
5. #21440662C2semidirect
6. #21440661C2semidirect
7. Q16×D241C2semidirect
8. SD16×D241C2semidirect
9. D8×D241C2semidirect
10. C2×C8×D241C2semidirect
11. #135×C241C2semidirect
12. #16838771C22semidirect
14. #16838770C22semidirect
16. #16838769C22semidirect
22. Dic964C22semidirect
23. #16838760C22semidirect
24. #16838759C22semidirect
25. #16838758C22semidirect
26. #16838757C22semidirect
27. D1928C22semidirect
28. #16838755C22semidirect
29. C2×#14515349C22semidirect
32. C8×D241C22semidirect
33. C8×D241C22semidirect
34. #41×C241C22semidirect
35. Q16×C241C22semidirect
36. SD16×C241C22semidirect
37. D8×C241C22semidirect
38. C2×C8×C241C22semidirect
84. D241#135direct (N × H)
85. C241C2×#135semidirect
93. #135D241direct (N × H)
103. #41C2×D241semidirect
111. Q16C2×D241semidirect
115. SD16C2×D241semidirect
118. D8C2×D241semidirect
124. C2×C8C2×D241semidirect

Maximal subgroups

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

subgroupindex
#41×D2412
#214406652
#214406642
#214406632
#214406622
#214406612
Q16×D2412
SD16×D2412
D8×D2412
C2×C8×D2412
#135×C2412
C2×#135241