Finite Group Databasebeta

← #135×D281 #31824241

Subgroups — #135×D281 #31824241

Order 17984.

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×D281C212yesmaximal, complemented
2#22582902C211yesmaximal, characteristic, complemented
3#22582901C211yesmaximal, characteristic, complemented
4#22582900C212yesmaximal, complemented
5#22582899C212yesmaximal, complemented
6#22582898C211yesmaximal, characteristic, complemented
7Q16×D281C211yesmaximal, characteristic, complemented
8SD16×D281C212yesmaximal, complemented
9D8×D281C211yesmaximal, characteristic, complemented
10C2×C8×D281C211yesmaximal, characteristic, complemented
11#135×C281C211yesmaximal, characteristic, complemented
12#17985139C2212yescomplemented
13#17985138—22no—
14#17985138C2212yescomplemented
15#17985137—22no—
16#17985137C2212yescomplemented
17Q8×D281C2212yes—
18D4×D281—22no—
19D4×D281C2212yes—
20C2×C4×D281—22no—
21C2×C4×D281C2211yescharacteristic
22Dic1124C2211yescharacteristic, complemented
23#17985128C2212yescomplemented
24#17985127C2212yescomplemented
25#17985126C2212yescomplemented
26#17985125C2212yescomplemented
27D2248C2211yescharacteristic, complemented
28#17985123C2212yescomplemented
29C2×#14555886C2211yescharacteristic, complemented
30C8×D281C2211yescharacteristic
31C8×D281C2211yescharacteristic
32C8×D281C2211yescharacteristic, complemented
33C8×D281C2211yescharacteristic, complemented
34#41×C281C2212yescomplemented
35Q16×C281C2211yescharacteristic, complemented
36SD16×C281C2212yescomplemented
37D8×C281C2211yescharacteristic, complemented
38C2×C8×C281C2211yescharacteristic, complemented
39C22×D281—22no—
40C22×D281D411yescharacteristic
41Dic562—22no—
42Dic562C2312yes—
43D1124—22no—
44D1124C2312yes—
45C22⋊D281—22no—
46C22⋊D281—22no—
47C22⋊D281—22no—
48C2×Dic281—22no—
49C2×Dic281D411yescharacteristic
50C4×D281—22no—
51C4×D281—22no—
52C4×D281—22no—
53C4×D281D411yescharacteristic
54C4×D281D411yescharacteristic
55C4×D281C2311yescharacteristic
56C4×D281C2311yescharacteristic
57#14555886C2311yescharacteristic
58#14555886C2311yescharacteristic
59Q8×C281C2312yes—
60D4×C281—22no—
61D4×C281C2312yes—
62C2×C4×C281—22no—
63C2×C4×C281C2311yescharacteristic
64C8×C281C2311yescharacteristic
65C8×C281C2311yescharacteristic
66C22×C281—22no—
67C22×C281C2×D411yescharacteristic
68C2×D281—22no—
69C2×D281—22no—
70C2×D281—44no—
71C2×D281C2×D411yescharacteristic
72C2×D281C2×D411yescharacteristic
73C4×C281—22no—
74C4×C281C2×D411yescharacteristic
75C4×C281C2411yescharacteristic
76Dic281—22no—
77Dic281C2×D411yescharacteristic
78Dic281C2×D411yescharacteristic
79C2×C281—21no—
80C2×C281—42no—
81C2×C281C22×D411yescharacteristic
82D281—21no—
83D281—42no—
84D281#13512yesdirect product factor
85C281C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2811nomaximal, Sylow
87C2×#41—2812no—
88#135—2812no—
89#135—2811no—
90#135—2812no—
91#135—2811no—
92#135—2811no—
93#135D28111yescharacteristic, direct product factor
94C2×Q16—2811no—
95C2×SD16—2812no—
96C2×D8—2811no—
97C22×C8—2811no—
98#41—2812no—
99#41—2812no—
100#41—2812no—
101#41—5622no—
102#41—5622no—
103#41C2×D28112yescomplemented
104C2×Q8—2812no—
105C2×D4—2812no—
106C2×D4—5622no—
107C22×C4—2811no—
108C22×C4—5622no—
109Q16—2812no—
110Q16—2811no—
111Q16C2×D28111yescharacteristic, complemented
112SD16—2812no—
113SD16—2812no—
114SD16—2812no—
115SD16C2×D28112yescomplemented
116D8—2812no—
117D8—2811no—
118D8C2×D28111yescharacteristic, complemented
119C2×C8—2811no—
120C2×C8—2811no—
121C2×C8—2811no—
122C2×C8—2811no—
123C2×C8—2811no—
124C2×C8C2×D28111yescharacteristic, complemented
125C23—2811no—
126C23—5622no—
127Q8—2812no—
128Q8—5622no—
129Q8C22×D28112yes—
130D4—22no—
131D4—2812no—
132D4—5622no—
133D4—5622no—
134D4—5622no—
135D4—5622no—
136D4C22×D28112yes—
137C2×C4—22no—
138C2×C4—2811no—
139C2×C4—2811no—
140C2×C4—2811no—
141C2×C4—2811no—
142C2×C4—2811no—
143C2×C4—5622no—
144C2×C4—5622no—
145C2×C4—5622no—
146C2×C4—5622no—
147C2×C4C22×D28111yescharacteristic
148C8—2811no—
149C8—2811no—
150C8C22×D28111yescharacteristic
151C8C22×D28111yescharacteristic
152C22—22no—
153C22—2811no—
154C22—2811no—
155C22—5622no—
156C22—5622no—
157C22—11244no—
158C22D4×D28111yescharacteristic
159C4—22no—
160C4—2811no—
161C4—2811no—
162C4—5622no—
163C4D4×D28111yescharacteristic
164C4C23×D28111yescharacteristic
165C2—21no—
166C2—42no—
167C2—2812no—
168C2—5621no—
169C2—11242no—
170C2C2×D4×D28111yescharacteristic
subgroupG/H
centerC4D4×D281
commutator, cosocleC4×C281C24
FrattiniC4C23×D281
Fitting#135×C281C2
socleC2×C281C22×D4
hypercenter#135D281
hypocenterC281C2×#135
LCSG ▷ C4×C281 ▷ C2×C281 ▷ C281
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D281C2semidirect
2. #22582902C2semidirect
3. #22582901C2semidirect
4. #22582900C2semidirect
5. #22582899C2semidirect
6. #22582898C2semidirect
7. Q16×D281C2semidirect
8. SD16×D281C2semidirect
9. D8×D281C2semidirect
10. C2×C8×D281C2semidirect
11. #135×C281C2semidirect
12. #17985139C22semidirect
14. #17985138C22semidirect
16. #17985137C22semidirect
22. Dic1124C22semidirect
23. #17985128C22semidirect
24. #17985127C22semidirect
25. #17985126C22semidirect
26. #17985125C22semidirect
27. D2248C22semidirect
28. #17985123C22semidirect
29. C2×#14555886C22semidirect
32. C8×D281C22semidirect
33. C8×D281C22semidirect
34. #41×C281C22semidirect
35. Q16×C281C22semidirect
36. SD16×C281C22semidirect
37. D8×C281C22semidirect
38. C2×C8×C281C22semidirect
84. D281#135direct (N × H)
85. C281C2×#135semidirect
93. #135D281direct (N × H)
103. #41C2×D281semidirect
111. Q16C2×D281semidirect
115. SD16C2×D281semidirect
118. D8C2×D281semidirect
124. C2×C8C2×D281semidirect

Maximal subgroups

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

subgroupindex
#41×D2812
#225829022
#225829012
#225829002
#225828992
#225828982
Q16×D2812
SD16×D2812
D8×D2812
C2×C8×D2812
#135×C2812
C2×#135281