Finite Group Databasebeta

← #31824272

Subgroups — #31824272

Order 17984.

258 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 99 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#22582927C211yesmaximal, characteristic, complemented
2#22582925C211yesmaximal, characteristic, complemented
3#22582911C212yesmaximal, complemented
4#22582907C211yesmaximal, characteristic, complemented
5#22582906C212yesmaximal, complemented
6#22582902C212yesmaximal, complemented
7#22582899C211yesmaximal, characteristic, complemented
8C2×#17985125C211yesmaximal, characteristic, complemented
9SD16×D281C212yesmaximal, complemented
10#22582887C211yesmaximal, characteristic, complemented
11#137×C281C211yesmaximal, characteristic, complemented
12#17985139—21no—
13#17985139C2212yes—
14#17985139C2212yescomplemented
15#17985138—21no—
16#17985138—22no—
17#17985138C2211yescharacteristic
18#17985138C2211yescharacteristic, complemented
19Q8×D281—21no—
20Q8×D281C2212yescomplemented
21C2×D1124—21no—
22C2×D1124C2211yescharacteristic
23D4×D281—22no—
24D4×D281C2212yes—
25#17985128C2211yescharacteristic, complemented
26#17985127C2212yescomplemented
27#17985126C2211yescharacteristic, complemented
28#17985125C2212yescomplemented
29#17985125C2212yescomplemented
30#17985125C2211yescharacteristic, complemented
31D2248C2212yescomplemented
32#17985123C2211yescharacteristic, complemented
33#17985121C2212yescomplemented
34C2×#14555886C2211yescharacteristic, complemented
35C8×D281C2212yescomplemented
36#41×C281C2211yescharacteristic
37C2×Q8×C281C2211yescharacteristic, complemented
38Q16×C281C2212yescomplemented
39SD16×C281C2212yescomplemented
40M16×C281C2211yescharacteristic, complemented
41C22×D281—21no—
42C22×D281—22no—
43Dic562—22no—
44Dic562D411yescharacteristic
45Dic562C2311yescharacteristic
46D1124—22no—
47D1124—22no—
48D1124—21no—
49D1124D411yescharacteristic
50D1124C2311yescharacteristic
51D1124C2312yes—
52C22⋊D281—21no—
53C22⋊D281—22no—
54C22⋊D281D412yes—
55C4×D281—22no—
56C4×D281—21no—
57C4×D281—22no—
58C4×D281—22no—
59C4×D281C2312yes—
60#14555886C2311yescharacteristic
61#14555886C2311yescharacteristic
62Q8×C281—21no—
63Q8×C281C2312yes—
64Q8×C281C2311yescharacteristic
65D4×C281—21no—
66D4×C281C2311yescharacteristic
67C2×C4×C281—21no—
68C2×C4×C281—21no—
69C2×C4×C281C2311yescharacteristic
70C8×C281C2312yes—
71C22×C281—21no—
72C22×C281C2×D411yescharacteristic
73C2×D281—21no—
74C2×D281—22no—
75C2×D281—82no—
76C2×D281—81no—
77C2×D281C2×D412yes—
78C4×C281—21no—
79C4×C281—22no—
80C4×C281C2×D411yescharacteristic
81C4×C281C2411yescharacteristic
82Dic281—21no—
83Dic281C2×D412yes—
84C2×C281—21no—
85C2×C281—41no—
86C2×C281C22×D411yescharacteristic
87D281—22no—
88D281—41no—
89D281—42no—
90C281#57711yescharacteristic, complemented, Sylow
91#577—2811nomaximal, Sylow
922−1+4—2811no—
93H(5,𝔽2)—2811no—
94#137—2812no—
95#137D28111yescharacteristic, complemented
96AGL(1,ℤ/8)—2812no—
97AGL(1,ℤ/8)—2811no—
98#135—2811no—
99#135—2812no—
100C2×SD16—2811no—
101C2×SD16—2812no—
102#131—2811no—
103#41—2811no—
104#41—2812no—
105#41—2812no—
106#41—2811no—
107#41—5622no—
108#41—5621no—
109#41—5621no—
110#41C2×D28111yescharacteristic
111C2×Q8—2812no—
112C2×Q8—5621no—
113C2×Q8C2×D28111yescharacteristic, complemented
114C2×D4—2812no—
115C2×D4—2811no—
116C2×D4—5621no—
117C2×D4—5622no—
118Q16—2811no—
119Q16C2×D28112yescomplemented
120SD16—2812no—
121SD16—2812no—
122SD16—2811no—
123SD16—2811no—
124SD16—2812no—
125SD16C2×D28112yescomplemented
126D8—2811no—
127D8—2812no—
128M16—2812no—
129M16C2×D28111yescharacteristic, complemented
130C2×C8—2811no—
131C2×C8—2812no—
132C23—5622no—
133C23—5621no—
134Q8—21no—
135Q8—2811no—
136Q8—2811no—
137Q8—5622no—
138Q8C22×D28112yes—
139Q8C22×D28111yescharacteristic
140D4—21no—
141D4—2812no—
142D4—2811no—
143D4—2811no—
144D4—2812no—
145D4—5622no—
146D4—5622no—
147D4—5622no—
148D4—5621no—
149D4—5621no—
150D4C22×D28111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—2812no—
154C2×C4—5622no—
155C2×C4—5622no—
156C2×C4—5622no—
157C2×C4—5621no—
158C2×C4C22×D28111yescharacteristic
159C8—2811no—
160C8—2811no—
161C8C22×D28112yes—
162C22—21no—
163C22—2812no—
164C22—5621no—
165C22—5622no—
166C22—22481no—
167C22—22482no—
168C22D4×D28111yescharacteristic
169C4—21no—
170C4—22no—
171C4—2812no—
172C4—5621no—
173C4D4×D28111yescharacteristic
174C4C23×D28111yescharacteristic
175C2—21no—
176C2—41no—
177C2—5622no—
178C2—11242no—
179C2—11241no—
180C2C2×D4×D28111yescharacteristic
subgroupG/H
centerC2C2×D4×D281
commutator, cosocleC4×C281C24
FrattiniC4C23×D281
Fitting#137×C281C2
socleC2×C281C22×D4
hypercenter#137D281
hypocenterC281#577
LCSG ▷ C4×C281 ▷ C2×C281 ▷ C281
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #22582927C2semidirect
2. #22582925C2semidirect
3. #22582911C2semidirect
4. #22582907C2semidirect
5. #22582906C2semidirect
6. #22582902C2semidirect
7. #22582899C2semidirect
8. C2×#17985125C2semidirect
9. SD16×D281C2semidirect
10. #22582887C2semidirect
11. #137×C281C2semidirect
14. #17985139C22semidirect
18. #17985138C22semidirect
20. Q8×D281C22semidirect
25. #17985128C22semidirect
26. #17985127C22semidirect
27. #17985126C22semidirect
28. #17985125C22semidirect
29. #17985125C22semidirect
30. #17985125C22semidirect
31. D2248C22semidirect
32. #17985123C22semidirect
33. #17985121C22semidirect
34. C2×#14555886C22semidirect
35. C8×D281C22semidirect
37. C2×Q8×C281C22semidirect
38. Q16×C281C22semidirect
39. SD16×C281C22semidirect
40. M16×C281C22semidirect
90. C281#577semidirect
95. #137D281semidirect
113. C2×Q8C2×D281semidirect
119. Q16C2×D281semidirect
125. SD16C2×D281semidirect
129. M16C2×D281semidirect

Maximal subgroups

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

subgroupindex
#225829272
#225829252
#225829112
#225829072
#225829062
#225829022
#225828992
C2×#179851252
SD16×D2812
#225828872
#137×C2812
#577281