Finite Group Databasebeta

← #29505581

Subgroups — #29505581

Order 15296.

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#21440008C211yesmaximal, characteristic, complemented
2#21440006C211yesmaximal, characteristic, complemented
3#21439992C212yesmaximal, complemented
4#21439988C211yesmaximal, characteristic, complemented
5#21439987C212yesmaximal, complemented
6#21439983C212yesmaximal, complemented
7#21439980C211yesmaximal, characteristic, complemented
8C2×#16838535C211yesmaximal, characteristic, complemented
9SD16×D239C212yesmaximal, complemented
10#21439968C211yesmaximal, characteristic, complemented
11#137×C239C211yesmaximal, characteristic, complemented
12#16838549—21no—
13#16838549C2212yes—
14#16838549C2212yescomplemented
15#16838548—21no—
16#16838548—22no—
17#16838548C2211yescharacteristic
18#16838548C2211yescharacteristic, complemented
19Q8×D239—21no—
20Q8×D239C2212yescomplemented
21C2×D956—21no—
22C2×D956C2211yescharacteristic
23D4×D239—22no—
24D4×D239C2212yes—
25#16838538C2211yescharacteristic, complemented
26#16838537C2212yescomplemented
27#16838536C2211yescharacteristic, complemented
28#16838535C2212yescomplemented
29#16838535C2212yescomplemented
30#16838535C2211yescharacteristic, complemented
31D1912C2212yescomplemented
32#16838533C2211yescharacteristic, complemented
33#16838531C2212yescomplemented
34C2×C239⋊C8C2211yescharacteristic, complemented
35C8×D239C2212yescomplemented
36#41×C239C2211yescharacteristic
37C2×Q8×C239C2211yescharacteristic, complemented
38Q16×C239C2212yescomplemented
39SD16×C239C2212yescomplemented
40M16×C239C2211yescharacteristic, complemented
41C22×D239—21no—
42C22×D239—22no—
43Dic478—22no—
44Dic478D411yescharacteristic
45Dic478C2311yescharacteristic
46D956—22no—
47D956—22no—
48D956—21no—
49D956D411yescharacteristic
50D956C2311yescharacteristic
51D956C2312yes—
52C22⋊D239—21no—
53C22⋊D239—22no—
54C22⋊D239D412yes—
55C4×D239—22no—
56C4×D239—21no—
57C4×D239—22no—
58C4×D239—22no—
59C4×D239C2312yes—
60C239⋊C8C2311yescharacteristic
61C239⋊C8C2311yescharacteristic
62Q8×C239—21no—
63Q8×C239C2312yes—
64Q8×C239C2311yescharacteristic
65D4×C239—21no—
66D4×C239C2311yescharacteristic
67C2×C4×C239—21no—
68C2×C4×C239—21no—
69C2×C4×C239C2311yescharacteristic
70C8×C239C2312yes—
71C22×C239—21no—
72C22×C239C2×D411yescharacteristic
73C2×D239—21no—
74C2×D239—22no—
75C2×D239—82no—
76C2×D239—81no—
77C2×D239C2×D412yes—
78C4×C239—21no—
79C4×C239—22no—
80C4×C239C2×D411yescharacteristic
81C4×C239C2411yescharacteristic
82Dic239—21no—
83Dic239C2×D412yes—
84C2×C239—21no—
85C2×C239—41no—
86C2×C239C22×D411yescharacteristic
87D239—22no—
88D239—41no—
89D239—42no—
90C239#57711yescharacteristic, complemented, Sylow
91#577—2391nomaximal, Sylow
922−1+4—2391no—
93H(5,𝔽2)—2391no—
94#137—2392no—
95#137D23911yescharacteristic, complemented
96AGL(1,ℤ/8)—2392no—
97AGL(1,ℤ/8)—2391no—
98#135—2391no—
99#135—2392no—
100C2×SD16—2391no—
101C2×SD16—2392no—
102#131—2391no—
103#41—4781no—
104#41—4782no—
105#41—4781no—
106#41—2392no—
107#41—2392no—
108#41—2391no—
109#41—2391no—
110#41C2×D23911yescharacteristic
111C2×Q8—4781no—
112C2×Q8—2392no—
113C2×Q8C2×D23911yescharacteristic, complemented
114C2×D4—4781no—
115C2×D4—4782no—
116C2×D4—2392no—
117C2×D4—2391no—
118Q16—2391no—
119Q16C2×D23912yescomplemented
120SD16—2392no—
121SD16—2392no—
122SD16—2391no—
123SD16—2391no—
124SD16—2392no—
125SD16C2×D23912yescomplemented
126D8—2391no—
127D8—2392no—
128M16—2392no—
129M16C2×D23911yescharacteristic, complemented
130C2×C8—2391no—
131C2×C8—2392no—
132C23—4782no—
133C23—4781no—
134Q8—21no—
135Q8—4782no—
136Q8—2391no—
137Q8—2391no—
138Q8C22×D23912yes—
139Q8C22×D23911yescharacteristic
140D4—21no—
141D4—4782no—
142D4—4782no—
143D4—4781no—
144D4—4782no—
145D4—4781no—
146D4—2392no—
147D4—2391no—
148D4—2391no—
149D4—2392no—
150D4C22×D23911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4782no—
154C2×C4—4782no—
155C2×C4—4782no—
156C2×C4—4781no—
157C2×C4—2392no—
158C2×C4C22×D23911yescharacteristic
159C8—2391no—
160C8—2391no—
161C8C22×D23912yes—
162C22—21no—
163C22—19121no—
164C22—19122no—
165C22—4781no—
166C22—4782no—
167C22—2392no—
168C22D4×D23911yescharacteristic
169C4—21no—
170C4—22no—
171C4—4781no—
172C4—2392no—
173C4D4×D23911yescharacteristic
174C4C23×D23911yescharacteristic
175C2—21no—
176C2—41no—
177C2—9562no—
178C2—9561no—
179C2—4782no—
180C2C2×D4×D23911yescharacteristic
subgroupG/H
centerC2C2×D4×D239
commutator, cosocleC4×C239C24
FrattiniC4C23×D239
Fitting#137×C239C2
socleC2×C239C22×D4
hypercenter#137D239
hypocenterC239#577
LCSG ▷ C4×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #21440008C2semidirect
2. #21440006C2semidirect
3. #21439992C2semidirect
4. #21439988C2semidirect
5. #21439987C2semidirect
6. #21439983C2semidirect
7. #21439980C2semidirect
8. C2×#16838535C2semidirect
9. SD16×D239C2semidirect
10. #21439968C2semidirect
11. #137×C239C2semidirect
14. #16838549C22semidirect
18. #16838548C22semidirect
20. Q8×D239C22semidirect
25. #16838538C22semidirect
26. #16838537C22semidirect
27. #16838536C22semidirect
28. #16838535C22semidirect
29. #16838535C22semidirect
30. #16838535C22semidirect
31. D1912C22semidirect
32. #16838533C22semidirect
33. #16838531C22semidirect
34. C2×C239⋊C8C22semidirect
35. C8×D239C22semidirect
37. C2×Q8×C239C22semidirect
38. Q16×C239C22semidirect
39. SD16×C239C22semidirect
40. M16×C239C22semidirect
90. C239#577semidirect
95. #137D239semidirect
113. C2×Q8C2×D239semidirect
119. Q16C2×D239semidirect
125. SD16C2×D239semidirect
129. M16C2×D239semidirect

Maximal subgroups

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

subgroupindex
#214400082
#214400062
#214399922
#214399882
#214399872
#214399832
#214399802
C2×#168385352
SD16×D2392
#214399682
#137×C2392
#577239