Finite Group Databasebeta

← #37527249

Subgroups — #37527249

Order 22976.

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#26040063C211yesmaximal, characteristic, complemented
2#26040060C211yesmaximal, characteristic, complemented
3#26040046C211yesmaximal, characteristic, complemented
4#26040044C212yesmaximal, complemented
5#26040042C212yesmaximal, complemented
6#26040041C211yesmaximal, characteristic, complemented
7#26040037C211yesmaximal, characteristic, complemented
8#26040036C212yesmaximal, complemented
9SD16×D359C212yesmaximal, complemented
10#26040021C211yesmaximal, characteristic, complemented
11C2×SD16×C359C211yesmaximal, characteristic, complemented
12#19153342—21no—
13#19153342C2212yescomplemented
14#19153341—22no—
15#19153341C2211yescharacteristic
16#19153341C2211yescharacteristic
17#19153341C2211yescharacteristic, complemented
18#19153340—21no—
19#19153340C2212yes—
20Q8×D359—21no—
21Q8×D359C2212yescomplemented
22C2×C22⋊D359—22no—
23D4×D359—21no—
24D4×D359C2212yes—
25Dic1436C2211yescharacteristic, complemented
26#19153331C2212yescomplemented
27#19153330C2211yescharacteristic, complemented
28#19153330C2211yescharacteristic, complemented
29#19153329C2212yescomplemented
30#19153328C2212yescomplemented
31D2872C2211yescharacteristic, complemented
32#19153326C2212yescomplemented
33#19153325C2211yescharacteristic, complemented
34#19153324C2212yescomplemented
35C8×D359C2212yescomplemented
36C2×Q8×C359C2211yescharacteristic, complemented
37C2×D4×C359C2211yescharacteristic
38SD16×C359C2212yescomplemented
39SD16×C359C2212yescomplemented
40C2×C8×C359C2211yescharacteristic, complemented
41C22×D359—22no—
42Dic718—22no—
43Dic718D411yescharacteristic
44Dic718C2311yescharacteristic
45Dic718C2311yescharacteristic
46D1436—22no—
47D1436D411yescharacteristic
48D1436C2311yescharacteristic
49D1436C2311yescharacteristic
50C22⋊D359—21no—
51C22⋊D359—21no—
52C22⋊D359—22no—
53C22⋊D359—22no—
54C22⋊D359D412yes—
55C2×Dic359—22no—
56C4×D359—22no—
57C4×D359—21no—
58C4×D359—21no—
59C4×D359—22no—
60C4×D359C2312yes—
61C359⋊C8C2312yes—
62C23×C359—21no—
63Q8×C359—21no—
64Q8×C359C2312yes—
65D4×C359—21no—
66D4×C359C2312yes—
67C2×C4×C359—21no—
68C2×C4×C359C2311yescharacteristic
69C8×C359C2311yescharacteristic
70C8×C359C2311yescharacteristic
71C22×C359—22no—
72C22×C359—81no—
73C22×C359C2×D411yescharacteristic
74C2×D359—21no—
75C2×D359—21no—
76C2×D359—82no—
77C2×D359C2×D412yes—
78C4×C359—22no—
79C4×C359C2×D411yescharacteristic
80C4×C359C2411yescharacteristic
81Dic359—21no—
82Dic359—21no—
83Dic359C2×D412yes—
84C2×C359—21no—
85C2×C359—42no—
86C2×C359C22×D411yescharacteristic
87D359—22no—
88D359—41no—
89D359—41no—
90C359#57711yescharacteristic, complemented, Sylow
91#577—3591nomaximal, Sylow
922−1+4—3591no—
93H(5,𝔽2)—3591no—
94#137—3591no—
95#137—3592no—
96AGL(1,ℤ/8)—3591no—
97AGL(1,ℤ/8)—3592no—
98#135—3592no—
99#135—3591no—
100C2×SD16—3592no—
101C2×SD16D35911yescharacteristic, complemented
102#131—3591no—
103#41—3591no—
104#41—3592no—
105#41—3592no—
106#41—3591no—
107#41—3591no—
108#41—7182no—
109#41—7181no—
110#41—7181no—
111C2×Q8—3592no—
112C2×Q8—7181no—
113C2×Q8C2×D35911yescharacteristic, complemented
114C2×D4—3592no—
115C2×D4—7181no—
116C2×D4—7182no—
117C2×D4C2×D35911yescharacteristic
118Q16—3592no—
119Q16—3591no—
120SD16—3592no—
121SD16—3592no—
122SD16—3591no—
123SD16—3591no—
124SD16C2×D35912yescomplemented
125SD16C2×D35912yescomplemented
126D8—3592no—
127D8—3591no—
128M16—3591no—
129M16—3592no—
130C2×C8—3592no—
131C2×C8C2×D35911yescharacteristic, complemented
132C23—21no—
133C23—7182no—
134Q8—21no—
135Q8—3591no—
136Q8—3591no—
137Q8—3591no—
138Q8—7182no—
139Q8C22×D35912yes—
140D4—21no—
141D4—3592no—
142D4—3591no—
143D4—3591no—
144D4—3591no—
145D4—7182no—
146D4—7182no—
147D4—7182no—
148D4—7181no—
149D4—7181no—
150D4C22×D35912yes—
151C2×C4—21no—
152C2×C4—3592no—
153C2×C4—7182no—
154C2×C4—7182no—
155C2×C4—7182no—
156C2×C4—7181no—
157C2×C4—7181no—
158C2×C4C22×D35911yescharacteristic
159C8—3592no—
160C8C22×D35911yescharacteristic
161C8C22×D35911yescharacteristic
162C22—22no—
163C22—81no—
164C22—28722no—
165C22—3592no—
166C22—7181no—
167C22—7181no—
168C22D4×D35911yescharacteristic
169C4—22no—
170C4—3592no—
171C4—7181no—
172C4—7181no—
173C4D4×D35911yescharacteristic
174C4C23×D35911yescharacteristic
175C2—21no—
176C2—42no—
177C2—14361no—
178C2—14361no—
179C2—7182no—
180C2C2×D4×D35911yescharacteristic
subgroupG/H
centerC2C2×D4×D359
commutator, cosocleC4×C359C24
FrattiniC4C23×D359
FittingC2×SD16×C359C2
socleC2×C359C22×D4
hypercenterC2×SD16D359
hypocenterC359#577
LCSG ▷ C4×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #26040063C2semidirect
2. #26040060C2semidirect
3. #26040046C2semidirect
4. #26040044C2semidirect
5. #26040042C2semidirect
6. #26040041C2semidirect
7. #26040037C2semidirect
8. #26040036C2semidirect
9. SD16×D359C2semidirect
10. #26040021C2semidirect
11. C2×SD16×C359C2semidirect
13. #19153342C22semidirect
17. #19153341C22semidirect
21. Q8×D359C22semidirect
25. Dic1436C22semidirect
26. #19153331C22semidirect
27. #19153330C22semidirect
28. #19153330C22semidirect
29. #19153329C22semidirect
30. #19153328C22semidirect
31. D2872C22semidirect
32. #19153326C22semidirect
33. #19153325C22semidirect
34. #19153324C22semidirect
35. C8×D359C22semidirect
36. C2×Q8×C359C22semidirect
38. SD16×C359C22semidirect
39. SD16×C359C22semidirect
40. C2×C8×C359C22semidirect
90. C359#577semidirect
101. C2×SD16D359semidirect
113. C2×Q8C2×D359semidirect
124. SD16C2×D359semidirect
125. SD16C2×D359semidirect
131. C2×C8C2×D359semidirect

Maximal subgroups

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

subgroupindex
#260400632
#260400602
#260400462
#260400442
#260400422
#260400412
#260400372
#260400362
SD16×D3592
#260400212
C2×SD16×C3592
#577359