Finite Group Databasebeta

← #37527252

Subgroups — #37527252

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#26040061C211yesmaximal, characteristic, complemented
3#26040047C212yesmaximal, complemented
4#26040043C211yesmaximal, characteristic, complemented
5#26040042C212yesmaximal, complemented
6#26040038C212yesmaximal, complemented
7#26040035C211yesmaximal, characteristic, complemented
8C2×#19153328C211yesmaximal, characteristic, complemented
9SD16×D359C212yesmaximal, complemented
10#26040023C211yesmaximal, characteristic, complemented
11#137×C359C211yesmaximal, characteristic, complemented
12#19153342—21no—
13#19153342C2212yes—
14#19153342C2212yescomplemented
15#19153341—21no—
16#19153341—22no—
17#19153341C2211yescharacteristic
18#19153341C2211yescharacteristic, complemented
19Q8×D359—21no—
20Q8×D359C2212yescomplemented
21C2×D1436—21no—
22C2×D1436C2211yescharacteristic
23D4×D359—22no—
24D4×D359C2212yes—
25#19153331C2211yescharacteristic, complemented
26#19153330C2212yescomplemented
27#19153329C2211yescharacteristic, complemented
28#19153328C2212yescomplemented
29#19153328C2212yescomplemented
30#19153328C2211yescharacteristic, complemented
31D2872C2212yescomplemented
32#19153326C2211yescharacteristic, complemented
33#19153324C2212yescomplemented
34C2×C359⋊C8C2211yescharacteristic, complemented
35C8×D359C2212yescomplemented
36#41×C359C2211yescharacteristic
37C2×Q8×C359C2211yescharacteristic, complemented
38Q16×C359C2212yescomplemented
39SD16×C359C2212yescomplemented
40M16×C359C2211yescharacteristic, complemented
41C22×D359—21no—
42C22×D359—22no—
43Dic718—22no—
44Dic718D411yescharacteristic
45Dic718C2311yescharacteristic
46D1436—22no—
47D1436—22no—
48D1436—21no—
49D1436D411yescharacteristic
50D1436C2311yescharacteristic
51D1436C2312yes—
52C22⋊D359—21no—
53C22⋊D359—22no—
54C22⋊D359D412yes—
55C4×D359—22no—
56C4×D359—21no—
57C4×D359—22no—
58C4×D359—22no—
59C4×D359C2312yes—
60C359⋊C8C2311yescharacteristic
61C359⋊C8C2311yescharacteristic
62Q8×C359—21no—
63Q8×C359C2312yes—
64Q8×C359C2311yescharacteristic
65D4×C359—21no—
66D4×C359C2311yescharacteristic
67C2×C4×C359—21no—
68C2×C4×C359—21no—
69C2×C4×C359C2311yescharacteristic
70C8×C359C2312yes—
71C22×C359—21no—
72C22×C359C2×D411yescharacteristic
73C2×D359—21no—
74C2×D359—22no—
75C2×D359—82no—
76C2×D359—81no—
77C2×D359C2×D412yes—
78C4×C359—21no—
79C4×C359—22no—
80C4×C359C2×D411yescharacteristic
81C4×C359C2411yescharacteristic
82Dic359—21no—
83Dic359C2×D412yes—
84C2×C359—21no—
85C2×C359—41no—
86C2×C359C22×D411yescharacteristic
87D359—22no—
88D359—41no—
89D359—42no—
90C359#57711yescharacteristic, complemented, Sylow
91#577—3591nomaximal, Sylow
922−1+4—3591no—
93H(5,𝔽2)—3591no—
94#137—3592no—
95#137D35911yescharacteristic, complemented
96AGL(1,ℤ/8)—3592no—
97AGL(1,ℤ/8)—3591no—
98#135—3591no—
99#135—3592no—
100C2×SD16—3591no—
101C2×SD16—3592no—
102#131—3591no—
103#41—3591no—
104#41—3592no—
105#41—3592no—
106#41—3591no—
107#41—7182no—
108#41—7181no—
109#41—7181no—
110#41C2×D35911yescharacteristic
111C2×Q8—3592no—
112C2×Q8—7181no—
113C2×Q8C2×D35911yescharacteristic, complemented
114C2×D4—3592no—
115C2×D4—3591no—
116C2×D4—7181no—
117C2×D4—7182no—
118Q16—3591no—
119Q16C2×D35912yescomplemented
120SD16—3592no—
121SD16—3592no—
122SD16—3591no—
123SD16—3591no—
124SD16—3592no—
125SD16C2×D35912yescomplemented
126D8—3591no—
127D8—3592no—
128M16—3592no—
129M16C2×D35911yescharacteristic, complemented
130C2×C8—3591no—
131C2×C8—3592no—
132C23—7182no—
133C23—7181no—
134Q8—21no—
135Q8—3591no—
136Q8—3591no—
137Q8—7182no—
138Q8C22×D35912yes—
139Q8C22×D35911yescharacteristic
140D4—21no—
141D4—3592no—
142D4—3591no—
143D4—3591no—
144D4—3592no—
145D4—7182no—
146D4—7182no—
147D4—7182no—
148D4—7181no—
149D4—7181no—
150D4C22×D35911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—3592no—
154C2×C4—7182no—
155C2×C4—7182no—
156C2×C4—7182no—
157C2×C4—7181no—
158C2×C4C22×D35911yescharacteristic
159C8—3591no—
160C8—3591no—
161C8C22×D35912yes—
162C22—21no—
163C22—28721no—
164C22—28722no—
165C22—3592no—
166C22—7181no—
167C22—7182no—
168C22D4×D35911yescharacteristic
169C4—21no—
170C4—22no—
171C4—3592no—
172C4—7181no—
173C4D4×D35911yescharacteristic
174C4C23×D35911yescharacteristic
175C2—21no—
176C2—41no—
177C2—14362no—
178C2—14361no—
179C2—7182no—
180C2C2×D4×D35911yescharacteristic
subgroupG/H
centerC2C2×D4×D359
commutator, cosocleC4×C359C24
FrattiniC4C23×D359
Fitting#137×C359C2
socleC2×C359C22×D4
hypercenter#137D359
hypocenterC359#577
LCSG ▷ C4×C359 ▷ C2×C359 ▷ C359
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #26040063C2semidirect
2. #26040061C2semidirect
3. #26040047C2semidirect
4. #26040043C2semidirect
5. #26040042C2semidirect
6. #26040038C2semidirect
7. #26040035C2semidirect
8. C2×#19153328C2semidirect
9. SD16×D359C2semidirect
10. #26040023C2semidirect
11. #137×C359C2semidirect
14. #19153342C22semidirect
18. #19153341C22semidirect
20. Q8×D359C22semidirect
25. #19153331C22semidirect
26. #19153330C22semidirect
27. #19153329C22semidirect
28. #19153328C22semidirect
29. #19153328C22semidirect
30. #19153328C22semidirect
31. D2872C22semidirect
32. #19153326C22semidirect
33. #19153324C22semidirect
34. C2×C359⋊C8C22semidirect
35. C8×D359C22semidirect
37. C2×Q8×C359C22semidirect
38. Q16×C359C22semidirect
39. SD16×C359C22semidirect
40. M16×C359C22semidirect
90. C359#577semidirect
95. #137D359semidirect
113. C2×Q8C2×D359semidirect
119. Q16C2×D359semidirect
125. SD16C2×D359semidirect
129. M16C2×D359semidirect

Maximal subgroups

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

subgroupindex
#260400632
#260400612
#260400472
#260400432
#260400422
#260400382
#260400352
C2×#191533282
SD16×D3592
#260400232
#137×C3592
#577359