Finite Group Databasebeta

← C22×#22575481 #46825620

Subgroups — C22×#22575481 #46825620

Order 34432.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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
1C22×#16864402C211yesmaximal, characteristic, complemented
2C23×D1076C211yesmaximal, characteristic, complemented
3C2×#22575481C2124yesmaximal, direct product factor
4C22×#16864391C212yesmaximal, complemented
5C22×D2152C212yesmaximal, complemented
6C22×M16×C269C211yesmaximal, characteristic, complemented
7C24×D269—21no—
8C2×#16864402—26no—
9C2×#16864402C22112yescomplemented
10C22×Dic538C2211yescharacteristic, complemented
11C22×D1076—21no—
12C22×D1076—26no—
13C22×D1076C2212yes—
14C22×D1076C22112yescomplemented
15C22×D1076C2211yescharacteristic, complemented
16C22×C22⋊D269—21no—
17C22×C4×D269—21no—
18#22575481C22164yesdirect product factor
19C2×#16864391C22124yescomplemented
20C2×D2152C22124yescomplemented
21C23×C4×C269C2211yescharacteristic
22C2×M16×C269C22112yescomplemented
23C22×C8×C269C2212yescomplemented
24C23×D269—21no—
25C23×D269—22no—
26C23×D269—212no—
27C23×D269—48no—
28#16864402—224no—
29#16864402C23116yescomplemented
30C2×Dic538—23no—
31C2×Dic538C2316yescomplemented
32C2×D1076—26no—
33C2×D1076—212no—
34C2×D1076—224no—
35C2×D1076—23no—
36C2×D1076C23112yes—
37C2×D1076C23116yescomplemented
38C2×D1076C2316yescomplemented
39C2×C22⋊D269—212no—
40C22×Dic269—21no—
41C2×C4×D269—212no—
42#16864391C23132yescomplemented
43D2152C23132yescomplemented
44C24×C269D411yescharacteristic
45C22×C4×C269D411yescharacteristic
46C22×C4×C269D416yes—
47C22×C4×C269C2311yescharacteristic
48C22×C4×C269C2316yes—
49M16×C269C23116yescomplemented
50C2×C8×C269C23112yescomplemented
51C22×D269—26no—
52C22×D269—212no—
53C22×D269—216no—
54C22×D269—448no—
55C22×D269—44no—
56C22×D269—48no—
57Dic538—26no—
58Dic538C2414yescomplemented
59D1076—212no—
60D1076—216no—
61D1076—26no—
62D1076C2418yes—
63D1076C2414yescomplemented
64C22⋊D269—216no—
65C2×Dic269—26no—
66C4×D269—216no—
67C23×C269—24no—
68C23×C269C2×D416yes—
69C23×C269C2×D411yescharacteristic
70C2×C4×C269C2×D416yes—
71C2×C4×C269C2×D413yes—
72C2×C4×C269C2×D4112yes—
73C2×C4×C269C2413yes—
74C2×C4×C269C2414yes—
75C8×C269C2418yescomplemented
76C22×C269—212no—
77C22×C269AGL(1,ℤ/8)14yescomplemented
78C22×C269C22×D414yes—
79C22×C269C22×D413yes—
80C2×D269—24no—
81C2×D269—28no—
82C2×D269—432no—
83C2×D269—412no—
84C2×D269—424no—
85C4×C269C22×D414yes—
86C4×C269C22×D413yes—
87C4×C269C2511yescharacteristic
88Dic269—24no—
89C2×C269—24no—
90C2×C269C2×AGL(1,ℤ/8)16yescomplemented
91C2×C269C23×D411yescharacteristic
92D269—44no—
93D269—48no—
94C269C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
95C22×AGL(1,ℤ/8)—2691nomaximal, Sylow
96C22×#41—2691no—
97C23×D4—2691no—
98C2×AGL(1,ℤ/8)—26924no—
99C22×SD16—2692no—
100C22×D8—2692no—
101C22×M16D26911yescharacteristic, complemented
102C25—5381no—
103C2×#41—26912no—
104C2×#41—5386no—
105C22×Q8—2691no—
106C22×D4—2691no—
107C22×D4—2692no—
108C22×D4—26912no—
109C22×D4—5381no—
110C22×D4—5386no—
111C22×D4—5381no—
112C23×C4—5381no—
113C23×C4C2×D26911yescharacteristic
114AGL(1,ℤ/8)—26964no—
115C2×SD16—26924no—
116C2×D8—26924no—
117C2×M16C2×D269112yescomplemented
118C22×C8C2×D26912yescomplemented
119C24—53812no—
120C24—5381no—
121C24—5382no—
122C24—10768no—
123C24D107611yescharacteristic
124#41—26916no—
125#41—53824no—
126C2×Q8—2696no—
127C2×Q8—5383no—
128C2×D4—2696no—
129C2×D4—26912no—
130C2×D4—26916no—
131C2×D4—53812no—
132C2×D4—53812no—
133C2×D4—5386no—
134C2×D4—53824no—
135C2×D4—5383no—
136C22×C4—5381no—
137C22×C4—53812no—
138C22×C4D107616yes—
139C22×C4D107611yescharacteristic
140C22×C4C22×D26916yes—
141C22×C4C22×D26911yescharacteristic
142SD16—26932no—
143D8—26932no—
144M16C22×D269116yescomplemented
145C2×C8C22×D269112yescomplemented
146C23—24no—
147C23—5386no—
148C23—53816no—
149C23—53812no—
150C23—107648no—
151C23—10768no—
152C23—10764no—
153C23C2×D107616yes—
154C23C2×D107611yescharacteristic
155Q8—2694no—
156Q8—5386no—
157D4—2694no—
158D4—2698no—
159D4—53816no—
160D4—53816no—
161D4—53812no—
162D4—5386no—
163C2×C4—5386no—
164C2×C4—53816no—
165C2×C4C2×D107616yes—
166C2×C4C2×D107613yes—
167C2×C4C2×D1076112yes—
168C2×C4C23×D26913yes—
169C2×C4C23×D26914yes—
170C8C23×D26918yescomplemented
171C22—212no—
172C22—5384no—
173C22—5388no—
174C22—107632no—
175C22—107624no—
176C22—107612no—
177C22#2257548114yesdirect product factor
178C22C22×D107614yes—
179C22C22×D107613yes—
180C4—5384no—
181C4C22×D107614yes—
182C4C22×D107613yes—
183C4C24×D26911yescharacteristic
184C2—24no—
185C2—10768no—
186C2—10764no—
187C2C2×#2257548116yesdirect product factor
188C2C23×D107611yescharacteristic
subgroupG/H
centerC23C2×D1076
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
FittingC22×M16×C269C2
socleC23×C269C2×D4
hypercenterC22×M16D269
hypocenterC269C22×AGL(1,ℤ/8)
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#16864402C2semidirect
2. C23×D1076C2semidirect
3. C2×#22575481C2direct (N × H)
4. C22×#16864391C2semidirect
5. C22×D2152C2semidirect
6. C22×M16×C269C2semidirect
9. C2×#16864402C22semidirect
10. C22×Dic538C22semidirect
14. C22×D1076C22semidirect
15. C22×D1076C22semidirect
18. #22575481C22direct (N × H)
19. C2×#16864391C22semidirect
20. C2×D2152C22semidirect
22. C2×M16×C269C22semidirect
23. C22×C8×C269C22semidirect
29. #16864402C23semidirect
31. C2×Dic538C23semidirect
37. C2×D1076C23semidirect
38. C2×D1076C23semidirect
42. #16864391C23semidirect
43. D2152C23semidirect
49. M16×C269C23semidirect
50. C2×C8×C269C23semidirect
58. Dic538C24semidirect
63. D1076C24semidirect
75. C8×C269C24semidirect
77. C22×C269AGL(1,ℤ/8)semidirect
90. C2×C269C2×AGL(1,ℤ/8)semidirect
94. C269C22×AGL(1,ℤ/8)semidirect
101. C22×M16D269semidirect
117. C2×M16C2×D269semidirect
118. C22×C8C2×D269semidirect
144. M16C22×D269semidirect
145. C2×C8C22×D269semidirect
170. C8C23×D269semidirect
177. C22#22575481direct (N × H)
187. C2C2×#22575481direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168644022
C23×D10762
C2×#225754812
C22×#168643912
C22×D21522
C22×M16×C2692
C22×AGL(1,ℤ/8)269