Finite Group Databasebeta

← C22×#22575477 #46825616

Subgroups — C22×#22575477 #46825616

Order 34432.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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×#16864401C211yesmaximal, characteristic, complemented
2C22×D4×D269C211yesmaximal, characteristic, complemented
3C2×#22575477C2124yesmaximal, direct product factor
4C22×#16864391C211yesmaximal, characteristic, complemented
5C22×#16864390C211yesmaximal, characteristic, complemented
6C22×#16864387C211yesmaximal, characteristic, complemented
7C22×#16864385C211yesmaximal, characteristic, complemented
8C22×D8×C269C211yesmaximal, characteristic, complemented
9C24×D269—21no—
10C2×#16864401—26no—
11C2×#16864401C22112yescomplemented
12C22×Dic538C2211yescharacteristic, complemented
13C22×D1076C2211yescharacteristic
14C22×C22⋊D269—21no—
15C22×C22⋊D269—21no—
16C2×D4×D269—26no—
17C2×D4×D269C22112yescomplemented
18C23×Dic269—21no—
19C22×C4×D269C2211yescharacteristic
20#22575477C22164yesdirect product factor
21C2×#16864391C22112yescomplemented
22C2×#16864390C22112yescomplemented
23C2×#16864387C22112yescomplemented
24C2×#16864385C22112yescomplemented
25C22×#14524130C2211yescharacteristic, complemented
26C22×D4×C269C2211yescharacteristic
27C22×D4×C269C2211yescharacteristic, complemented
28C2×D8×C269C22112yescomplemented
29C22×C8×C269C2211yescharacteristic, complemented
30C23×D269—212no—
31C23×D269—21no—
32C23×D269—48no—
33C23×D269D411yescharacteristic
34#16864401—224no—
35#16864401C23116yescomplemented
36C2×Dic538—23no—
37C2×Dic538C2316yescomplemented
38C2×D1076—23no—
39C2×D1076C2316yes—
40C2×C22⋊D269—212no—
41C2×C22⋊D269—212no—
42D4×D269—224no—
43D4×D269C23116yescomplemented
44C22×Dic269—212no—
45C22×Dic269—21no—
46C22×Dic269D411yescharacteristic
47C2×C4×D269D416yes—
48C2×C4×D269C2316yes—
49#16864391C23116yescomplemented
50#16864390C23116yescomplemented
51#16864387C23116yescomplemented
52#16864385C23116yescomplemented
53C2×#14524130C2316yescomplemented
54C24×C269—21no—
55C24×C269—21no—
56C2×D4×C269—23no—
57C2×D4×C269—23no—
58C2×D4×C269C2316yes—
59C2×D4×C269C2316yescomplemented
60C22×C4×C269C2311yescharacteristic
61D8×C269C23116yescomplemented
62C2×C8×C269C2316yescomplemented
63C22×D269—24no—
64C22×D269—26no—
65C22×D269—216no—
66C22×D269—448no—
67C22×D269—44no—
68C22×D269C2×D416yes—
69Dic538—26no—
70Dic538C2414yescomplemented
71D1076—26no—
72D1076C2414yes—
73C22⋊D269—216no—
74C22⋊D269—216no—
75C2×Dic269—26no—
76C2×Dic269—216no—
77C2×Dic269C2×D416yes—
78C4×D269C2×D4112yes—
79C4×D269C2414yes—
80#14524130C2414yescomplemented
81C23×C269—26no—
82C23×C269—26no—
83C23×C269—44no—
84C23×C269—44no—
85C23×C269C2×D411yescharacteristic
86D4×C269—26no—
87D4×C269—26no—
88D4×C269C2414yes—
89D4×C269C2414yescomplemented
90C2×C4×C269C2×D413yes—
91C2×C4×C269C2413yes—
92C8×C269C2414yescomplemented
93C22×C269—24no—
94C22×C269—24no—
95C22×C269—412no—
96C22×C269—412no—
97C22×C269AGL(1,ℤ/8)14yescomplemented
98C22×C269C22×D413yes—
99C2×D269—212no—
100C2×D269—24no—
101C2×D269—432no—
102C2×D269—412no—
103C2×D269C22×D414yes—
104C4×C269C22×D413yes—
105C4×C269C2511yescharacteristic
106Dic269—24no—
107Dic269C22×D414yes—
108C2×C269—44no—
109C2×C269—44no—
110C2×C269C2×AGL(1,ℤ/8)16yescomplemented
111C2×C269C23×D411yescharacteristic
112D269—24no—
113D269—44no—
114C269C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
115C22×AGL(1,ℤ/8)—2691nomaximal, Sylow
116C22×#41—2691no—
117C23×D4—2691no—
118C2×AGL(1,ℤ/8)—26924no—
119C22×SD16—2691no—
120C22×SD16—2691no—
121C22×D8—2691no—
122C22×D8D26911yescharacteristic, complemented
123C22×M16—2691no—
124C25—5381no—
125C2×#41—26912no—
126C2×#41—5386no—
127C22×Q8—2691no—
128C22×D4—26912no—
129C22×D4—2691no—
130C22×D4—5386no—
131C22×D4—5381no—
132C22×D4—5381no—
133C22×D4C2×D26911yescharacteristic
134C22×D4C2×D26911yescharacteristic, complemented
135C23×C4—2691no—
136C23×C4—5381no—
137AGL(1,ℤ/8)—26964no—
138C2×SD16—26912no—
139C2×SD16—26912no—
140C2×D8—26912no—
141C2×D8C2×D269112yescomplemented
142C2×M16—26912no—
143C22×C8—2691no—
144C22×C8C2×D26911yescharacteristic, complemented
145C24—21no—
146C24—21no—
147C24—2691no—
148C24—53812no—
149C24—5381no—
150C24—10768no—
151#41—26916no—
152#41—53824no—
153C2×Q8—2696no—
154C2×Q8—5383no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—26916no—
158C2×D4—2696no—
159C2×D4—53824no—
160C2×D4—53812no—
161C2×D4—53812no—
162C2×D4—5383no—
163C2×D4C22×D26916yes—
164C2×D4C22×D26916yescomplemented
165C22×C4—2691no—
166C22×C4—2696no—
167C22×C4—2696no—
168C22×C4—53812no—
169C22×C4—5381no—
170C22×C4C22×D26911yescharacteristic
171SD16—26916no—
172SD16—26916no—
173D8—26916no—
174D8C22×D269116yescomplemented
175M16—26916no—
176C2×C8—2696no—
177C2×C8C22×D26916yescomplemented
178C23—26no—
179C23—26no—
180C23—44no—
181C23—44no—
182C23—2696no—
183C23—53816no—
184C23—5384no—
185C23—5386no—
186C23—107648no—
187C23—10764no—
188C23D4×D26911yescharacteristic
189Q8—2694no—
190Q8—5386no—
191D4—26no—
192D4—26no—
193D4—2694no—
194D4—53816no—
195D4—53816no—
196D4—5386no—
197D4C23×D26914yes—
198D4C23×D26914yescomplemented
199C2×C4—2696no—
200C2×C4—26912no—
201C2×C4—2694no—
202C2×C4—53816no—
203C2×C4—5386no—
204C2×C4D4×D26913yes—
205C2×C4C23×D26913yes—
206C8—2694no—
207C8C23×D26914yescomplemented
208C22—24no—
209C22—24no—
210C22—412no—
211C22—412no—
212C22—2694no—
213C22—53812no—
214C22—5384no—
215C22—107632no—
216C22—107612no—
217C22#2257547714yesdirect product factor
218C22C2×D4×D26913yes—
219C4—2694no—
220C4—5384no—
221C4C2×D4×D26913yes—
222C4C24×D26911yescharacteristic
223C2—44no—
224C2—44no—
225C2—5384no—
226C2—10764no—
227C2C2×#2257547716yesdirect product factor
228C2C22×D4×D26911yescharacteristic
subgroupG/H
centerC23D4×D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
FittingC22×D8×C269C2
socleC23×C269C2×D4
hypercenterC22×D8D269
hypocenterC269C22×AGL(1,ℤ/8)
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#16864401C2semidirect
2. C22×D4×D269C2semidirect
3. C2×#22575477C2direct (N × H)
4. C22×#16864391C2semidirect
5. C22×#16864390C2semidirect
6. C22×#16864387C2semidirect
7. C22×#16864385C2semidirect
8. C22×D8×C269C2semidirect
11. C2×#16864401C22semidirect
12. C22×Dic538C22semidirect
17. C2×D4×D269C22semidirect
20. #22575477C22direct (N × H)
21. C2×#16864391C22semidirect
22. C2×#16864390C22semidirect
23. C2×#16864387C22semidirect
24. C2×#16864385C22semidirect
25. C22×#14524130C22semidirect
27. C22×D4×C269C22semidirect
28. C2×D8×C269C22semidirect
29. C22×C8×C269C22semidirect
35. #16864401C23semidirect
37. C2×Dic538C23semidirect
43. D4×D269C23semidirect
49. #16864391C23semidirect
50. #16864390C23semidirect
51. #16864387C23semidirect
52. #16864385C23semidirect
53. C2×#14524130C23semidirect
59. C2×D4×C269C23semidirect
61. D8×C269C23semidirect
62. C2×C8×C269C23semidirect
70. Dic538C24semidirect
80. #14524130C24semidirect
89. D4×C269C24semidirect
92. C8×C269C24semidirect
97. C22×C269AGL(1,ℤ/8)semidirect
110. C2×C269C2×AGL(1,ℤ/8)semidirect
114. C269C22×AGL(1,ℤ/8)semidirect
122. C22×D8D269semidirect
134. C22×D4C2×D269semidirect
141. C2×D8C2×D269semidirect
144. C22×C8C2×D269semidirect
164. C2×D4C22×D269semidirect
174. D8C22×D269semidirect
177. C2×C8C22×D269semidirect
198. D4C23×D269semidirect
207. C8C23×D269semidirect
217. C22#22575477direct (N × H)
227. C2C2×#22575477direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168644012
C22×D4×D2692
C2×#225754772
C22×#168643912
C22×#168643902
C22×#168643872
C22×#168643852
C22×D8×C2692
C22×AGL(1,ℤ/8)269