Finite Group Databasebeta

← C22×#22575480 #46825619

Subgroups — C22×#22575480 #46825619

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×#16864403C211yesmaximal, characteristic, complemented
2C22×D4×D269C211yesmaximal, characteristic, complemented
3C2×#22575480C2124yesmaximal, direct product factor
4C22×#16864389C211yesmaximal, characteristic, complemented
5C22×D2152C211yesmaximal, characteristic, complemented
6C22×#16864387C211yesmaximal, characteristic, complemented
7C22×#16864385C211yesmaximal, characteristic, complemented
8C22×SD16×C269C211yesmaximal, characteristic, complemented
9C24×D269—21no—
10C2×#16864403—26no—
11C2×#16864403C22112yescomplemented
12C22×D1076—21no—
13C22×D1076C2211yescharacteristic
14C22×D1076C2211yescharacteristic, complemented
15C22×C22⋊D269—21no—
16C2×D4×D269—26no—
17C2×D4×D269C22112yescomplemented
18C22×C4×D269—21no—
19C22×C4×D269C2211yescharacteristic
20#22575480C22164yesdirect product factor
21C2×#16864389C22112yescomplemented
22C2×D2152C22112yescomplemented
23C2×#16864387C22112yescomplemented
24C2×#16864385C22112yescomplemented
25C22×#14524130C2211yescharacteristic, complemented
26C22×Q8×C269C2211yescharacteristic, complemented
27C22×D4×C269C2211yescharacteristic
28C2×SD16×C269C22112yescomplemented
29C22×C8×C269C2211yescharacteristic, complemented
30C23×D269—212no—
31C23×D269—21no—
32C23×D269—21no—
33C23×D269—48no—
34C23×D269D411yescharacteristic
35#16864403—224no—
36#16864403C23116yescomplemented
37C2×D1076—212no—
38C2×D1076—23no—
39C2×D1076—23no—
40C2×D1076C2316yes—
41C2×D1076C2316yescomplemented
42C2×C22⋊D269—212no—
43D4×D269—224no—
44D4×D269C23116yescomplemented
45C22×Dic269D411yescharacteristic
46C2×C4×D269—212no—
47C2×C4×D269D416yes—
48C2×C4×D269C2316yes—
49#16864389C23116yescomplemented
50D2152C23116yescomplemented
51#16864387C23116yescomplemented
52#16864385C23116yescomplemented
53C2×#14524130C2316yescomplemented
54C24×C269—21no—
55C2×Q8×C269—23no—
56C2×Q8×C269C2316yescomplemented
57C2×D4×C269—23no—
58C2×D4×C269C2316yes—
59C22×C4×C269—21no—
60C22×C4×C269C2311yescharacteristic
61SD16×C269C23116yescomplemented
62C2×C8×C269C2316yescomplemented
63C22×D269—24no—
64C22×D269—26no—
65C22×D269—26no—
66C22×D269—216no—
67C22×D269—448no—
68C22×D269—44no—
69C22×D269—44no—
70C22×D269C2×D416yes—
71D1076—216no—
72D1076—26no—
73D1076—26no—
74D1076C2414yes—
75D1076C2414yescomplemented
76C22⋊D269—216no—
77C2×Dic269C2×D416yes—
78C4×D269—216no—
79C4×D269C2×D4112yes—
80C4×D269C2414yes—
81#14524130C2414yescomplemented
82C23×C269—26no—
83C23×C269—44no—
84C23×C269C2×D411yescharacteristic
85Q8×C269—26no—
86Q8×C269C2414yescomplemented
87D4×C269—26no—
88D4×C269C2414yes—
89C2×C4×C269—26no—
90C2×C4×C269C2×D413yes—
91C2×C4×C269C2413yes—
92C8×C269C2414yescomplemented
93C22×C269—24no—
94C22×C269—412no—
95C22×C269AGL(1,ℤ/8)14yescomplemented
96C22×C269C22×D413yes—
97C2×D269—212no—
98C2×D269—24no—
99C2×D269—24no—
100C2×D269—432no—
101C2×D269—412no—
102C2×D269—412no—
103C2×D269C22×D414yes—
104C4×C269—24no—
105C4×C269C22×D413yes—
106C4×C269C2511yescharacteristic
107Dic269C22×D414yes—
108C2×C269—44no—
109C2×C269C2×AGL(1,ℤ/8)16yescomplemented
110C2×C269C23×D411yescharacteristic
111D269—24no—
112D269—44no—
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×SD16D26911yescharacteristic, complemented
121C22×D8—2691no—
122C22×D8—2691no—
123C22×M16—2691no—
124C25—5381no—
125C2×#41—26912no—
126C2×#41—5386no—
127C22×Q8C2×D26911yescharacteristic, complemented
128C22×D4—26912no—
129C22×D4—2691no—
130C22×D4—2691no—
131C22×D4—5381no—
132C22×D4—5386no—
133C22×D4—5381no—
134C22×D4C2×D26911yescharacteristic
135C23×C4—2691no—
136C23×C4—5381no—
137AGL(1,ℤ/8)—26964no—
138C2×SD16—26912no—
139C2×SD16C2×D269112yescomplemented
140C2×D8—26912no—
141C2×D8—26912no—
142C2×M16—26912no—
143C22×C8—2691no—
144C22×C8C2×D26911yescharacteristic, complemented
145C24—21no—
146C24—2691no—
147C24—53812no—
148C24—5381no—
149C24—5381no—
150C24—10768no—
151#41—26916no—
152#41—53824no—
153C2×Q8—23no—
154C2×Q8C22×D26916yescomplemented
155C2×D4—23no—
156C2×D4—26916no—
157C2×D4—2696no—
158C2×D4—2696no—
159C2×D4—53812no—
160C2×D4—53824no—
161C2×D4—53812no—
162C2×D4—5383no—
163C2×D4—5383no—
164C2×D4C22×D26916yes—
165C22×C4—21no—
166C22×C4—2691no—
167C22×C4—2696no—
168C22×C4—2696no—
169C22×C4—53812no—
170C22×C4C22×D26911yescharacteristic
171SD16—26916no—
172SD16C22×D269116yescomplemented
173D8—26916no—
174D8—26916no—
175M16—26916no—
176C2×C8—2696no—
177C2×C8C22×D26916yescomplemented
178C23—26no—
179C23—44no—
180C23—2696no—
181C23—53816no—
182C23—5384no—
183C23—5386no—
184C23—5386no—
185C23—107648no—
186C23—10764no—
187C23—10764no—
188C23D4×D26911yescharacteristic
189Q8—26no—
190Q8C23×D26914yescomplemented
191D4—26no—
192D4—2694no—
193D4—2694no—
194D4—53816no—
195D4—53816no—
196D4—5386no—
197D4—5386no—
198D4C23×D26914yes—
199C2×C4—26no—
200C2×C4—2696no—
201C2×C4—26912no—
202C2×C4—2694no—
203C2×C4—53816no—
204C2×C4D4×D26913yes—
205C2×C4C23×D26913yes—
206C8—2694no—
207C8C23×D26914yescomplemented
208C22—24no—
209C22—412no—
210C22—2694no—
211C22—53812no—
212C22—5384no—
213C22—5384no—
214C22—107632no—
215C22—107612no—
216C22—107612no—
217C22#2257548014yesdirect product factor
218C22C2×D4×D26913yes—
219C4—24no—
220C4—2694no—
221C4C2×D4×D26913yes—
222C4C24×D26911yescharacteristic
223C2—44no—
224C2—5384no—
225C2—10764no—
226C2—10764no—
227C2C2×#2257548016yesdirect product factor
228C2C22×D4×D26911yescharacteristic
subgroupG/H
centerC23D4×D269
commutator, cosocleC4×C269C25
FrattiniC4C24×D269
FittingC22×SD16×C269C2
socleC23×C269C2×D4
hypercenterC22×SD16D269
hypocenterC269C22×AGL(1,ℤ/8)
LCSG ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#16864403C2semidirect
2. C22×D4×D269C2semidirect
3. C2×#22575480C2direct (N × H)
4. C22×#16864389C2semidirect
5. C22×D2152C2semidirect
6. C22×#16864387C2semidirect
7. C22×#16864385C2semidirect
8. C22×SD16×C269C2semidirect
11. C2×#16864403C22semidirect
14. C22×D1076C22semidirect
17. C2×D4×D269C22semidirect
20. #22575480C22direct (N × H)
21. C2×#16864389C22semidirect
22. C2×D2152C22semidirect
23. C2×#16864387C22semidirect
24. C2×#16864385C22semidirect
25. C22×#14524130C22semidirect
26. C22×Q8×C269C22semidirect
28. C2×SD16×C269C22semidirect
29. C22×C8×C269C22semidirect
36. #16864403C23semidirect
41. C2×D1076C23semidirect
44. D4×D269C23semidirect
49. #16864389C23semidirect
50. D2152C23semidirect
51. #16864387C23semidirect
52. #16864385C23semidirect
53. C2×#14524130C23semidirect
56. C2×Q8×C269C23semidirect
61. SD16×C269C23semidirect
62. C2×C8×C269C23semidirect
75. D1076C24semidirect
81. #14524130C24semidirect
86. Q8×C269C24semidirect
92. C8×C269C24semidirect
95. C22×C269AGL(1,ℤ/8)semidirect
109. C2×C269C2×AGL(1,ℤ/8)semidirect
114. C269C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D269semidirect
127. C22×Q8C2×D269semidirect
139. C2×SD16C2×D269semidirect
144. C22×C8C2×D269semidirect
154. C2×Q8C22×D269semidirect
172. SD16C22×D269semidirect
177. C2×C8C22×D269semidirect
190. Q8C23×D269semidirect
207. C8C23×D269semidirect
217. C22#22575480direct (N × H)
227. C2C2×#22575480direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#168644032
C22×D4×D2692
C2×#225754802
C22×#168643892
C22×D21522
C22×#168643872
C22×#168643852
C22×SD16×C2692
C22×AGL(1,ℤ/8)269