Finite Group Databasebeta

← C22×#15668036 #26034983

Subgroups — C22×#15668036 #26034983

Order 11392.

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×#13083573C211yesmaximal, characteristic, complemented
2C22×D4×D89C211yesmaximal, characteristic, complemented
3C2×#15668036C2124yesmaximal, direct product factor
4C22×#13083563C211yesmaximal, characteristic, complemented
5C22×#13083562C211yesmaximal, characteristic, complemented
6C22×#13083559C211yesmaximal, characteristic, complemented
7C22×#13083557C211yesmaximal, characteristic, complemented
8C22×D8×C89C211yesmaximal, characteristic, complemented
9C24×D89—21no—
10C2×#13083573—26no—
11C2×#13083573C22112yescomplemented
12C22×Dic178C2211yescharacteristic, complemented
13C22×D356C2211yescharacteristic
14C22×C22⋊D89—21no—
15C22×C22⋊D89—21no—
16C2×D4×D89—26no—
17C2×D4×D89C22112yescomplemented
18C23×Dic89—21no—
19C22×C4×D89C2211yescharacteristic
20#15668036C22164yesdirect product factor
21C2×#13083563C22112yescomplemented
22C2×#13083562C22112yescomplemented
23C2×#13083559C22112yescomplemented
24C2×#13083557C22112yescomplemented
25C22×#10623494C2211yescharacteristic, complemented
26C22×D4×C89C2211yescharacteristic
27C22×D4×C89C2211yescharacteristic, complemented
28C2×D8×C89C22112yescomplemented
29C22×C8×C89C2211yescharacteristic, complemented
30C23×D89—21no—
31C23×D89—212no—
32C23×D89—48no—
33C23×D89D411yescharacteristic
34#13083573—224no—
35#13083573C23116yescomplemented
36C2×Dic178—23no—
37C2×Dic178C2316yescomplemented
38C2×D356—23no—
39C2×D356C2316yes—
40C2×C22⋊D89—212no—
41C2×C22⋊D89—212no—
42D4×D89—224no—
43D4×D89C23116yescomplemented
44C22×Dic89—21no—
45C22×Dic89—212no—
46C22×Dic89D411yescharacteristic
47C2×C4×D89D416yes—
48C2×C4×D89C2316yes—
49#13083563C23116yescomplemented
50#13083562C23116yescomplemented
51#13083559C23116yescomplemented
52#13083557C23116yescomplemented
53C2×#10623494C2316yescomplemented
54C24×C89—21no—
55C24×C89—21no—
56C2×D4×C89—23no—
57C2×D4×C89—23no—
58C2×D4×C89C2316yes—
59C2×D4×C89C2316yescomplemented
60C22×C4×C89C2311yescharacteristic
61D8×C89C23116yescomplemented
62C2×C8×C89C2316yescomplemented
63C23×C89—26no—
64C23×C89—26no—
65C23×C89—44no—
66C23×C89—44no—
67C23×C89C2×D411yescharacteristic
68C22×D89—24no—
69C22×D89—26no—
70C22×D89—216no—
71C22×D89—448no—
72C22×D89—44no—
73C22×D89C2×D416yes—
74D4×C89—26no—
75D4×C89—26no—
76D4×C89C2414yes—
77D4×C89C2414yescomplemented
78C2×C4×C89C2×D413yes—
79C2×C4×C89C2413yes—
80C22⋊D89—216no—
81C22⋊D89—216no—
82C2×Dic89—26no—
83C2×Dic89—216no—
84C2×Dic89C2×D416yes—
85D356—26no—
86D356C2414yes—
87C4×D89C2×D4112yes—
88C4×D89C2414yes—
89Dic178—26no—
90Dic178C2414yescomplemented
91C8×C89C2414yescomplemented
92#10623494C2414yescomplemented
93C22×C89—24no—
94C22×C89—24no—
95C22×C89—412no—
96C22×C89—412no—
97C22×C89AGL(1,ℤ/8)14yescomplemented
98C22×C89C22×D413yes—
99C2×D89—212no—
100C2×D89—24no—
101C2×D89—432no—
102C2×D89—412no—
103C2×D89C22×D414yes—
104C4×C89C22×D413yes—
105C4×C89C2511yescharacteristic
106Dic89—24no—
107Dic89C22×D414yes—
108C2×C89—44no—
109C2×C89—44no—
110C2×C89C2×AGL(1,ℤ/8)16yescomplemented
111C2×C89C23×D411yescharacteristic
112D89—24no—
113D89—44no—
114C22×AGL(1,ℤ/8)—891nomaximal, Sylow
115C89C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
116C22×#41—891no—
117C23×D4—891no—
118C2×AGL(1,ℤ/8)—8924no—
119C22×SD16—891no—
120C22×SD16—891no—
121C22×D8—891no—
122C22×D8D8911yescharacteristic, complemented
123C22×M16—891no—
124C25—1781no—
125C2×#41—8912no—
126C2×#41—1786no—
127C22×Q8—891no—
128C22×D4—8912no—
129C22×D4—891no—
130C22×D4—1786no—
131C22×D4—1781no—
132C22×D4—1781no—
133C22×D4C2×D8911yescharacteristic
134C22×D4C2×D8911yescharacteristic, complemented
135C23×C4—891no—
136C23×C4—1781no—
137AGL(1,ℤ/8)—8964no—
138C2×SD16—8912no—
139C2×SD16—8912no—
140C2×D8—8912no—
141C2×D8C2×D89112yescomplemented
142C2×M16—8912no—
143C22×C8—891no—
144C22×C8C2×D8911yescharacteristic, complemented
145C24—21no—
146C24—21no—
147C24—891no—
148C24—3568no—
149C24—17812no—
150C24—1781no—
151#41—8916no—
152#41—17824no—
153C2×Q8—896no—
154C2×Q8—1783no—
155C2×D4—23no—
156C2×D4—23no—
157C2×D4—8916no—
158C2×D4—896no—
159C2×D4—17824no—
160C2×D4—17812no—
161C2×D4—17812no—
162C2×D4—1783no—
163C2×D4C22×D8916yes—
164C2×D4C22×D8916yescomplemented
165C22×C4—891no—
166C22×C4—896no—
167C22×C4—896no—
168C22×C4—17812no—
169C22×C4—1781no—
170C22×C4C22×D8911yescharacteristic
171SD16—8916no—
172SD16—8916no—
173D8—8916no—
174D8C22×D89116yescomplemented
175M16—8916no—
176C2×C8—896no—
177C2×C8C22×D8916yescomplemented
178C23—26no—
179C23—26no—
180C23—44no—
181C23—44no—
182C23—896no—
183C23—35648no—
184C23—3564no—
185C23—17816no—
186C23—1786no—
187C23—1784no—
188C23D4×D8911yescharacteristic
189Q8—894no—
190Q8—1786no—
191D4—26no—
192D4—26no—
193D4—894no—
194D4—17816no—
195D4—17816no—
196D4—1786no—
197D4C23×D8914yes—
198D4C23×D8914yescomplemented
199C2×C4—896no—
200C2×C4—8912no—
201C2×C4—894no—
202C2×C4—17816no—
203C2×C4—1786no—
204C2×C4D4×D8913yes—
205C2×C4C23×D8913yes—
206C8—894no—
207C8C23×D8914yescomplemented
208C22—24no—
209C22—24no—
210C22—412no—
211C22—412no—
212C22—894no—
213C22—35632no—
214C22—35612no—
215C22—1784no—
216C22—17812no—
217C22#1566803614yesdirect product factor
218C22C2×D4×D8913yes—
219C4—894no—
220C4—1784no—
221C4C2×D4×D8913yes—
222C4C24×D8911yescharacteristic
223C2—44no—
224C2—44no—
225C2—3564no—
226C2—1784no—
227C2C2×#1566803616yesdirect product factor
228C2C22×D4×D8911yescharacteristic
subgroupG/H
centerC23D4×D89
commutator, cosocleC4×C89C25
FrattiniC4C24×D89
FittingC22×D8×C89C2
socleC23×C89C2×D4
hypercenterC22×D8D89
hypocenterC89C22×AGL(1,ℤ/8)
LCSG ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#13083573C2semidirect
2. C22×D4×D89C2semidirect
3. C2×#15668036C2direct (N × H)
4. C22×#13083563C2semidirect
5. C22×#13083562C2semidirect
6. C22×#13083559C2semidirect
7. C22×#13083557C2semidirect
8. C22×D8×C89C2semidirect
11. C2×#13083573C22semidirect
12. C22×Dic178C22semidirect
17. C2×D4×D89C22semidirect
20. #15668036C22direct (N × H)
21. C2×#13083563C22semidirect
22. C2×#13083562C22semidirect
23. C2×#13083559C22semidirect
24. C2×#13083557C22semidirect
25. C22×#10623494C22semidirect
27. C22×D4×C89C22semidirect
28. C2×D8×C89C22semidirect
29. C22×C8×C89C22semidirect
35. #13083573C23semidirect
37. C2×Dic178C23semidirect
43. D4×D89C23semidirect
49. #13083563C23semidirect
50. #13083562C23semidirect
51. #13083559C23semidirect
52. #13083557C23semidirect
53. C2×#10623494C23semidirect
59. C2×D4×C89C23semidirect
61. D8×C89C23semidirect
62. C2×C8×C89C23semidirect
77. D4×C89C24semidirect
90. Dic178C24semidirect
91. C8×C89C24semidirect
92. #10623494C24semidirect
97. C22×C89AGL(1,ℤ/8)semidirect
110. C2×C89C2×AGL(1,ℤ/8)semidirect
115. C89C22×AGL(1,ℤ/8)semidirect
122. C22×D8D89semidirect
134. C22×D4C2×D89semidirect
141. C2×D8C2×D89semidirect
144. C22×C8C2×D89semidirect
164. C2×D4C22×D89semidirect
174. D8C22×D89semidirect
177. C2×C8C22×D89semidirect
198. D4C23×D89semidirect
207. C8C23×D89semidirect
217. C22#15668036direct (N × H)
227. C2C2×#15668036direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130835732
C22×D4×D892
C2×#156680362
C22×#130835632
C22×#130835622
C22×#130835592
C22×#130835572
C22×D8×C892
C22×AGL(1,ℤ/8)89