Finite Group Databasebeta

← C22×#576×C89 #36440932

Subgroups — C22×#576×C89 #36440932

Order 22784.

11462 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×H(5,𝔽2)×C89C212yesmaximal, complemented
2C2×#576×C89C2148yesmaximal, direct product factor
3C22×AGL(1,ℤ/8)×C89C216yesmaximal, complemented
4C22×#135×C89C213yesmaximal, complemented
5C23×D8×C89C213yesmaximal, complemented
6C22×#131×C89C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C89—224no—
8C2×H(5,𝔽2)×C89C22148yescomplemented
9C22×#41×C89—22no—
10C22×#41×C89C2211yescharacteristic
11C22×#41×C89C2216yescomplemented
12C23×D4×C89—26no—
13C23×D4×C89C2216yescomplemented
14#576×C89C221256yesdirect product factor
15C2×AGL(1,ℤ/8)×C89C221144yescomplemented
16C2×#135×C89C22172yescomplemented
17C22×Q16×C89C2211yescharacteristic, complemented
18C22×SD16×C89C2216yescomplemented
19C22×D8×C89C2213yescomplemented
20C22×D8×C89C2216yescomplemented
21C22×D8×C89C22172yescomplemented
22C2×#131×C89C22124yescomplemented
23C22×M16×C89C2213yescomplemented
24C23×C8×C89C2213yescomplemented
25C25×C89—26no—
26H(5,𝔽2)×C89—2192no—
27H(5,𝔽2)×C89C231128yescomplemented
28C2×#41×C89—248no—
29C2×#41×C89—236no—
30C2×#41×C89D4112yes—
31C2×#41×C89C23112yes—
32C2×#41×C89C23172yescomplemented
33C22×Q8×C89D411yescharacteristic
34C22×Q8×C89C2312yes—
35C22×D4×C89—26no—
36C22×D4×C89—2144no—
37C22×D4×C89—26no—
38C22×D4×C89—236no—
39C22×D4×C89D413yes—
40C22×D4×C89C2316yes—
41C22×D4×C89C23172yescomplemented
42C23×C4×C89—26no—
43C23×C4×C89C2313yes—
44AGL(1,ℤ/8)×C89C231384yescomplemented
45#135×C89C231192yescomplemented
46C2×Q16×C89C23112yescomplemented
47C2×SD16×C89C23172yescomplemented
48C2×D8×C89C23136yescomplemented
49C2×D8×C89C23172yescomplemented
50C2×D8×C89C231192yescomplemented
51#131×C89C23164yescomplemented
52C2×M16×C89C23136yescomplemented
53C22×C8×C89C2311yescharacteristic
54C22×C8×C89C2313yes—
55C22×C8×C89C23136yescomplemented
56C24×C89—26no—
57C24×C89—272no—
58C24×C89—824no—
59C24×C89C2×D413yes—
60#41×C89—2128no—
61#41×C89—2144no—
62#41×C89C2×D4148yes—
63#41×C89C24116yes—
64#41×C89C24196yescomplemented
65C2×Q8×C89—26no—
66C2×Q8×C89C2×D4112yes—
67C2×Q8×C89C24112yes—
68C2×D4×C89—272no—
69C2×D4×C89—2384no—
70C2×D4×C89—218no—
71C2×D4×C89—272no—
72C2×D4×C89—2144no—
73C2×D4×C89C2×D4136yes—
74C2×D4×C89C24136yes—
75C2×D4×C89C24196yescomplemented
76C22×C4×C89—22no—
77C22×C4×C89—272no—
78C22×C4×C89C2×D4118yes—
79C22×C4×C89C2×D413yes—
80C22×C4×C89C24118yes—
81C22×C4×C89C2411yescharacteristic
82Q16×C89C24116yescomplemented
83SD16×C89C24196yescomplemented
84D8×C89C24148yescomplemented
85D8×C89C24196yescomplemented
86M16×C89C24148yescomplemented
87C2×C8×C89C2416yes—
88C2×C8×C89C24118yes—
89C2×C8×C89C24148yescomplemented
90C23×C89—212no—
91C23×C89—236no—
92C23×C89—296no—
93C23×C89—424no—
94C23×C89—8144no—
95C23×C89C22×D4118yes—
96C23×C89C22×D411yescharacteristic
97Q8×C89—212no—
98Q8×C89C22×D4116yes—
99Q8×C89C2518yes—
100D4×C89—296no—
101D4×C89—236no—
102D4×C89—296no—
103D4×C89C22×D4148yes—
104D4×C89C25124yes—
105C2×C4×C89—212no—
106C2×C4×C89—296no—
107C2×C4×C89C22×D4118yes—
108C2×C4×C89C22×D413yes—
109C2×C4×C89C22×D4136yes—
110C2×C4×C89C2513yes—
111C2×C4×C89C25112yes—
112C8×C89C2514yes—
113C8×C89C25112yes—
114C22×C89—236no—
115C22×C89—224no—
116C22×C89—472no—
117C22×C89—896no—
118C22×C89#57614yesdirect product factor
119C22×C89C23×D4112yes—
120C22×C89C23×D413yes—
121C4×C89—28no—
122C4×C89C23×D4112yes—
123C4×C89C23×D413yes—
124C4×C89C2611yescharacteristic
125C22×#576C8911yesmaximal, characteristic, direct product factor, Sylow
126C2×C89—212no—
127C2×C89—424no—
128C2×C89C2×#57616yesdirect product factor
129C2×C89C24×D411yescharacteristic
130C22×H(5,𝔽2)C2×C8912yescomplemented
131C2×#576C2×C89148yesdirect product factor
132C22×AGL(1,ℤ/8)C2×C8916yescomplemented
133C22×#135C2×C8913yescomplemented
134C23×D8C2×C8913yescomplemented
135C22×#131C2×C8911yescharacteristic, complemented
136C89C22×#57611yescharacteristic, direct product factor, Sylow
137C2×H(5,𝔽2)—224no—
138C2×H(5,𝔽2)C22×C89148yescomplemented
139C22×#41—22no—
140C22×#41C22×C8911yescharacteristic
141C22×#41C22×C8916yescomplemented
142C23×D4—26no—
143C23×D4C22×C8916yescomplemented
144#576C22×C891256yesdirect product factor
145C2×AGL(1,ℤ/8)C22×C891144yescomplemented
146C2×#135C22×C89172yescomplemented
147C22×Q16C22×C8911yescharacteristic, complemented
148C22×SD16C22×C8916yescomplemented
149C22×D8C22×C8913yescomplemented
150C22×D8C22×C8916yescomplemented
151C22×D8C22×C89172yescomplemented
152C2×#131C22×C89124yescomplemented
153C22×M16C22×C8913yescomplemented
154C23×C8C22×C8913yescomplemented
155C25—26no—
156H(5,𝔽2)—2192no—
157H(5,𝔽2)C23×C891128yescomplemented
158C2×#41—248no—
159C2×#41—236no—
160C2×#41D4×C89112yes—
161C2×#41C23×C89112yes—
162C2×#41C23×C89172yescomplemented
163C22×Q8D4×C8911yescharacteristic
164C22×Q8C23×C8912yes—
165C22×D4—26no—
166C22×D4—2144no—
167C22×D4—26no—
168C22×D4—236no—
169C22×D4D4×C8913yes—
170C22×D4C23×C8916yes—
171C22×D4C23×C89172yescomplemented
172C23×C4—26no—
173C23×C4C23×C8913yes—
174AGL(1,ℤ/8)C23×C891384yescomplemented
175#135C23×C891192yescomplemented
176C2×Q16C23×C89112yescomplemented
177C2×SD16C23×C89172yescomplemented
178C2×D8C23×C89136yescomplemented
179C2×D8C23×C89172yescomplemented
180C2×D8C23×C891192yescomplemented
181#131C23×C89164yescomplemented
182C2×M16C23×C89136yescomplemented
183C22×C8C23×C8911yescharacteristic
184C22×C8C23×C8913yes—
185C22×C8C23×C89136yescomplemented
186C24—26no—
187C24—272no—
188C24—824no—
189C24C2×D4×C8913yes—
190#41—2128no—
191#41—2144no—
192#41C2×D4×C89148yes—
193#41C24×C89116yes—
194#41C24×C89196yescomplemented
195C2×Q8—26no—
196C2×Q8C2×D4×C89112yes—
197C2×Q8C24×C89112yes—
198C2×D4—272no—
199C2×D4—2384no—
200C2×D4—218no—
201C2×D4—272no—
202C2×D4—2144no—
203C2×D4C2×D4×C89136yes—
204C2×D4C24×C89136yes—
205C2×D4C24×C89196yescomplemented
206C22×C4—22no—
207C22×C4—272no—
208C22×C4C2×D4×C8913yes—
209C22×C4C2×D4×C89118yes—
210C22×C4C24×C8911yescharacteristic
211C22×C4C24×C89118yes—
212Q16C24×C89116yescomplemented
213SD16C24×C89196yescomplemented
214D8C24×C89148yescomplemented
215D8C24×C89196yescomplemented
216M16C24×C89148yescomplemented
217C2×C8C24×C8916yes—
218C2×C8C24×C89118yes—
219C2×C8C24×C89148yescomplemented
220C23—212no—
221C23—236no—
222C23—296no—
223C23—424no—
224C23—8144no—
225C23C22×D4×C89118yes—
226C23C22×D4×C8911yescharacteristic
227Q8—212no—
228Q8C22×D4×C89116yes—
229Q8C25×C8918yes—
230D4—296no—
231D4—236no—
232D4—296no—
233D4C22×D4×C89148yes—
234D4C25×C89124yes—
235C2×C4—212no—
236C2×C4—296no—
237C2×C4C22×D4×C89118yes—
238C2×C4C22×D4×C8913yes—
239C2×C4C22×D4×C89136yes—
240C2×C4C25×C8913yes—
241C2×C4C25×C89112yes—
242C8C25×C8914yes—
243C8C25×C89112yes—
244C22—236no—
245C22—224no—
246C22—472no—
247C22—896no—
248C22#576×C8914yesdirect product factor
249C22C23×D4×C89112yes—
250C22C23×D4×C8913yes—
251C4—28no—
252C4C23×D4×C89112yes—
253C4C23×D4×C8913yes—
254C4C26×C8911yescharacteristic
255C2—212no—
256C2—424no—
257C2C2×#576×C8916yesdirect product factor
258C2C24×D4×C8911yescharacteristic
subgroupG/H
center, socleC23×C89C22×D4
commutator, Frattini, cosocleC4C26×C89
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C89 ◁ C22×#41×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C89C2semidirect
2. C2×#576×C89C2direct (N × H)
3. C22×AGL(1,ℤ/8)×C89C2semidirect
4. C22×#135×C89C2semidirect
5. C23×D8×C89C2semidirect
6. C22×#131×C89C2semidirect
8. C2×H(5,𝔽2)×C89C22semidirect
11. C22×#41×C89C22semidirect
13. C23×D4×C89C22semidirect
14. #576×C89C22direct (N × H)
15. C2×AGL(1,ℤ/8)×C89C22semidirect
16. C2×#135×C89C22semidirect
17. C22×Q16×C89C22semidirect
18. C22×SD16×C89C22semidirect
19. C22×D8×C89C22semidirect
20. C22×D8×C89C22semidirect
21. C22×D8×C89C22semidirect
22. C2×#131×C89C22semidirect
23. C22×M16×C89C22semidirect
24. C23×C8×C89C22semidirect
27. H(5,𝔽2)×C89C23semidirect
32. C2×#41×C89C23semidirect
41. C22×D4×C89C23semidirect
44. AGL(1,ℤ/8)×C89C23semidirect
45. #135×C89C23semidirect
46. C2×Q16×C89C23semidirect
47. C2×SD16×C89C23semidirect
48. C2×D8×C89C23semidirect
49. C2×D8×C89C23semidirect
50. C2×D8×C89C23semidirect
51. #131×C89C23semidirect
52. C2×M16×C89C23semidirect
55. C22×C8×C89C23semidirect
64. #41×C89C24semidirect
75. C2×D4×C89C24semidirect
82. Q16×C89C24semidirect
83. SD16×C89C24semidirect
84. D8×C89C24semidirect
85. D8×C89C24semidirect
86. M16×C89C24semidirect
89. C2×C8×C89C24semidirect
118. C22×C89#576direct (N × H)
125. C22×#576C89direct (N × H)
128. C2×C89C2×#576direct (N × H)
130. C22×H(5,𝔽2)C2×C89semidirect
131. C2×#576C2×C89direct (N × H)
132. C22×AGL(1,ℤ/8)C2×C89semidirect
133. C22×#135C2×C89semidirect
134. C23×D8C2×C89semidirect
135. C22×#131C2×C89semidirect
136. C89C22×#576direct (N × H)
138. C2×H(5,𝔽2)C22×C89semidirect
141. C22×#41C22×C89semidirect
143. C23×D4C22×C89semidirect
144. #576C22×C89direct (N × H)
145. C2×AGL(1,ℤ/8)C22×C89semidirect
146. C2×#135C22×C89semidirect
147. C22×Q16C22×C89semidirect
148. C22×SD16C22×C89semidirect
149. C22×D8C22×C89semidirect
150. C22×D8C22×C89semidirect
151. C22×D8C22×C89semidirect
152. C2×#131C22×C89semidirect
153. C22×M16C22×C89semidirect
154. C23×C8C22×C89semidirect
157. H(5,𝔽2)C23×C89semidirect
162. C2×#41C23×C89semidirect
171. C22×D4C23×C89semidirect
174. AGL(1,ℤ/8)C23×C89semidirect
175. #135C23×C89semidirect
176. C2×Q16C23×C89semidirect
177. C2×SD16C23×C89semidirect
178. C2×D8C23×C89semidirect
179. C2×D8C23×C89semidirect
180. C2×D8C23×C89semidirect
181. #131C23×C89semidirect
182. C2×M16C23×C89semidirect
185. C22×C8C23×C89semidirect
194. #41C24×C89semidirect
205. C2×D4C24×C89semidirect
212. Q16C24×C89semidirect
213. SD16C24×C89semidirect
214. D8C24×C89semidirect
215. D8C24×C89semidirect
216. M16C24×C89semidirect
219. C2×C8C24×C89semidirect
248. C22#576×C89direct (N × H)
257. C2C2×#576×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C892
C2×#576×C892
C22×AGL(1,ℤ/8)×C892
C22×#135×C892
C23×D8×C892
C22×#131×C892
C22×#57689