Finite Group Databasebeta

← C2×#26028249 #36760546

Subgroups — C2×#26028249 #36760546

Order 22784.

4438 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 4438 conjugacy classes of subgroups — too many to draw. Only its 667 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×#15667981C212yesmaximal, complemented
2#26028249C2116yesmaximal, direct product factor
3C2×#19151523C213yesmaximal, complemented
4C2×#19151514C216yesmaximal, complemented
5C2×#19151507C213yesmaximal, complemented
6C2×#379×C89C211yesmaximal, characteristic, complemented
7C23×D356C2211yescharacteristic, complemented
8C23×C22⋊D89C2216yes—
9C2×#15667981—216no—
10C2×#15667981C22116yescomplemented
11C22×C4⋊Dic89C2213yescomplemented
12C22×C22⋊Dic89C2216yes—
13C22×#13083551C2216yescomplemented
14#19151523C22124yescomplemented
15#19151514C22148yescomplemented
16#19151507C22124yescomplemented
17C2×#15667919C2213yescomplemented
18C2×#15667919C2213yescomplemented
19C26×C89C2211yescharacteristic
20C22×C22⋊C4×C89C2213yes—
21C22×C22⋊C4×C89C2213yes—
22#379×C89C2218yescomplemented
23C24×D89C2312yes—
24C22×D356—46no—
25C22×D356C2314yescomplemented
26C22×C22⋊D89—224no—
27C22×C22⋊D89—412no—
28C22×C22⋊D89—412no—
29C22×C22⋊D89C23124yes—
30C23×Dic89C2316yes—
31#15667981—264no—
32C2×C4⋊Dic89—46no—
33C2×C4⋊Dic89C23112yescomplemented
34C2×C22⋊Dic89—224no—
35C2×C22⋊Dic89C23124yes—
36C2×#13083551—412no—
37C2×#13083551C23124yescomplemented
38#15667919C23112yescomplemented
39#15667919C23112yescomplemented
40C25×C89—46no—
41C25×C89—46no—
42C25×C89D418yescomplemented
43C25×C89C2313yes—
44C25×C89C2314yes—
45C23×C4×C89C2311yescharacteristic
46C23×C4×C89C2313yes—
47C2×C22⋊C4×C89—46no—
48C2×C22⋊C4×C89—46no—
49C2×C22⋊C4×C89C23112yes—
50C2×C22⋊C4×C89C23112yes—
51C23×D89—24no—
52C23×D89—212no—
53C23×D89—212no—
54C23×D89C2414yes—
55C2×D356—412no—
56C2×D356—416no—
57C2×C22⋊D89—248no—
58C2×C22⋊D89—424no—
59C2×C22⋊D89—424no—
60C2×C22⋊D89—496no—
61C22×Dic89—212no—
62C22×Dic89—212no—
63C22×Dic89—212no—
64C22×Dic89C24112yes—
65C4⋊Dic89—412no—
66C22⋊Dic89—248no—
67#13083551—424no—
68C24×C89—212no—
69C24×C89—212no—
70C24×C89—26no—
71C24×C89—26no—
72C24×C89—448no—
73C24×C89—412no—
74C24×C89—412no—
75C24×C89—448no—
76C24×C89—416no—
77C24×C89C2×D4112yes—
78C24×C89C2×D4116yescomplemented
79C24×C89C2416yes—
80C24×C89C2411yescharacteristic
81C22×C4×C89—26no—
82C22×C4×C89—212no—
83C22×C4×C89—26no—
84C22×C4×C89C2412yes—
85C22×C4×C89C2416yes—
86C22⋊C4×C89—412no—
87C22⋊C4×C89—412no—
88C23×C89—224no—
89C23×C89—248no—
90C23×C89—26no—
91C23×C89—26no—
92C23×C89—212no—
93C23×C89—212no—
94C23×C89—448no—
95C23×C89—448no—
96C23×C89—416no—
97C23×C89—4128no—
98C23×C89—412no—
99C23×C89—424no—
100C23×C89—412no—
101C23×C89C22×D4112yes—
102C23×C89C22×D412yes—
103C23×C89H(5,𝔽2)16yes—
104C23×C89H(5,𝔽2)16yes—
105C23×C89C2511yescharacteristic
106C22×D89—24no—
107C22×D89—212no—
108C22×D89—212no—
109C22×D89—424no—
110C22×D89—424no—
111C22×D89—48no—
112C2×C4×C89—212no—
113C2×C4×C89—26no—
114C2×C4×C89—26no—
115C2×C4×C89—412no—
116C2×C4×C89—44no—
117C22⋊D89—496no—
118C2×Dic89—212no—
119C2×Dic89—212no—
120C2×Dic89—212no—
121C2×Dic89—424no—
122D356—416no—
123C22×C89—224no—
124C22×C89—212no—
125C22×C89—464no—
126C22×C89—412no—
127C22×C89—46no—
128C22×C89—46no—
129C22×C89—448no—
130C22×C89#534112yes—
131C22×C89#560112yes—
132C22×C89H(3,𝔽4)14yes—
133C22×C89C23×D411yescharacteristic
134C22×C89C2×H(5,𝔽2)13yes—
135C22×C89C2×H(5,𝔽2)13yes—
136C2×D89—412no—
137C2×D89—412no—
138C2×D89—44no—
139C2×D89—816no—
140C4×C89—46no—
141C4×C89—42no—
142Dic89—412no—
143C2×#2812—891nomaximal, Sylow
144C2×C89—412no—
145C2×C89#281218yescomplemented
146C2×C89C2×#53413yes—
147C2×C89C2×#56013yes—
148C2×C89C2×H(3,𝔽4)11yescharacteristic
149D89—84no—
150C22×#120—892no—
151#2812—8916no—
152C2×#397—893no—
153C2×#394—896no—
154C2×#392—893no—
155C2×#379D8911yescharacteristic, complemented
156C89C2×#281211yescharacteristic, complemented, Sylow
157C26C2×D8911yescharacteristic
158C23×D4—896no—
159C23×D4—891no—
160C2×#120—8916no—
161C2×#120—17816no—
162C22×C4⋊C4—893no—
163C22×C22⋊C4—896no—
164C22×C22⋊C4—896no—
165C22×C22⋊C4C2×D8913yes—
166C22×C22⋊C4C2×D8913yes—
167#397—8924no—
168#394—8948no—
169#392—8924no—
170#379C2×D8918yescomplemented
171C2×#95—893no—
172C2×#95—893no—
173C25—46no—
174C25—46no—
175C25—892no—
176C25D35618yescomplemented
177C25C22×D8913yes—
178C25C22×D8914yes—
179C22×D4—8924no—
180C22×D4—894no—
181C22×D4—3566no—
182C22×D4—35612no—
183C22×D4—35612no—
184C22×D4—17824no—
185C23×C4—896no—
186C23×C4C22×D8913yes—
187C23×C4C22×D8911yescharacteristic
188#120—17864no—
189C2×C4⋊C4—8912no—
190C2×C4⋊C4—3566no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—8924no—
194C2×C22⋊C4—8924no—
195C2×C22⋊C4—35612no—
196C2×C22⋊C4—17824no—
197C2×C22⋊C4C22×D89112yes—
198C2×C22⋊C4C22×D89112yes—
199#95—8912no—
200#95—8912no—
201C24—26no—
202C24—26no—
203C24—212no—
204C24—212no—
205C24—448no—
206C24—412no—
207C24—412no—
208C24—448no—
209C24—416no—
210C24—894no—
211C24—1784no—
212C24—17812no—
213C24—17812no—
214C24C2×D356112yes—
215C24C2×D356116yescomplemented
216C24C23×D8916yes—
217C24C23×D8911yescharacteristic
218C2×D4—35612no—
219C2×D4—35616no—
220C2×D4—35696no—
221C2×D4—35624no—
222C2×D4—35624no—
223C2×D4—17848no—
224C22×C4—212no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—8912no—
228C22×C4—17812no—
229C22×C4—17812no—
230C22×C4—17812no—
231C22×C4C23×D8912yes—
232C22×C4C23×D8916yes—
233C4⋊C4—35612no—
234C22⋊C4—412no—
235C22⋊C4—412no—
236C22⋊C4—35624no—
237C22⋊C4—17848no—
238C23—248no—
239C23—224no—
240C23—26no—
241C23—26no—
242C23—212no—
243C23—212no—
244C23—448no—
245C23—448no—
246C23—416no—
247C23—4128no—
248C23—424no—
249C23—412no—
250C23—412no—
251C23—35624no—
252C23—35624no—
253C23—3568no—
254C23—1784no—
255C23—17812no—
256C23—17812no—
257C23C22×D356112yes—
258C23C22×D35612yes—
259C23#1566805716yes—
260C23#1566805716yes—
261C23C24×D8911yescharacteristic
262D4—35616no—
263D4—35696no—
264C2×C4—212no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—412no—
268C2×C4—44no—
269C2×C4—35624no—
270C2×C4—17812no—
271C2×C4—17812no—
272C2×C4—17812no—
273C22—224no—
274C22—212no—
275C22—464no—
276C22—412no—
277C22—46no—
278C22—46no—
279C22—448no—
280C22—35612no—
281C22—35612no—
282C22—3564no—
283C22—71216no—
284C22#19152160112yes—
285C22#19152337112yes—
286C22#1915234114yes—
287C22C23×D35611yescharacteristic
288C22C2×#1566805713yes—
289C22C2×#1566805713yes—
290C4—46no—
291C4—42no—
292C4—35612no—
293C2—412no—
294C2—7124no—
295C2#2602824918yesdirect product factor
296C2C2×#1915216013yes—
297C2C2×#1915233713yes—
298C2C2×#1915234111yescharacteristic
subgroupG/H
centerC24C23×D89
commutator, cosocleC23×C89C25
FrattiniC23C24×D89
FittingC2×#379×C89C2
socleC24×C89C24
hypercenterC2×#379D89
hypocenterC89C2×#2812
LCSG ▷ C23×C89 ▷ C89
UCSC1 ◁ C24 ◁ C2×#379

Complements

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

NH ≅ G/Ntype
1. C22×#15667981C2semidirect
2. #26028249C2direct (N × H)
3. C2×#19151523C2semidirect
4. C2×#19151514C2semidirect
5. C2×#19151507C2semidirect
6. C2×#379×C89C2semidirect
7. C23×D356C22semidirect
10. C2×#15667981C22semidirect
11. C22×C4⋊Dic89C22semidirect
13. C22×#13083551C22semidirect
14. #19151523C22semidirect
15. #19151514C22semidirect
16. #19151507C22semidirect
17. C2×#15667919C22semidirect
18. C2×#15667919C22semidirect
22. #379×C89C22semidirect
25. C22×D356C23semidirect
33. C2×C4⋊Dic89C23semidirect
37. C2×#13083551C23semidirect
38. #15667919C23semidirect
39. #15667919C23semidirect
42. C25×C89D4semidirect
78. C24×C89C2×D4semidirect
145. C2×C89#2812semidirect
155. C2×#379D89semidirect
156. C89C2×#2812semidirect
170. #379C2×D89semidirect
176. C25D356semidirect
215. C24C2×D356semidirect
295. C2#26028249direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156679812
#260282492
C2×#191515232
C2×#191515142
C2×#191515072
C2×#379×C892
C2×#281289