Finite Group Databasebeta

← #469×D251 #44559957

Subgroups — #469×D251 #44559957

Order 32128.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×D8×D251C211yesmaximal, characteristic, complemented
2C2×M16×D251C211yesmaximal, characteristic, complemented
3#121×D251C212yesmaximal, complemented
4#30631941C211yesmaximal, characteristic, complemented
5#30631940C211yesmaximal, characteristic, complemented
6#30631939C212yesmaximal, complemented
7#30631938C212yesmaximal, complemented
8#30631937C211yesmaximal, characteristic, complemented
9#106×D251C211yesmaximal, characteristic, complemented
10#102×D251C212yesmaximal, complemented
11#469×C251C211yesmaximal, characteristic, complemented
12C2×D4×D251—22no—
13C2×D4×D251C2212yes—
14C22×C4×D251C2211yescharacteristic
15C2×D2008C2211yescharacteristic, complemented
16C2×#16860795C2212yescomplemented
17D8×D251—24no—
18C2×#16860794C2211yescharacteristic, complemented
19C2×#16860793C2211yescharacteristic, complemented
20C2×#16860793C2211yescharacteristic, complemented
21M16×D251—24no—
22C2×C8×D251C2211yescharacteristic
23C2×C8×D251C2211yescharacteristic
24#22548690C2212yescomplemented
25#22548689C2212yescomplemented
26#22548688—22no—
27#22548687—22no—
28#22548686C2212yescomplemented
29C4⋊C4×D251C2212yes—
30C22⋊C4×D251—22no—
31#22548647C2211yescharacteristic, complemented
32#22548646C2212yescomplemented
33#22548637C2212yescomplemented
34#22548636C2212yescomplemented
35#22548635C2212yescomplemented
36C2×D8×C251C2211yescharacteristic, complemented
37C2×M16×C251C2211yescharacteristic, complemented
38#121×C251C2212yescomplemented
39#106×C251C2211yescharacteristic, complemented
40#102×C251C2212yescomplemented
41C23×D251—22no—
42C23×D251D411yescharacteristic
43C2×D1004—22no—
44C2×D1004C2312yes—
45C2×C22⋊D251—22no—
46C2×C22⋊D251—22no—
47C2×C22⋊D251—22no—
48D4×D251—24no—
49D4×D251—24no—
50D4×D251—48no—
51C22×Dic251D411yescharacteristic
52C2×C4×D251—22no—
53C2×C4×D251—22no—
54C2×C4×D251—22no—
55C2×C4×D251D411yescharacteristic
56C2×C4×D251D411yescharacteristic
57C2×C4×D251C2311yescharacteristic
58C2×C4×D251C2311yescharacteristic
59D2008—22no—
60#16860795—24no—
61#16860794—22no—
62#16860793—22no—
63#16860793—22no—
64C2×C251⋊C8C2311yescharacteristic
65C2×C251⋊C8C2311yescharacteristic
66C8×D251—22no—
67C8×D251D414yescomplemented
68C4⋊Dic251C2312yes—
69Dic251⋊C4—22no—
70C22⋊Dic251—22no—
71(C2×D251)⋊C4—22no—
72(C2×D251)⋊C4—22no—
73C2×D4×C251—22no—
74C2×D4×C251C2312yes—
75C22×C4×C251C2311yescharacteristic
76D8×C251—22no—
77M16×C251—22no—
78C2×C8×C251C2311yescharacteristic
79C2×C8×C251C2311yescharacteristic
80C4⋊C4×C251C2312yes—
81C22⋊C4×C251—22no—
82C22×D251—22no—
83C22×D251—24no—
84C22×D251—22no—
85C22×D251—44no—
86C22×D251—44no—
87C22×D251—44no—
88C22×D251C2×D411yescharacteristic
89C22×D251C2×D411yescharacteristic
90D1004—22no—
91D1004—22no—
92D1004—44no—
93C22⋊D251—44no—
94C22⋊D251—44no—
95C22⋊D251—44no—
96C2×Dic251—22no—
97C2×Dic251—22no—
98C2×Dic251C2×D411yescharacteristic
99C2×Dic251C2×D411yescharacteristic
100C4×D251—22no—
101C4×D251—21no—
102C4×D251—21no—
103C4×D251—22no—
104C4×D251—44no—
105C4×D251C2×D412yes—
106C4×D251#4112yes—
107C251⋊C8—21no—
108C251⋊C8C2×D412yescomplemented
109C23×C251—22no—
110C23×C251C2×D411yescharacteristic
111D4×C251—22no—
112D4×C251—22no—
113D4×C251—44no—
114C2×C4×C251—21no—
115C2×C4×C251—21no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D411yescharacteristic
118C2×C4×C251C2411yescharacteristic
119C8×C251—21no—
120C8×C251C2×D412yescomplemented
121C22×C251—21no—
122C22×C251—22no—
123C22×C251—42no—
124C22×C251—42no—
125C22×C251—42no—
126C22×C251C22×D411yescharacteristic
127C2×D251—21no—
128C2×D251—22no—
129C2×D251—44no—
130C2×D251—42no—
131C2×D251—42no—
132C2×D251—42no—
133C2×D251—88no—
134C2×D251#12112yes—
135C2×D251AGL(1,ℤ/8)14yes—
136C4×C251—21no—
137C4×C251—42no—
138C4×C251C22×D411yescharacteristic
139C4×C251C2×#4111yescharacteristic
140Dic251—21no—
141Dic251—42no—
142Dic251#12112yes—
143C2×C251—41no—
144C2×C251—82no—
145C2×C251C2×#12111yescharacteristic
146C2×C251C2×AGL(1,ℤ/8)12yes—
147D251—41no—
148D251—82no—
149D251#46914yesdirect product factor
150C251C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—2511nomaximal, Sylow
152C22×D8—2511no—
153C22×M16—2511no—
154C2×#121—2512no—
155#469—2511no—
156#469—2511no—
157#469—2512no—
158#469—2511no—
159#469—2512no—
160#469D25111yescharacteristic, direct product factor
161C2×#106—2511no—
162C2×#102—2512no—
163C22×D4—5022no—
164C22×D4—2512no—
165C23×C4—2511no—
166C2×D8—5024no—
167C2×D8—2512no—
168C2×D8—2511no—
169C2×D8C2×D25111yescharacteristic, complemented
170C2×M16—5024no—
171C2×M16—2511no—
172C2×M16—2511no—
173C2×M16—2511no—
174C2×M16C2×D25111yescharacteristic, complemented
175C22×C8—2511no—
176C22×C8—2511no—
177#121—5022no—
178#121—5022no—
179#121—2512no—
180#121—2512no—
181#121—2512no—
182#121C2×D25112yescomplemented
183C2×C4⋊C4—2512no—
184C2×C22⋊C4—5022no—
185#106—2511no—
186#106—2512no—
187#106C2×D25111yescharacteristic, complemented
188#102—2512no—
189#102—2512no—
190#102—2512no—
191#102C2×D25112yescomplemented
192C24—5022no—
193C24—2511no—
194C2×D4—22no—
195C2×D4—10048no—
196C2×D4—5022no—
197C2×D4—5022no—
198C2×D4—5022no—
199C2×D4—5024no—
200C2×D4—5024no—
201C2×D4—5022no—
202C2×D4—2512no—
203C2×D4C22×D25112yes—
204C22×C4—5022no—
205C22×C4—5022no—
206C22×C4—5022no—
207C22×C4—2511no—
208C22×C4—2511no—
209C22×C4—2511no—
210C22×C4—2511no—
211C22×C4—2511no—
212C22×C4C22×D25111yescharacteristic
213D8—22no—
214D8—5024no—
215D8—5022no—
216M16—22no—
217M16—5022no—
218M16—5022no—
219M16—5022no—
220C2×C8—5022no—
221C2×C8—2511no—
222C2×C8—2511no—
223C2×C8—2514no—
224C2×C8C22×D25111yescharacteristic
225C2×C8C22×D25111yescharacteristic
226C4⋊C4—5022no—
227C4⋊C4—2512no—
228C4⋊C4C22×D25112yes—
229C22⋊C4—22no—
230C22⋊C4—5022no—
231C22⋊C4—5022no—
232C22⋊C4—5022no—
233C23—22no—
234C23—10044no—
235C23—10044no—
236C23—10044no—
237C23—5022no—
238C23—5024no—
239C23—5022no—
240C23—2511no—
241C23—2511no—
242C23D4×D25111yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—10044no—
247D4—10044no—
248D4—10044no—
249D4—10044no—
250D4—5022no—
251D4—5022no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—10044no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4—5021no—
261C2×C4—5021no—
262C2×C4—2511no—
263C2×C4—2511no—
264C2×C4—2512no—
265C2×C4—2512no—
266C2×C4D4×D25111yescharacteristic
267C2×C4C23×D25111yescharacteristic
268C8—21no—
269C8—5021no—
270C8—2512no—
271C8D4×D25112yescomplemented
272C22—22no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—20088no—
278C22—10044no—
279C22—10042no—
280C22—10042no—
281C22—10042no—
282C22—5022no—
283C22—5021no—
284C22—2514no—
285C22—2512no—
286C22C2×D4×D25111yescharacteristic
287C4—21no—
288C4—42no—
289C4—10042no—
290C4—5021no—
291C4—2512no—
292C4C2×D4×D25111yescharacteristic
293C4#41×D25111yescharacteristic
294C2—41no—
295C2—82no—
296C2—20082no—
297C2—10041no—
298C2—2514no—
299C2#121×D25111yescharacteristic
300C2AGL(1,ℤ/8)×D25112yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#469×C251C2
socleC22×C251C22×D4
hypercenter#469D251
hypocenterC251C2×#469
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D251C2semidirect
2. C2×M16×D251C2semidirect
3. #121×D251C2semidirect
4. #30631941C2semidirect
5. #30631940C2semidirect
6. #30631939C2semidirect
7. #30631938C2semidirect
8. #30631937C2semidirect
9. #106×D251C2semidirect
10. #102×D251C2semidirect
11. #469×C251C2semidirect
15. C2×D2008C22semidirect
16. C2×#16860795C22semidirect
18. C2×#16860794C22semidirect
19. C2×#16860793C22semidirect
20. C2×#16860793C22semidirect
24. #22548690C22semidirect
25. #22548689C22semidirect
28. #22548686C22semidirect
31. #22548647C22semidirect
32. #22548646C22semidirect
33. #22548637C22semidirect
34. #22548636C22semidirect
35. #22548635C22semidirect
36. C2×D8×C251C22semidirect
37. C2×M16×C251C22semidirect
38. #121×C251C22semidirect
39. #106×C251C22semidirect
40. #102×C251C22semidirect
67. C8×D251D4semidirect
108. C251⋊C8C2×D4semidirect
120. C8×C251C2×D4semidirect
149. D251#469direct (N × H)
150. C251C2×#469semidirect
160. #469D251direct (N × H)
169. C2×D8C2×D251semidirect
174. C2×M16C2×D251semidirect
182. #121C2×D251semidirect
187. #106C2×D251semidirect
191. #102C2×D251semidirect
271. C8D4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D2512
C2×M16×D2512
#121×D2512
#306319412
#306319402
#306319392
#306319382
#306319372
#106×D2512
#102×D2512
#469×C2512
C2×#469251