Finite Group Databasebeta

← #44560499

Subgroups — #44560499

Order 32128.

450 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 125 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×C4×D1004C211yesmaximal, characteristic, complemented
2#30632015C211yesmaximal, characteristic, complemented
3#30631998C211yesmaximal, characteristic, complemented
4#30631984C212yesmaximal, complemented
5#30631899C212yesmaximal, complemented
6#30631837C211yesmaximal, characteristic, complemented
7#30631820C211yesmaximal, characteristic
8#30631797C212yesmaximal, complemented
9#30631785C212yesmaximal, complemented
10#30631744C211yesmaximal, characteristic, complemented
11#529×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860797C2212yes—
15C2×#16860795C2212yes—
16C4×D1004—21no—
17C4×D1004—21no—
18C4×D1004C2212yes—
19C4×D1004C2212yes—
20C2×C4⋊Dic251C2211yescharacteristic
21C2×(C2×D251)⋊C4—21no—
22#22548648C2212yescomplemented
23#22548646C2212yescomplemented
24C4⋊(C251⋊C8)C2212yescomplemented
25#22548639C2212yescomplemented
26#22548636C2212yescomplemented
27#22548635C2212yes—
28#22548635C2212yes—
29C22⋊(C251⋊C8)C2211yescharacteristic
30C22⋊(C251⋊C8)C2211yescharacteristic
31#22548626C2212yescomplemented
32#126×C251—21no—
33#125×C251C2211yescharacteristic, complemented
34#124×C251C2211yescharacteristic, complemented
35#123×C251—21no—
36C22⋊Q8×C251C2211yescharacteristic
37#121×C251C2211yescharacteristic
38C4×Q8×C251C2211yescharacteristic, complemented
39C4×D4×C251—21no—
40C4×D4×C251C2211yescharacteristic, complemented
41#117×C251—21no—
42C2×C42×C251C2211yescharacteristic
43C23×D251—21no—
44C2×D1004—24no—
45C2×D1004—21no—
46C2×D1004C2312yes—
47C22×Dic251—21no—
48C2×C4×D251—22no—
49C2×C4×D251—22no—
50C2×C4×D251—44no—
51#16860797—24no—
52#16860795—24no—
53C2×C251⋊C8C2312yes—
54C2×C251⋊C8C2312yes—
55C4⋊Dic251—21no—
56C4⋊Dic251C2312yes—
57(C2×D251)⋊C4—22no—
58(C2×D251)⋊C4—22no—
59C2×Q8×C251C2311yescharacteristic
60C2×D4×C251—21no—
61C2×D4×C251C2311yescharacteristic
62C22×C4×C251—21no—
63C22×C4×C251D411yescharacteristic
64C22×C4×C251D411yescharacteristic
65C22×C4×C251C2311yescharacteristic
66C4⋊C4×C251—21no—
67C4⋊C4×C251—21no—
68C4⋊C4×C251—21no—
69C4⋊C4×C251—21no—
70C4⋊C4×C251C2311yescharacteristic
71C4⋊C4×C251C2311yescharacteristic
72C22⋊C4×C251—21no—
73C22⋊C4×C251—21no—
74C22⋊C4×C251—21no—
75C22⋊C4×C251—21no—
76C22⋊C4×C251—21no—
77C42×C251—21no—
78C42×C251D411yescharacteristic
79C42×C251D411yescharacteristic
80C42×C251C2311yescharacteristic
81C42×C251C2311yescharacteristic
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—42no—
85C22×D251—42no—
86D1004—24no—
87D1004—22no—
88D1004#4114yes—
89C2×Dic251—22no—
90C2×Dic251—42no—
91C4×D251—44no—
92C4×D251—44no—
93C251⋊C8—22no—
94C251⋊C8—22no—
95C23×C251—21no—
96C23×C251C2×D411yescharacteristic
97Q8×C251—22no—
98D4×C251—22no—
99D4×C251—42no—
100C2×C4×C251—22no—
101C2×C4×C251—22no—
102C2×C4×C251—21no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—41no—
106C2×C4×C251—42no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2×D411yescharacteristic
111C2×C4×C251C2×D411yescharacteristic
112C2×C4×C251C2411yescharacteristic
113C22×C251—21no—
114C22×C251—21no—
115C22×C251—41no—
116C22×C251—41no—
117C22×C251—41no—
118C22×C251AGL(1,ℤ/8)12yes—
119C22×C251C22×D411yescharacteristic
120C2×D251—24no—
121C2×D251—48no—
122C2×D251—42no—
123C2×D251—42no—
124C4×C251—21no—
125C4×C251—21no—
126C4×C251—21no—
127C4×C251—21no—
128C4×C251—21no—
129C4×C251—41no—
130C4×C251—41no—
131C4×C251—41no—
132C4×C251C2×#4112yes—
133Dic251—42no—
134C2×C251—22no—
135C2×C251—81no—
136C2×C251#52511yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C251#57511yescharacteristic
139D251—44no—
140C251#311411yescharacteristic, complemented, Sylow
141#3114—2511nomaximal, Sylow
142#529D25111yescharacteristic, complemented
143C2×C4×D4—2511no—
144#483—2511no—
145#480—2511no—
146#478—2512no—
147#463—2512no—
148#450—2511no—
149#447—2511no—
150#442—2512no—
151#440—2512no—
152#432—2511no—
153C22×D4—2511no—
154C23×C4—5021no—
155C2×SD16—2512no—
156C2×D8—2512no—
157#126—21no—
158#125C2×D25111yescharacteristic, complemented
159#124C2×D25111yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D25111yescharacteristic
162#121C2×D25111yescharacteristic
163C4×Q8C2×D25111yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—5021no—
166C4×D4—5021no—
167C4×D4—2512no—
168C4×D4—2512no—
169C4×D4C2×D25111yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—2511no—
172C2×C22⋊C4—5021no—
173C2×C42C2×D25111yescharacteristic
174#107—2512no—
175#106—2512no—
176C4⋊C8—2512no—
177#103—2512no—
178#102—2512no—
179#102—2512no—
180#102—2512no—
181C22⋊C8—2511no—
182C22⋊C8—2511no—
183#97—2512no—
184C24—5021no—
185C2×Q8C22×D25111yescharacteristic
186C2×D4—21no—
187C2×D4—5024no—
188C2×D4—5021no—
189C2×D4—2512no—
190C2×D4C22×D25111yescharacteristic
191C22×C4—21no—
192C22×C4—10044no—
193C22×C4—5021no—
194C22×C4—5022no—
195C22×C4—5022no—
196C22×C4C22⋊D25111yescharacteristic
197C22×C4C22⋊D25111yescharacteristic
198C22×C4C22×D25111yescharacteristic
199SD16—5024no—
200D8—5024no—
201C2×C8—2512no—
202C2×C8—2512no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—5021no—
208C4⋊C4—2512no—
209C4⋊C4C22×D25111yescharacteristic
210C4⋊C4C22×D25111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—5022no—
217C22⋊C4—5022no—
218C42—21no—
219C42C22⋊D25111yescharacteristic
220C42C22⋊D25111yescharacteristic
221C42C22×D25111yescharacteristic
222C42C22×D25111yescharacteristic
223C23—21no—
224C23—10042no—
225C23—10042no—
226C23—5024no—
227C23—5022no—
228C23C2×C22⋊D25111yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—5024no—
233D4—5022no—
234D4—2514no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—10044no—
243C2×C4—10044no—
244C2×C4—10042no—
245C2×C4—5022no—
246C2×C4C2×C22⋊D25111yescharacteristic
247C2×C4C2×C22⋊D25111yescharacteristic
248C2×C4C2×C22⋊D25111yescharacteristic
249C2×C4C2×C22⋊D25111yescharacteristic
250C2×C4C2×C22⋊D25111yescharacteristic
251C2×C4C23×D25111yescharacteristic
252C8—5022no—
253C8—5022no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—10048no—
260C22—10042no—
261C22—10042no—
262C22—5024no—
263C22#2254874012yes—
264C22C22×C22⋊D25111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—10042no—
274C4#41×D25112yes—
275C2—22no—
276C2—81no—
277C2—10044no—
278C2#3063219411yescharacteristic
279C2C2×#2254874011yescharacteristic
280C2#3063248511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#529×C251C2
socleC22×C251C22×D4
prime radical#30631820C2
hypercenter#529D251
hypocenterC251#3114
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D1004C2semidirect
2. #30632015C2semidirect
3. #30631998C2semidirect
4. #30631984C2semidirect
5. #30631899C2semidirect
6. #30631837C2semidirect
8. #30631797C2semidirect
9. #30631785C2semidirect
10. #30631744C2semidirect
11. #529×C251C2semidirect
22. #22548648C22semidirect
23. #22548646C22semidirect
24. C4⋊(C251⋊C8)C22semidirect
25. #22548639C22semidirect
26. #22548636C22semidirect
31. #22548626C22semidirect
33. #125×C251C22semidirect
34. #124×C251C22semidirect
38. C4×Q8×C251C22semidirect
40. C4×D4×C251C22semidirect
140. C251#3114semidirect
142. #529D251semidirect
158. #125C2×D251semidirect
159. #124C2×D251semidirect
163. C4×Q8C2×D251semidirect
169. C4×D4C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D10042
#306320152
#306319982
#306319842
#306318992
#306318372
#306318202
#306317972
#306317852
#306317442
#529×C2512
#3114251