Finite Group Databasebeta

← #12069492

Subgroups — #12069492

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#436C211yesmaximal, characteristic, complemented
2#10614039C211yesmaximal, characteristic
3#10610189C211yesmaximal, characteristic, complemented
4#10602391C211yesmaximal, characteristic, complemented
5#10601655C214yesmaximal, complemented
6#10601652C214yesmaximal, complemented
7#10601219C211yesmaximal, characteristic, complemented
8#10601215C211yesmaximal, characteristic, complemented
9#10600517C211yesmaximal, characteristic, complemented
10C22×C5×M16—21no—
11C2×C4×C5×D4C2211yescharacteristic
12C2×#4047—21no—
13#65811C2211yescharacteristic
14C5×#436—21no—
15C5×#436—21no—
16C5×#436C2212yes—
17C5×#436C2212yes—
18C2×C5×C4⋊C8C2211yescharacteristic
19C2×C5×C22⋊C8—21no—
20C2×C4×C5×C8C2211yescharacteristic
21#65506C412yes—
22#65399—21no—
23#65304C412yes—
24C4×#686C2214yes—
25#65296C412yes—
26#65256C412yes—
27#65214C2211yescharacteristic
28#65214C2211yescharacteristic
29#65213C2211yescharacteristic
30#64977—21no—
31#64977C2212yescomplemented
32C8×D20—21no—
33C8×D20C2212yescomplemented
34D4⋊#168C2214yes—
35#64703C2214yescomplemented
36#64699C2214yescomplemented
37#64675C2212yescomplemented
38C8×#168C2212yescomplemented
39#17928—51nomaximal, Sylow
40C22×C5×D4C2×C411yescharacteristic
41C23×C4×C5—21no—
42C2×#709—21no—
43C2×C5×M16—22no—
44C2×C5×M16—22no—
45C2×C5×M16—44no—
46C22×C5×C8—21no—
47C22×C5×C8D411yescharacteristic
48C22×C5×C8D411yescharacteristic
49C4×C5×D4—21no—
50C4×C5×D4—21no—
51C4×C5×D4C2×C412yes—
52C4×C5×D4C2312yes—
53C2×C5×C4⋊C4C2×C411yescharacteristic
54C2×C5×C22⋊C4—21no—
55C2×C42×C5C2311yescharacteristic
56C2×#686C2×C414yes—
57#4076—21no—
58#4076C2×C412yes—
59C2×#681—21no—
60#4047—44no—
61C2×#676—22no—
62C2×C8×D5—22no—
63#4021—21no—
64#4020C2×C411yescharacteristic
65C4×D20—21no—
66C4×D20C2312yes—
67#4015C2×C411yescharacteristic
68C5×C4⋊C8—21no—
69C5×C4⋊C8C2312yes—
70C5×C22⋊C8—22no—
71C5×C22⋊C8—22no—
72C4×C5×C8D411yescharacteristic
73C4×C5×C8D411yescharacteristic
74C4×C5×C8C2311yescharacteristic
75C4×C5×C8C2311yescharacteristic
76#3964C2×C414yes—
77C22⋊#168—21no—
78C22⋊#168C2×C412yes—
79#3952—22no—
80#3952—22no—
81#3941C2×C414yes—
82#3939C2×C414yes—
83C4⋊#168—21no—
84C4⋊#168—21no—
85C4⋊#168C2×C412yes—
86C4⋊#168C2312yes—
87C4×#168C2312yes—
88C4×#168C2312yes—
89#3100—51no—
90#2931—51no—
91C2×#436D511yescharacteristic, complemented
92#1589—54no—
93#1581—54no—
94#1571—51no—
95#1486—51no—
96#1480—51no—
97#1450—51no—
98C24×C5—21no—
99C2×C5×D4—21no—
100C2×C5×D4—42no—
101C2×C5×D4C22×C412yes—
102C22×C4×C5—21no—
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—42no—
106C22×C4×C5—42no—
107C22×C4×C5C22×C411yescharacteristic
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5#4111yescharacteristic
110C2×C22⋊D5—22no—
111#709—42no—
112#709—42no—
113C2×D20—21no—
114C2×D20—21no—
115C2×D20C22×C412yes—
116C2×C4×D5—22no—
117C2×Dic10—21no—
118C5×M16—44no—
119C5×M16—44no—
120C2×C5×C8—22no—
121C2×C5×C8—22no—
122C2×C5×C8—42no—
123C2×C5×C8D814yescomplemented
124C2×C5×C8C2×D411yescharacteristic
125C2×C5×C8C2×D411yescharacteristic
126C2×C5×C8C2×D411yescharacteristic
127C2×C5×C8C2×D411yescharacteristic
128C5×C4⋊C4—21no—
129C5×C4⋊C4C22×C412yes—
130C5×C22⋊C4—22no—
131C5×C22⋊C4—22no—
132C42×C5C22×C411yescharacteristic
133C42×C5C2×D411yescharacteristic
134C42×C5#4111yescharacteristic
135C42×C5C2411yescharacteristic
136#686—44no—
137#685—22no—
138#685—22no—
139C4⋊Dic5—21no—
140C4⋊Dic5C22×C412yes—
141#681—42no—
142C2×#168—22no—
143C2×#168—22no—
144C2×#168C22×C414yes—
145#676—44no—
146C8×D5—44no—
147C2×#131—101no—
148C22×M16—21no—
149#532—51no—
150C2×C4×D4C2×D511yescharacteristic
151#480—52no—
152#474—52no—
153#459—52no—
154#447—52no—
155C4×D8—54no—
156#436—21no—
157#436—21no—
158#436—52no—
159#436—101no—
160#436C2×D512yes—
161#436C2×D512yes—
162C8×D4—52no—
163C8×D4—101no—
164#431—51no—
165#431—51no—
166#423—51no—
167C2×C4⋊C8C2×D511yescharacteristic
168#408—101no—
169C2×C22⋊C8—21no—
170C2×C4×C8C2×D511yescharacteristic
171#335—54no—
172D4⋊C8—54no—
173D4⋊C8—54no—
174#322—52no—
175C82—52no—
176C23×C5—22no—
177C23×C5—42no—
178C23×C5—42no—
179C23×C5—42no—
180C23×C5C4×D411yescharacteristic
181C22×D5—22no—
182C5×D4—22no—
183C5×D4—22no—
184C5×D4—42no—
185C2×C4×C5—22no—
186C2×C4×C5—22no—
187C2×C4×C5—41no—
188C2×C4×C5—42no—
189C2×C4×C5—42no—
190C2×C4×C5—42no—
191C2×C4×C5—44no—
192C2×C4×C5C4×D411yescharacteristic
193C2×C4×C5C4×D411yescharacteristic
194C2×C4×C5C4×D411yescharacteristic
195C2×C4×C5C2×D812yes—
196C2×C4×C5#13512yes—
197C2×C4×C5C23×C411yescharacteristic
198C2×C4×C5C22×D411yescharacteristic
199C2×C4×C5C2×#4111yescharacteristic
200C22⋊D5—44no—
201C2×Dic5—22no—
202D20—24no—
203D20—42no—
204D20—42no—
205C4×D5—42no—
206C4×D5—42no—
207Dic10—42no—
208C5×C8—21no—
209C5×C8—21no—
210C5×C8—42no—
211C5×C8C2×D812yes—
212C5×C8C2×D812yes—
213#168—24no—
214#168—42no—
215#168—42no—
216C2×#41—101no—
217C22×D4C4×D511yescharacteristic
218C23×C4—21no—
219C2×D8—54no—
220#131—204no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—44no—
224C2×M16—101no—
225C2×M16—102no—
226C22×C8—21no—
227C22×C8—102no—
228C22×C8C22⋊D511yescharacteristic
229C22×C8C22⋊D511yescharacteristic
230C4⋊Q8—51no—
231#127—51no—
232#124—101no—
233#121—52no—
234#121—101no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—52no—
238C4×D4—101no—
239C4×D4C4×D512yes—
240C4×D4C22×D512yes—
241C2×C4⋊C4C4×D511yescharacteristic
242C2×C22⋊C4—21no—
243C2×C42C22×D511yescharacteristic
244#107—54no—
245C4⋊C8—21no—
246C4⋊C8—52no—
247C4⋊C8—52no—
248C4⋊C8—101no—
249C4⋊C8—101no—
250C4⋊C8C22×D512yes—
251#102—54no—
252#102—54no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—52no—
256C22⋊C8—101no—
257C22⋊C8—102no—
258C22⋊C8—102no—
259C4×C8—52no—
260C4×C8—52no—
261C4×C8C22⋊D511yescharacteristic
262C4×C8C22⋊D511yescharacteristic
263C4×C8C22×D511yescharacteristic
264C4×C8C22×D511yescharacteristic
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—84no—
271C22×C5C4×D812yes—
272C22×C5C2×C4×D411yescharacteristic
273C2×D5—42no—
274C2×D5—42no—
275C2×D5—42no—
276C4×C5—21no—
277C4×C5—21no—
278C4×C5—21no—
279C4×C5—42no—
280C4×C5C4×D812yes—
281C4×C5#56812yes—
282C4×C5C22×D811yescharacteristic
283C4×C5C2×#13511yescharacteristic
284Dic5—42no—
285C24—21no—
286#41—202no—
287#41—202no—
288C2×Q8—101no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—52no—
292C2×D4—102no—
293C2×D4—101no—
294C2×D4—101no—
295C2×D4C2×C4×D512yes—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—102no—
302C22×C4C2×C4×D511yescharacteristic
303C22×C4#70911yescharacteristic
304C22×C4C2×C22⋊D511yescharacteristic
305D8—204no—
306M16—44no—
307M16—44no—
308M16—202no—
309M16—204no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—42no—
313C2×C8—54no—
314C2×C8—102no—
315C2×C8—102no—
316C2×C8—204no—
317C2×C8#68614yescomplemented
318C2×C8C2×C22⋊D511yescharacteristic
319C2×C8C2×C22⋊D511yescharacteristic
320C2×C8C2×C22⋊D511yescharacteristic
321C2×C8C2×C22⋊D511yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—52no—
324C4⋊C4—101no—
325C4⋊C4C2×C4×D512yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—102no—
329C22⋊C4—102no—
330C42C2×C4×D511yescharacteristic
331C42#70911yescharacteristic
332C42C2×C22⋊D511yescharacteristic
333C42C23×D511yescharacteristic
334C2×C5—22no—
335C2×C5—82no—
336C2×C5#293011yescharacteristic
337C2×C5C2×C4×D811yescharacteristic
338C2×C5#295511yescharacteristic
339D5—82no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—102no—
345C23C4×C22⋊D511yescharacteristic
346Q8—202no—
347D4—22no—
348D4—22no—
349D4—42no—
350D4—104no—
351D4—202no—
352D4—204no—
353D4—202no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—102no—
362C2×C4—202no—
363C2×C4—202no—
364C2×C4C4×C22⋊D511yescharacteristic
365C2×C4C4×C22⋊D511yescharacteristic
366C2×C4C4×C22⋊D511yescharacteristic
367C2×C4C2×#68612yes—
368C2×C4#409612yes—
369C2×C4C22×C4×D511yescharacteristic
370C2×C4C2×#70911yescharacteristic
371C2×C4C22×C22⋊D511yescharacteristic
372C8—21no—
373C8—21no—
374C8—42no—
375C8—104no—
376C8—202no—
377C8—202no—
378C8C2×#68612yes—
379C8C2×#68612yes—
380C5#1792811yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—84no—
387C22—202no—
388C22—202no—
389C22—202no—
390C22C4×#68612yes—
391C22C2×C4×C22⋊D511yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—202no—
397C4C4×#68612yes—
398C4#6607912yes—
399C4C22×#68611yescharacteristic
400C4C2×#409611yescharacteristic
401C2—22no—
402C2—82no—
403C2—402no—
404C2C2×C4×#68611yescharacteristic
405C2#1061548711yescharacteristic
406C2#1061552311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC2×C5×#436C2
socleC22×C5C2×C4×D4
prime radical#10614039C2
cosocleC42×C5C24
hypercenterC2×#436D5
hypocenterC5#17928
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×#436

Complements

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

NH ≅ G/Ntype
1. C2×C5×#436C2semidirect
3. #10610189C2semidirect
4. #10602391C2semidirect
5. #10601655C2semidirect
6. #10601652C2semidirect
7. #10601219C2semidirect
8. #10601215C2semidirect
9. #10600517C2semidirect
31. #64977C22semidirect
33. C8×D20C22semidirect
35. #64703C22semidirect
36. #64699C22semidirect
37. #64675C22semidirect
38. C8×#168C22semidirect
91. C2×#436D5semidirect
123. C2×C5×C8D8semidirect
317. C2×C8#686semidirect
380. C5#17928semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4362
#106140392
#106101892
#106023912
#106016552
#106016522
#106012192
#106012152
#106005172
#179285