Finite Group Databasebeta

← #12071048

Subgroups — #12071048

Order 1280.

652 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2990C211yesmaximal, characteristic, complemented
2C4×#4096C211yesmaximal, characteristic
3#10611024C211yesmaximal, characteristic, complemented
4#10602434C211yesmaximal, characteristic, complemented
5#10601746C212yesmaximal, complemented
6#10601745C212yesmaximal, complemented
7#10601635C212yesmaximal, complemented
8#10601633C212yesmaximal, complemented
9#10601534C211yesmaximal, characteristic, complemented
10#10601290C211yesmaximal, characteristic, complemented
11#10601283C211yesmaximal, characteristic, complemented
12C4×C5×#41C2211yescharacteristic
13C2×#4096C411yescharacteristic
14C4×#709C2211yescharacteristic
15C5×#446C2212yes—
16C5×#435—22no—
17C5×C8×D4C2212yes—
18C5×#432—21no—
19C5×#431C2211yescharacteristic
20C5×#423C2211yescharacteristic
21C2×C5×C4⋊C8—21no—
22#65522C411yescharacteristic
23C4×#689C411yescharacteristic
24C4×#689C2211yescharacteristic
25C4×#688C411yescharacteristic
26C4×#688C2211yescharacteristic
27C4×#687C411yescharacteristic
28C4×#687C2211yescharacteristic
29C4×#686C411yescharacteristic
30C4×#686C2211yescharacteristic
31#65291C411yescharacteristic
32#65287C411yescharacteristic
33#65213C2211yescharacteristic
34C4×#681C2211yescharacteristic
35C2×C4×#168C2211yescharacteristic
36#65124—21no—
37D5×C4⋊C8—21no—
38#65017—22no—
39C8×D20C2212yes—
40#64968C2212yes—
41Q8⋊#168C2212yes—
42D4⋊#168C2212yes—
43#64704C2212yes—
44#64703C2212yes—
45#64700C2212yes—
46#64700C2212yes—
47#64675C2212yes—
48#64674C2212yes—
49#18065—51nomaximal, Sylow
50C2×C5×#41C2×C411yescharacteristic
51C2×#709C2×C411yescharacteristic
52C2×C5×M16—21no—
53C22×C5×C8—22no—
54C4×C5×Q8C2×C411yescharacteristic
55C4×C5×Q8C2311yescharacteristic
56C4×C5×D4—21no—
57C4×C5×D4—21no—
58C4×C5×D4C2×C411yescharacteristic
59C4×C5×D4C2311yescharacteristic
60C5×#117—21no—
61C5×#117C2×C411yescharacteristic
62C2×C42×C5—21no—
63C2×C42×C5C2311yescharacteristic
64#4096—24no—
65C2×#689C2×C411yescharacteristic
66C2×#688C2×C411yescharacteristic
67C2×#687C2×C411yescharacteristic
68C2×#686C2×C411yescharacteristic
69C4×C22⋊D5—21no—
70C4×C22⋊D5—21no—
71#4072C2×C411yescharacteristic
72C2×#681—21no—
73C2×#681—21no—
74C22×#168C2×C411yescharacteristic
75C22×#168C2×C411yescharacteristic
76C2×C8×D5—22no—
77C4×D20C2×C411yescharacteristic
78C4×D20C2311yescharacteristic
79#4018—21no—
80C42×D5—21no—
81C4×Dic10C2×C411yescharacteristic
82C4×Dic10C2311yescharacteristic
83C5×C4⋊C8—22no—
84C5×C4⋊C8—21no—
85C5×C4⋊C8D411yescharacteristic
86C5×C4⋊C8D411yescharacteristic
87C5×C4⋊C8C2312yes—
88C5×C22⋊C8—22no—
89C5×C22⋊C8D412yes—
90C5×#97—22no—
91C4×C5×C8C2312yes—
92#3967C2×C411yescharacteristic
93#3967C2×C411yescharacteristic
94#3964C2×C411yescharacteristic
95#3964C2×C411yescharacteristic
96#3952—22no—
97#3947—22no—
98#3946—22no—
99#3942C2×C411yescharacteristic
100#3942C2×C411yescharacteristic
101#3941C2×C411yescharacteristic
102#3941C2×C411yescharacteristic
103#3940C2×C411yescharacteristic
104#3940C2×C411yescharacteristic
105#3939C2×C411yescharacteristic
106#3939C2×C411yescharacteristic
107C4⋊#168—21no—
108C4⋊#168C2312yes—
109#3935—21no—
110C4×#168C2×C411yescharacteristic
111C4×#168C2×C411yescharacteristic
112C4×#168C2312yes—
113C4×#168C2311yescharacteristic
114C4×#168C2311yescharacteristic
115#2990—51no—
116#2990D511yescharacteristic, complemented
117C4×#135—51no—
118#1593—52no—
119#1582—52no—
120#1582—52no—
121#1580—52no—
122#1566—51no—
123#1489—51no—
124#1489—51no—
125#1451—51no—
126C5×#41—22no—
127C5×#41—41no—
128C2×C5×Q8C22×C411yescharacteristic
129C2×C5×D4—21no—
130C2×C5×D4C22×C411yescharacteristic
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5C22×C411yescharacteristic
135C22×C4×C5C2×D411yescharacteristic
136C22×C4×C5#4111yescharacteristic
137C2×C22⋊D5—21no—
138#709—22no—
139#709—41no—
140C2×D20C22×C411yescharacteristic
141C2×C4×D5—21no—
142C2×C4×D5—21no—
143C2×C4×D5—21no—
144C2×Dic10C22×C411yescharacteristic
145C5×M16—81no—
146C2×C5×C8—22no—
147C2×C5×C8—82no—
148C2×C5×C8C2×D412yes—
149C2×C5×C8C2×D412yes—
150C5×C4⋊C4—21no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4C22×C411yescharacteristic
153C5×C4⋊C4C22×C411yescharacteristic
154C5×C22⋊C4—21no—
155C5×C22⋊C4—21no—
156C5×C22⋊C4—21no—
157C42×C5—21no—
158C42×C5—21no—
159C42×C5—21no—
160C42×C5C22×C411yescharacteristic
161C42×C5C2×D411yescharacteristic
162C42×C5#4111yescharacteristic
163C42×C5C2411yescharacteristic
164C22⋊Dic5—21no—
165#689—22no—
166#688—22no—
167#687—22no—
168#686M1614yescomplemented
169#685—21no—
170#685—21no—
171C4⋊Dic5C22×C411yescharacteristic
172C4⋊Dic5C22×C411yescharacteristic
173#683—21no—
174#683—21no—
175C4×Dic5—21no—
176C4×Dic5—21no—
177C4×Dic5—21no—
178#681—42no—
179#681—81no—
180C2×#168—22no—
181C2×#168—22no—
182C2×#168—22no—
183C2×#168—41no—
184C2×#168C22×C411yescharacteristic
185C2×#168C22×C411yescharacteristic
186C2×#168C22×C411yescharacteristic
187C2×#168C22×C411yescharacteristic
188C8×D5—82no—
189C2×#135—51no—
190C4×#41—51no—
191C4×#41C2×D511yescharacteristic
192#446—52no—
193#446C2×D512yes—
194C4×Q16—51no—
195C4×Q16—51no—
196C4×SD16—51no—
197C4×SD16—51no—
198C4×SD16—51no—
199C4×SD16—51no—
200C4×D8—51no—
201C4×D8—51no—
202#435—22no—
203#435—102no—
204C8×D4—52no—
205C8×D4C2×D512yes—
206#432—21no—
207#432—101no—
208#431C2×D511yescharacteristic
209#427—51no—
210#423—51no—
211#423C2×D511yescharacteristic
212C2×C4⋊C8—21no—
213C2×C4⋊C8—101no—
214#416—51no—
215#416—51no—
216C4×M16—51no—
217C2×C4×C8—51no—
218#334—52no—
219#334—52no—
220Q8⋊C8—52no—
221Q8⋊C8—52no—
222D4⋊C8—52no—
223D4⋊C8—52no—
224#322—52no—
225#322—52no—
226C23×C5—21no—
227C23×C5C4×D411yescharacteristic
228C22×D5—21no—
229C5×Q8—21no—
230C5×Q8—21no—
231C5×D4—21no—
232C5×D4—42no—
233C5×D4C2×M1612yes—
234C2×C4×C5—21no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—22no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—42no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—42no—
246C2×C4×C5C4×D411yescharacteristic
247C2×C4×C5C4×D411yescharacteristic
248C2×C4×C5C4×D411yescharacteristic
249C2×C4×C5C23×C411yescharacteristic
250C2×C4×C5C22×D411yescharacteristic
251C2×C4×C5C2×#4111yescharacteristic
252C22⋊D5—42no—
253C2×Dic5—21no—
254C2×Dic5—21no—
255C2×Dic5—21no—
256D20—21no—
257D20C2×M1612yes—
258C4×D5—42no—
259C4×D5—41no—
260C4×D5—41no—
261C4×D5—42no—
262Dic10—21no—
263Dic10—21no—
264C5×C8—22no—
265C5×C8—22no—
266C5×C8—42no—
267#168—21no—
268#168—21no—
269#168—21no—
270#168—42no—
271#168—42no—
272#168C2×M1612yes—
273C2×#41—51no—
274C2×#41C4×D511yescharacteristic
275#135—104no—
276C2×Q16—51no—
277C2×SD16—51no—
278C2×SD16—51no—
279C2×D8—51no—
280C2×M16—21no—
281C2×M16—101no—
282C2×M16—101no—
283C22×C8—22no—
284C22×C8—51no—
285C22×C8—51no—
286C22×C8—102no—
287C4×Q8—51no—
288C4×Q8—51no—
289C4×Q8C4×D511yescharacteristic
290C4×Q8C22×D511yescharacteristic
291C4×D4—21no—
292C4×D4—21no—
293C4×D4—51no—
294C4×D4—51no—
295C4×D4—101no—
296C4×D4—101no—
297C4×D4C4×D511yescharacteristic
298C4×D4C22×D511yescharacteristic
299#117—21no—
300#117—51no—
301#117—101no—
302#117C4×D511yescharacteristic
303C2×C42—21no—
304C2×C42—101no—
305C2×C42C22×D511yescharacteristic
306#107—51no—
307#107—51no—
308#106—51no—
309#106—51no—
310C4⋊C8—22no—
311C4⋊C8—21no—
312C4⋊C8—52no—
313C4⋊C8—102no—
314C4⋊C8—101no—
315C4⋊C8C22⋊D511yescharacteristic
316C4⋊C8C22⋊D511yescharacteristic
317C4⋊C8C22×D512yes—
318#103—51no—
319#103—51no—
320#103—51no—
321#103—51no—
322#102—51no—
323#102—51no—
324#102—51no—
325#102—51no—
326C22⋊C8—22no—
327C22⋊C8—102no—
328C22⋊C8C22⋊D512yes—
329#97—22no—
330#97—102no—
331#97—101no—
332C4×C8—51no—
333C4×C8—51no—
334C4×C8—51no—
335C4×C8—51no—
336C4×C8—52no—
337C4×C8C22×D512yes—
338C22×C5—21no—
339C22×C5—22no—
340C22×C5—21no—
341C22×C5—21no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5C2×C4×D411yescharacteristic
345C2×D5—22no—
346C2×D5—41no—
347C2×D5—41no—
348C4×C5—21no—
349C4×C5—21no—
350C4×C5—21no—
351C4×C5—41no—
352C4×C5—41no—
353C4×C5—41no—
354C4×C5#44412yes—
355C4×C5C22×M1611yescharacteristic
356C4×C5#56811yescharacteristic
357C4×C5#57711yescharacteristic
358C4×C5#57711yescharacteristic
359Dic5—41no—
360Dic5—41no—
361Dic5—41no—
362#41—22no—
363#41—41no—
364#41—102no—
365#41—201no—
366C2×Q8—51no—
367C2×Q8C2×C4×D511yescharacteristic
368C2×D4—21no—
369C2×D4—51no—
370C2×D4—101no—
371C2×D4C2×C4×D511yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—101no—
376C22×C4—101no—
377C22×C4—101no—
378C22×C4C2×C4×D511yescharacteristic
379C22×C4#70911yescharacteristic
380C22×C4C2×C22⋊D511yescharacteristic
381Q16—102no—
382SD16—102no—
383SD16—102no—
384D8—54no—
385M16—81no—
386M16—202no—
387M16—401no—
388C2×C8—22no—
389C2×C8—51no—
390C2×C8—51no—
391C2×C8—51no—
392C2×C8—51no—
393C2×C8—82no—
394C2×C8—102no—
395C2×C8—102no—
396C2×C8—102no—
397C2×C8—201no—
398C2×C8—402no—
399C2×C8C2×C22⋊D512yes—
400C2×C8C2×C22⋊D512yes—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—51no—
404C4⋊C4—51no—
405C4⋊C4—101no—
406C4⋊C4—101no—
407C4⋊C4C2×C4×D511yescharacteristic
408C4⋊C4C2×C4×D511yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—101no—
413C22⋊C4—101no—
414C22⋊C4—101no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—101no—
419C42—101no—
420C42—101no—
421C42C2×C4×D511yescharacteristic
422C42#70911yescharacteristic
423C42C2×C22⋊D511yescharacteristic
424C42C23×D511yescharacteristic
425C2×C5—41no—
426C2×C5—42no—
427C2×C5D4×M1611yescharacteristic
428C2×C5#295211yescharacteristic
429C2×C5C2×#44411yescharacteristic
430D5—42no—
431C23—21no—
432C23—101no—
433C23C4×C22⋊D511yescharacteristic
434Q8—21no—
435Q8—21no—
436Q8—101no—
437Q8—101no—
438D4—21no—
439D4—42no—
440D4—52no—
441D4—101no—
442D4—202no—
443D4D5×M1612yes—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—101no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—202no—
460C2×C4—201no—
461C2×C4—201no—
462C2×C4—202no—
463C2×C4C4×C22⋊D511yescharacteristic
464C2×C4C4×C22⋊D511yescharacteristic
465C2×C4C4×C22⋊D511yescharacteristic
466C2×C4C22×C4×D511yescharacteristic
467C2×C4C2×#70911yescharacteristic
468C2×C4C22×C22⋊D511yescharacteristic
469C8—22no—
470C8—22no—
471C8—42no—
472C8—52no—
473C8—101no—
474C8—101no—
475C8—101no—
476C8—202no—
477C8—202no—
478C5#1806511yescharacteristic, complemented, Sylow
479C22—21no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—102no—
486C22—201no—
487C22—201no—
488C22C2×C4×C22⋊D511yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—201no—
496C4—201no—
497C4—201no—
498C4#6543412yes—
499C4C2×D5×M1611yescharacteristic
500C4#6607911yescharacteristic
501C4#6617011yescharacteristic
502C4#6617111yescharacteristic
503C2—41no—
504C2—42no—
505C2—202no—
506C2#1061428911yescharacteristic
507C2M16×C22⋊D511yescharacteristic
508C2C2×#6543411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#2990C2
socleC22×C5C2×C4×D4
prime radicalC2×#4096C4
cosocleC42×C5C24
hypercenter#2990D5
hypocenterC5#18065
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2990

Complements

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

NH ≅ G/Ntype
1. C5×#2990C2semidirect
3. #10611024C2semidirect
4. #10602434C2semidirect
5. #10601746C2semidirect
6. #10601745C2semidirect
7. #10601635C2semidirect
8. #10601633C2semidirect
9. #10601534C2semidirect
10. #10601290C2semidirect
11. #10601283C2semidirect
116. #2990D5semidirect
168. #686M16semidirect
478. C5#18065semidirect

Maximal subgroups

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

subgroupindex
C5×#29902
C4×#40962
#106110242
#106024342
#106017462
#106017452
#106016352
#106016332
#106015342
#106012902
#106012832
#180655