Finite Group Databasebeta

← #12133144

Subgroups — #12133144

Order 1280.

1068 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#552C211yesmaximal, characteristic
2C2×#65332C211yesmaximal, characteristic, complemented
3#10614105C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5#10602532C212yesmaximal, complemented
6#10602490C211yesmaximal, characteristic, complemented
7#10602488C212yesmaximal, complemented
8#10602422C212yesmaximal, complemented
9#10602404C211yesmaximal, characteristic
10#10602368C211yesmaximal, characteristic, complemented
11#10602349C212yesmaximal, complemented
12C5×#552—24no—
13C5×#552—24no—
14C2×C5×#126—21no—
15C2×C5×#125—21no—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#123—21no—
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C5×C22⋊Q8C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C5×#117—21no—
23C2×C5×#117C2211yescharacteristic
24C22×C5×C4⋊C4C2211yescharacteristic
25C22×C4⋊Dic5C2211yescharacteristic
26C22×#681C2211yescharacteristic
27C2×#3967C2212yes—
28C2×#3967C2212yes—
29#65509—21no—
30#65501C2212yes—
31#65501C2211yescharacteristic
32C2×C22⋊#168C2212yes—
33C2×C22⋊#168C2211yescharacteristic
34#65332—24no—
35#65284—24no—
36#65275—21no—
37C2×#3940C2212yes—
38C2×#3940C2212yes—
39C2×#3939C2212yes—
40#65238—21no—
41#65235—21no—
42#65217C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#64744C2212yes—
47#24111—51nomaximal, Sylow
48C22×C5×Q8C2311yescharacteristic
49C22×C5×D4D411yescharacteristic
50C23×C4×C5—21no—
51C23×C4×C5C2311yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C23×Dic5—21no—
54C5×#126—28no—
55C5×#125—28no—
56C5×#125—22no—
57C5×#125—22no—
58C5×#123—28no—
59C5×C22⋊Q8—22no—
60C5×C22⋊Q8—22no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8D414yes—
63C4×C5×Q8—22no—
64C4×C5×Q8—22no—
65C4×C5×D4—22no—
66C4×C5×D4—22no—
67C5×#117—28no—
68C5×#117—22no—
69C5×#117D414yes—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—22no—
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—21no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4D411yescharacteristic
86C2×C5×C22⋊C4D411yescharacteristic
87C2×C42×C5—21no—
88C2×C42×C5—21no—
89C2×C42×C5C2311yescharacteristic
90C2×C22⋊Dic5—21no—
91C2×C22⋊Dic5—22no—
92C2×C4⋊Dic5—22no—
93C2×C4⋊Dic5—22no—
94C2×C4⋊Dic5—21no—
95C2×C4⋊Dic5C2312yes—
96C2×C4⋊Dic5C2312yes—
97C2×#683—22no—
98C2×#681—22no—
99C2×#681—24no—
100C22×#168C2312yes—
101C22×#168C2312yes—
102#3967—24no—
103#3967—24no—
104#3963—22no—
105#3963—22no—
106#3963—21no—
107C22⋊#168—22no—
108C22⋊#168—24no—
109#3940—24no—
110#3940—24no—
111#3939—24no—
112C2×#552D511yescharacteristic
113C2×#483—51no—
114C2×#428—51no—
115#2487—51no—
116#2059—52no—
117#2036—52no—
118#1894—51no—
119#1870—51no—
120#1865—52no—
121#1853—52no—
122#1825—51no—
123C24×C5C2×D411yescharacteristic
124C24×C5C2×D411yescharacteristic
125C2×C5×Q8—21no—
126C2×C5×Q8—22no—
127C2×C5×Q8—22no—
128C2×C5×Q8C2×D412yes—
129C2×C5×D4—22no—
130C2×C5×D4—22no—
131C2×C5×D4—41no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—24no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—42no—
144C22×C4×C5—42no—
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D412yes—
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2411yescharacteristic
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—42no—
155C22×Dic5—82no—
156C5×C4⋊C4—24no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4C2×D412yes—
167C5×C4⋊C4C2×D412yes—
168C5×C4⋊C4C2×D412yes—
169C5×C22⋊C4—24no—
170C5×C22⋊C4—24no—
171C5×C22⋊C4—24no—
172C5×C22⋊C4—22no—
173C5×C22⋊C4—22no—
174C42×C5—24no—
175C42×C5—24no—
176C42×C5—21no—
177C42×C5—21no—
178C22⋊Dic5—42no—
179C22⋊Dic5—82no—
180C4⋊Dic5—22no—
181C4⋊Dic5—22no—
182C4⋊Dic5—22no—
183C4⋊Dic5—22no—
184C4⋊Dic5—42no—
185#683—82no—
186#681—28no—
187C2×#168—22no—
188C2×#168—22no—
189C2×#168—22no—
190C2×#168—22no—
191C2×#168—22no—
192C2×#168#4114yes—
193C22×M16—51no—
194#552—24no—
195#552—24no—
196C2×#126—21no—
197C2×#125—21no—
198C2×#125C2×D511yescharacteristic
199C2×#123—21no—
200C2×C22⋊Q8C2×D511yescharacteristic
201C2×C22⋊Q8C2×D511yescharacteristic
202C2×C4×Q8C2×D511yescharacteristic
203C2×C4×D4C2×D511yescharacteristic
204C2×#117—21no—
205C2×#117C2×D511yescharacteristic
206C22×C4⋊C4—51no—
207C22×C4⋊C4C2×D511yescharacteristic
208#483—104no—
209#428—104no—
210C2×#107—52no—
211C2×#106—52no—
212C2×#106—52no—
213C2×#103—52no—
214C2×#103—52no—
215C2×C22⋊C8—51no—
216C2×C22⋊C8—52no—
217#400—101no—
218#399—101no—
219#389—52no—
220#386—101no—
221#385—101no—
222#380—51no—
223#380—52no—
224#340—52no—
225#340—52no—
226#340—52no—
227#336—52no—
228C23×C5—21no—
229C23×C5—22no—
230C23×C5—22no—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—21no—
234C23×C5—22no—
235C23×C5—22no—
236C23×C5#12012yes—
237C23×C5#13712yes—
238C23×C5C22×D411yescharacteristic
239C5×Q8—24no—
240C5×Q8—41no—
241C5×Q8—41no—
242C5×D4—44no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—24no—
247C2×C4×C5—21no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—21no—
256C2×C4×C5—21no—
257C2×C4×C5—22no—
258C2×C4×C5—48no—
259C2×C4×C5—42no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—42no—
268C2×C4×C5#12012yes—
269C2×C4×C5C22×D411yescharacteristic
270C2×C4×C5C22×D411yescharacteristic
271C2×C4×C5C2×#4112yes—
272C2×C4×C52−1+412yes—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—84no—
280#168—24no—
281#168—42no—
282C22×Q8C22×D511yescharacteristic
283C22×D4C22⋊D511yescharacteristic
284C23×C4—21no—
285C23×C4—101no—
286C23×C4C22×D511yescharacteristic
287C23×C4C22×D511yescharacteristic
288C2×M16—104no—
289C2×M16—102no—
290C22×C8—52no—
291C22×C8—52no—
292#126—28no—
293#125—28no—
294#125—22no—
295#125—22no—
296#123—28no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8C22⋊D514yes—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×D4—22no—
304C4×D4—22no—
305#117—28no—
306#117—22no—
307#117C22⋊D514yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—52no—
318C2×C4⋊C4—102no—
319C2×C4⋊C4—102no—
320C2×C4⋊C4—101no—
321C2×C4⋊C4—102no—
322C2×C4⋊C4C22⋊D511yescharacteristic
323C2×C4⋊C4C22×D511yescharacteristic
324C2×C4⋊C4C22×D511yescharacteristic
325C2×C4⋊C4C22×D511yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—101no—
330C2×C22⋊C4—102no—
331C2×C22⋊C4C22⋊D511yescharacteristic
332C2×C22⋊C4C22⋊D511yescharacteristic
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D511yescharacteristic
336#107—104no—
337#106—104no—
338#106—104no—
339#103—104no—
340#103—104no—
341C22⋊C8—104no—
342C22⋊C8—102no—
343#95—102no—
344#95—101no—
345#95—102no—
346C22×C5—24no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5—42no—
352C22×C5—42no—
353C22×C5—42no—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5C2×#12011yescharacteristic
357C22×C5#54712yes—
358C22×C5C2×#13711yescharacteristic
359C22×C5#57511yescharacteristic
360C22×C5#57712yes—
361C4×C5—22no—
362C4×C5—41no—
363C4×C5—42no—
364C4×C5—41no—
365C4×C5—42no—
366C4×C5—42no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5#52511yescharacteristic
370C4×C5#53611yescharacteristic
371C4×C5#54712yes—
372Dic5—82no—
373Dic5—82no—
374C24C2×C22⋊D511yescharacteristic
375C24C2×C22⋊D511yescharacteristic
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8C2×C22⋊D512yes—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—41no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—102no—
397C22×C4—102no—
398C22×C4—202no—
399C22×C4—402no—
400C22×C4C2×C22⋊D511yescharacteristic
401C22×C4C2×C22⋊D511yescharacteristic
402C22×C4C2×C22⋊D511yescharacteristic
403C22×C4C2×C22⋊D511yescharacteristic
404C22×C4C2×C22⋊D512yes—
405C22×C4C2×C22⋊D512yes—
406C22×C4C23×D511yescharacteristic
407M16—108no—
408C2×C8—54no—
409C2×C8—102no—
410C2×C8—102no—
411C2×C8—102no—
412C2×C8—102no—
413C2×C8—102no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—102no—
425C4⋊C4—102no—
426C4⋊C4—102no—
427C4⋊C4—102no—
428C4⋊C4—202no—
429C4⋊C4—402no—
430C4⋊C4C2×C22⋊D512yes—
431C4⋊C4C2×C22⋊D512yes—
432C4⋊C4C2×C22⋊D512yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—202no—
439C22⋊C4—402no—
440C42—24no—
441C42—24no—
442C42—21no—
443C42—21no—
444C2×C5—42no—
445C2×C5—42no—
446C2×C5#240611yescharacteristic
447C2×C5#300711yescharacteristic
448C2×C5#301011yescharacteristic
449C2×C5#334412yes—
450C2×C5#335112yes—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23#409712yes—
460C23#409912yes—
461C23C22×C22⋊D511yescharacteristic
462Q8—24no—
463Q8—41no—
464Q8—41no—
465D4—44no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—48no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—202no—
492C2×C4—202no—
493C2×C4—202no—
494C2×C4—202no—
495C2×C4—202no—
496C2×C4—202no—
497C2×C4—404no—
498C2×C4#409912yes—
499C2×C4D5×#4112yes—
500C2×C4#415012yes—
501C2×C4C22×C22⋊D511yescharacteristic
502C2×C4C22×C22⋊D511yescharacteristic
503C8—104no—
504C8—202no—
505C5#2411111yescharacteristic, complemented, Sylow
506C22—22no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—22no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22#6615611yescharacteristic
517C22C2×#409711yescharacteristic
518C22#6615812yes—
519C22#6617112yes—
520C22C2×#409911yescharacteristic
521C4—22no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—402no—
530C4—402no—
531C4#6615812yes—
532C4#6615911yescharacteristic
533C4#6616211yescharacteristic
534C2—42no—
535C2—42no—
536C2#1061532412yes—
537C2#1061532712yes—
538C2#1061581511yescharacteristic
539C2#1061582111yescharacteristic
540C2#1061584411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#552C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×#552D5
hypocenterC5#24111
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#65332C2semidirect
4. C2×#65284C2semidirect
5. #10602532C2semidirect
6. #10602490C2semidirect
7. #10602488C2semidirect
8. #10602422C2semidirect
10. #10602368C2semidirect
11. #10602349C2semidirect
505. C5#24111semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5522
C2×#653322
#106141052
C2×#652842
#106025322
#106024902
#106024882
#106024222
#106024042
#106023682
#106023492
#241115