Finite Group Databasebeta

← #12109025

Subgroups — #12109025

Order 1280.

1114 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65956C211yesmaximal, characteristic
2C2×#65266C211yesmaximal, characteristic
3C2×#65008C211yesmaximal, characteristic
4C5×#1872C211yesmaximal, characteristic
5#10602485C211yesmaximal, characteristic
6#10602352C211yesmaximal, characteristic
7#10601021C212yesmaximal, complemented
8#10601015C212yesmaximal, complemented
9#10600723C211yesmaximal, characteristic
10#10599698C212yesmaximal, complemented
11#10599599C212yesmaximal, complemented
12C2×C5×C22⋊Q8C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C2×Q8×Dic5C2211yescharacteristic
15C2×#4090C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C2×#4070C2211yescharacteristic
18C22×C4×Dic5C2211yescharacteristic
19#65956—24no—
20#65956C418yescomplemented
21C2×#4035—21no—
22C2×#4034C2211yescharacteristic
23C2×#4033—21no—
24C2×#4033C2211yescharacteristic
25C2×#4024—21no—
26C2×#4023—21no—
27C2×C5×#103C2212yes—
28C2×C5×C22⋊C8C2211yescharacteristic
29C2×#3967C2212yes—
30C2×#3963—21no—
31C2×#3963—21no—
32C2×C22⋊#168C2211yescharacteristic
33C2×#3948C2212yes—
34C2×#3947C2212yes—
35C2×#3946C2212yes—
36#65266—21no—
37#65266—21no—
38#65266C2212yes—
39#65266C2212yes—
40C2×#3942C2212yes—
41#65008—24no—
42C5×#340C2212yes—
43#64771C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#21866—51nomaximal, Sylow
47C22×C5×Q8C2311yescharacteristic
48C23×C4×C5—21no—
49C23×C4×C5C2311yescharacteristic
50C23×Dic5—21no—
51C23×Dic5D411yescharacteristic
52C23×Dic5D411yescharacteristic
53C22×Dic10—21no—
54C22×Dic10C2311yescharacteristic
55C22×C5×C8C2312yes—
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8C2×C414yes—
58C2×C5×C4⋊C4—21no—
59C2×C5×C4⋊C4—21no—
60C2×C5×C4⋊C4—42no—
61C2×C5×C4⋊C4C2312yes—
62C2×C5×C4⋊C4C2311yescharacteristic
63C2×C5×C22⋊C4—21no—
64C2×C22⋊Dic5—21no—
65Q8×Dic5—22no—
66Q8×Dic5C2×C414yescomplemented
67#4090—22no—
68#4090C2×C414yescomplemented
69C2×C4⋊Dic5—21no—
70C2×C4⋊Dic5—21no—
71C2×C4⋊Dic5—42no—
72C2×C4⋊Dic5C2312yes—
73C2×C4⋊Dic5C2311yescharacteristic
74#4070—22no—
75#4070C2×C414yes—
76C2×#683—21no—
77C2×#683—21no—
78C2×#683—21no—
79C2×#683—21no—
80C2×#683—21no—
81C2×C4×Dic5—21no—
82C2×C4×Dic5—21no—
83C2×C4×Dic5C2×C414yes—
84C2×C4×Dic5D411yescharacteristic
85C2×C4×Dic5D411yescharacteristic
86C2×C4×Dic5D414yes—
87C2×C4×Dic5C2311yescharacteristic
88C2×C4×Dic5C2311yescharacteristic
89C22×#168C2312yes—
90#4035—28no—
91#4034—22no—
92#4034C2×C414yescomplemented
93#4033—28no—
94#4033—22no—
95#4033C2×C414yescomplemented
96#4024—28no—
97#4023—28no—
98C5×#103—42no—
99C5×C22⋊C8—22no—
100#3967—42no—
101#3963—22no—
102#3963—22no—
103#3963—22no—
104#3963—22no—
105C22⋊#168—22no—
106#3948—42no—
107#3947—24no—
108#3946—24no—
109#3942—42no—
110C2×#533—51no—
111C2×#432—51no—
112C2×#382—51no—
113#1993—52no—
114#1960—52no—
115#1872—51no—
116#1872—51no—
117#1872—51no—
118#1872D511yescharacteristic
119#1805—52no—
120#1805—52no—
121C24×C5C2×D411yescharacteristic
122C2×C5×Q8—21no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8C22×C412yes—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—21no—
129C22×C4×C5—21no—
130C22×C4×C5—22no—
131C22×C4×C5—42no—
132C22×C4×C5—42no—
133C22×C4×C5—42no—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D411yescharacteristic
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2411yescharacteristic
138C22×Dic5—22no—
139C22×Dic5—22no—
140C22×Dic5—21no—
141C22×Dic5—21no—
142C22×Dic5—22no—
143C22×Dic5—41no—
144C22×Dic5—41no—
145C22×Dic5—42no—
146C22×Dic5—42no—
147C22×Dic5—42no—
148C22×Dic5C22⋊C414yes—
149C22×Dic5C22⋊C414yes—
150C22×Dic5C2×D411yescharacteristic
151C22×Dic5C2×D411yescharacteristic
152C22×Dic5C2×D411yescharacteristic
153C22×Dic5C2×D411yescharacteristic
154C2×Dic10—22no—
155C2×Dic10—22no—
156C2×Dic10—21no—
157C2×Dic10—24no—
158C2×Dic10—42no—
159C2×Dic10—42no—
160C2×Dic10C22×C412yes—
161C2×C5×C8—22no—
162C2×C5×C8—22no—
163C2×C5×C8—22no—
164C5×C4⋊C4—24no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—42no—
169C5×C4⋊C4C22×C412yes—
170C5×C22⋊C4—24no—
171C22⋊Dic5—24no—
172C4⋊Dic5—21no—
173C4⋊Dic5—24no—
174C4⋊Dic5—22no—
175C4⋊Dic5—22no—
176C4⋊Dic5—42no—
177C4⋊Dic5C22×C412yes—
178#683—24no—
179#683—24no—
180#683—24no—
181#683—24no—
182#683—24no—
183C4×Dic5—24no—
184C4×Dic5—24no—
185C4×Dic5C22⋊C414yes—
186C4×Dic5C22⋊C414yes—
187C4×Dic5C22×C412yes—
188C4×Dic5C22×C412yes—
189C4×Dic5C2×D412yes—
190C4×Dic5C2×D412yes—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168—22no—
194#533—58no—
195#533—104no—
196C2×C4⋊Q8—51no—
197C2×C4⋊Q8—51no—
198C2×#125—101no—
199C2×C22⋊Q8—51no—
200C2×C22⋊Q8—101no—
201C2×C22⋊Q8C2×D511yescharacteristic
202C2×C4×Q8—51no—
203C2×C4×Q8—51no—
204C2×C4×Q8—101no—
205C2×#117—101no—
206C22×C4⋊C4—51no—
207C22×C4⋊C4C2×D511yescharacteristic
208C22×C42—51no—
209#432—104no—
210C2×C4⋊C8—52no—
211C2×#103—52no—
212C2×#103—52no—
213C2×#103—52no—
214C2×#103C2×D512yes—
215C2×C22⋊C8—51no—
216C2×C22⋊C8C2×D511yescharacteristic
217C2×#97—52no—
218#382—52no—
219#382—52no—
220#382—101no—
221#382—101no—
222C2×#95—101no—
223C2×#95—101no—
224#340—52no—
225#340—52no—
226#340—52no—
227#340C2×D512yes—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5—21no—
232C23×C5C2×C22⋊C412yes—
233C23×C5#13712yes—
234C23×C5#13712yes—
235C23×C5C22×D411yescharacteristic
236C5×Q8—24no—
237C5×Q8—42no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5—42no—
250C2×C4×C5—42no—
251C2×C4×C5—84no—
252C2×C4×C5C2×C22⋊C412yes—
253C2×C4×C5C23×C411yescharacteristic
254C2×C4×C5C22×D411yescharacteristic
255C2×C4×C5H(5,𝔽2)12yes—
256C2×C4×C52−1+412yes—
257C2×Dic5—21no—
258C2×Dic5—21no—
259C2×Dic5—21no—
260C2×Dic5—21no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—21no—
264C2×Dic5—21no—
265C2×Dic5—28no—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—44no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—41no—
272C2×Dic5—41no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—84no—
277C2×Dic5C2×C22⋊C412yes—
278C2×Dic5C2×C22⋊C412yes—
279C2×Dic5C2×C22⋊C412yes—
280C2×Dic5C2×C22⋊C412yes—
281Dic10—24no—
282Dic10—48no—
283Dic10—42no—
284C5×C8—42no—
285#168—42no—
286C22×Q8—51no—
287C22×Q8—101no—
288C22×Q8C22×D511yescharacteristic
289C23×C4—21no—
290C23×C4—51no—
291C23×C4—51no—
292C23×C4—101no—
293C23×C4C22×D511yescharacteristic
294C22×C8—52no—
295C22×C8C22×D512yes—
296C4⋊Q8—54no—
297C4⋊Q8—54no—
298C4⋊Q8—102no—
299C4⋊Q8—102no—
300#125—108no—
301C22⋊Q8—22no—
302C22⋊Q8—54no—
303C22⋊Q8—108no—
304C22⋊Q8—102no—
305C22⋊Q8C4×D514yes—
306C4×Q8—54no—
307C4×Q8—54no—
308C4×Q8—102no—
309C4×Q8—108no—
310C4×Q8—102no—
311#117—108no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—51no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—101no—
318C2×C4⋊C4—101no—
319C2×C4⋊C4—101no—
320C2×C4⋊C4—101no—
321C2×C4⋊C4—101no—
322C2×C4⋊C4—101no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—202no—
325C2×C4⋊C4C22×D512yes—
326C2×C4⋊C4C22×D511yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—101no—
329C2×C42—54no—
330C2×C42—51no—
331C2×C42—51no—
332C2×C42—54no—
333C2×C42—51no—
334C2×C42—51no—
335C2×C42—101no—
336C2×C42—101no—
337C4⋊C8—104no—
338#103—42no—
339#103—202no—
340#103—202no—
341#103—202no—
342C22⋊C8—22no—
343C22⋊C8—102no—
344#97—104no—
345#95—102no—
346#95—102no—
347#95—102no—
348#95—102no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5#41914yes—
356C22×C5C22×C22⋊C411yescharacteristic
357C22×C5#53511yescharacteristic
358C22×C5#53611yescharacteristic
359C22×C5C2×#13711yescharacteristic
360C22×C5C2×#13711yescharacteristic
361C22×C5#57511yescharacteristic
362C22×C5#57511yescharacteristic
363C4×C5—22no—
364C4×C5—42no—
365C4×C5—42no—
366C4×C5—82no—
367C4×C5#52012yes—
368C4×C5#53712yes—
369Dic5—22no—
370Dic5—22no—
371Dic5—22no—
372Dic5—22no—
373Dic5—42no—
374Dic5—42no—
375Dic5—82no—
376C24D4×D511yescharacteristic
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—52no—
381C2×Q8—104no—
382C2×Q8—101no—
383C2×Q8—102no—
384C2×Q8—102no—
385C2×Q8—202no—
386C2×Q8—202no—
387C2×Q8C2×C4×D512yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—51no—
397C22×C4—51no—
398C22×C4—51no—
399C22×C4—51no—
400C22×C4—54no—
401C22×C4—54no—
402C22×C4—102no—
403C22×C4—102no—
404C22×C4—101no—
405C22×C4—101no—
406C22×C4—102no—
407C22×C4—201no—
408C22×C4—201no—
409C22×C4—202no—
410C22×C4—202no—
411C22×C4—202no—
412C22×C4C2×C4×D512yes—
413C22×C4D4×D512yes—
414C22×C4D4×D511yescharacteristic
415C22×C4C23×D511yescharacteristic
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—102no—
420C2×C8—102no—
421C2×C8—102no—
422C4⋊C4—21no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—42no—
427C4⋊C4—52no—
428C4⋊C4—104no—
429C4⋊C4—104no—
430C4⋊C4—104no—
431C4⋊C4—104no—
432C4⋊C4—104no—
433C4⋊C4—101no—
434C4⋊C4—104no—
435C4⋊C4—102no—
436C4⋊C4—102no—
437C4⋊C4—202no—
438C4⋊C4C2×C4×D512yes—
439C22⋊C4—24no—
440C22⋊C4—104no—
441C42—52no—
442C42—52no—
443C42—52no—
444C42—52no—
445C42—54no—
446C42—54no—
447C42—104no—
448C42—104no—
449C2×C5—24no—
450C2×C5#231111yescharacteristic
451C2×C5C2×#41911yescharacteristic
452C2×C5#289611yescharacteristic
453C2×C5#321511yescharacteristic
454C2×C5#321511yescharacteristic
455C2×C5#323611yescharacteristic
456C2×C5#323611yescharacteristic
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23D5×C22⋊C412yes—
462C23#406112yes—
463C23#406412yes—
464C23C2×D4×D511yescharacteristic
465Q8—24no—
466Q8—42no—
467Q8—104no—
468Q8—208no—
469Q8—202no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—52no—
484C2×C4—52no—
485C2×C4—52no—
486C2×C4—52no—
487C2×C4—84no—
488C2×C4—101no—
489C2×C4—101no—
490C2×C4—101no—
491C2×C4—101no—
492C2×C4—101no—
493C2×C4—101no—
494C2×C4—101no—
495C2×C4—101no—
496C2×C4—108no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—204no—
500C2×C4—201no—
501C2×C4—201no—
502C2×C4—201no—
503C2×C4—201no—
504C2×C4—202no—
505C2×C4—202no—
506C2×C4—202no—
507C2×C4—404no—
508C2×C4D5×C22⋊C412yes—
509C2×C4C22×C4×D511yescharacteristic
510C2×C4C2×D4×D511yescharacteristic
511C2×C4#414412yes—
512C2×C4#415012yes—
513C8—42no—
514C8—202no—
515C5#2186611yescharacteristic, complemented, Sylow
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22#6509114yes—
523C22C2×D5×C22⋊C411yescharacteristic
524C22#6592611yescharacteristic
525C22#6598211yescharacteristic
526C22C2×#406111yescharacteristic
527C22C2×#406411yescharacteristic
528C22#6610711yescharacteristic
529C22#6611111yescharacteristic
530C4—22no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—102no—
535C4—102no—
536C4—102no—
537C4—102no—
538C4—202no—
539C4—202no—
540C4—402no—
541C4#6588012yes—
542C4#6593112yes—
543C2—24no—
544C2#1060639311yescharacteristic
545C2C2×#6509111yescharacteristic
546C2#1061061211yescharacteristic
547C2#1061187111yescharacteristic
548C2#1061193311yescharacteristic
549C2#1061315111yescharacteristic
550C2#1061319411yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1872C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenter#1872D5
hypocenterC5#21866
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
7. #10601021C2semidirect
8. #10601015C2semidirect
10. #10599698C2semidirect
11. #10599599C2semidirect
20. #65956C4semidirect
66. Q8×Dic5C2×C4semidirect
68. #4090C2×C4semidirect
92. #4034C2×C4semidirect
95. #4033C2×C4semidirect
515. C5#21866semidirect

Maximal subgroups

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

subgroupindex
C2×#659562
C2×#652662
C2×#650082
C5×#18722
#106024852
#106023522
#106010212
#106010152
#106007232
#105996982
#105995992
#218665