Finite Group Databasebeta

← #12098829

Subgroups — #12098829

Order 1280.

1462 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2C2×C5×#424C211yesmaximal, characteristic, complemented
3C2×#65422C211yesmaximal, characteristic
4C2×#65402C211yesmaximal, characteristic
5#10613964C211yesmaximal, characteristic, complemented
6#10602778C211yesmaximal, characteristic, complemented
7C2×#64771C211yesmaximal, characteristic, complemented
8#10601279C212yesmaximal, complemented
9#10601278C212yesmaximal, complemented
10#10601273C212yesmaximal, complemented
11#10601272C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C23×C5×C8C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16C2×#4072C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19#65857—24no—
20#65857C418yescomplemented
21C2×#4038—21no—
22C2×D5×C4⋊C4—21no—
23C2×#4027—22no—
24C2×C4×D20C2212yes—
25C2×C4×Dic10C2212yes—
26C5×#424—24no—
27C2×C5×C4⋊C8C2212yes—
28C2×C5×C4⋊C8C2212yes—
29#65422—24no—
30#65402—24no—
31C2×#3953C2211yescharacteristic
32C2×#3953C2212yes—
33C2×#3953C2211yescharacteristic
34C2×#3948C2211yescharacteristic
35C2×#3948C2212yes—
36C2×#3948C2211yescharacteristic
37#65274—21no—
38#65274—21no—
39#65239—22no—
40#65225C2212yes—
41#65224C2212yes—
42#64771C2212yes—
43#64771C2212yes—
44#64771C2212yes—
45#64771C2212yes—
46#20652—51nomaximal, Sylow
47C23×C4×C5C2311yescharacteristic
48C22×C22⋊D5—21no—
49C23×Dic5—21no—
50C23×Dic5—21no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709C2×C414yes—
55C2×#709D414yes—
56C2×#709Q814yes—
57C22×D20C2311yescharacteristic
58C22×C4×D5—21no—
59C22×C4×D5—22no—
60C22×Dic10C2311yescharacteristic
61C2×C5×M16—22no—
62C2×C5×M16—22no—
63C2×C5×M16—22no—
64C22×C5×C8—22no—
65C22×C5×C8—22no—
66C22×C5×C8—22no—
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2312yes—
70C5×#117C2×C414yes—
71C5×#117D414yes—
72C2×C5×C4⋊C4D411yescharacteristic
73C2×C5×C4⋊C4D411yescharacteristic
74C2×C5×C22⋊C4D412yes—
75C2×C42×C5C2312yes—
76C2×C22⋊Dic5—21no—
77C2×#685—22no—
78#4072—22no—
79#4072—22no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—22no—
83C2×C4⋊Dic5—22no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—22no—
86C2×C4⋊Dic5—22no—
87C2×C4⋊Dic5C2×C414yes—
88C2×C4⋊Dic5C2312yes—
89C2×C4⋊Dic5C2311yescharacteristic
90C2×C4⋊Dic5C2311yescharacteristic
91C2×C4⋊Dic5C2312yes—
92C2×#683—22no—
93C2×C4×Dic5—22no—
94C2×C4×Dic5—21no—
95#4038—28no—
96D5×C4⋊C4—28no—
97#4027—216no—
98C4×D20—24no—
99C4×D20C2×C418yescomplemented
100C4×Dic10—24no—
101C4×Dic10C2×C418yescomplemented
102C5×C4⋊C8—24no—
103C5×C4⋊C8—24no—
104#3963—22no—
105#3963—22no—
106#3953—22no—
107#3953—22no—
108#3953—24no—
109#3948—22no—
110#3948—22no—
111#3948—24no—
112C2×#520—51no—
113C2×#424D511yescharacteristic, complemented
114C2×#417—51no—
115C2×#416—51no—
116#2317—51no—
117#1805—52no—
118#1804—52no—
119#1803—52no—
120#1802—52no—
121#1736—51no—
122C2×#340—51no—
123C24×C5C2×D411yescharacteristic
124C23×D5—21no—
125C22×C4×C5C22×C412yes—
126C22×C4×C5C2×D412yes—
127C22×C4×C5C2×D412yes—
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×D412yes—
130C22×C4×C5C2×D412yes—
131C22×C4×C5C2×Q812yes—
132C22×C4×C5#4112yes—
133C22×C4×C5#4112yes—
134C22×C4×C5C2411yescharacteristic
135C2×C22⋊D5—24no—
136C2×C22⋊D5—24no—
137C2×C22⋊D5—24no—
138C22×Dic5—21no—
139C22×Dic5—22no—
140C22×Dic5—24no—
141C22×Dic5—21no—
142C22×Dic5—21no—
143C22×Dic5—22no—
144C22×Dic5—42no—
145C22×Dic5—81no—
146C22×Dic5—81no—
147C22×Dic5—81no—
148C22×Dic5—81no—
149#709—28no—
150#709—28no—
151#709—28no—
152#709C4⋊C4116yescomplemented
153C2×D20—21no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20C22×C412yes—
157C2×D20C2×D412yes—
158C2×D20C2×Q812yes—
159C2×C4×D5—24no—
160C2×C4×D5—24no—
161C2×C4×D5—24no—
162C2×C4×D5—28no—
163C2×C4×D5—82no—
164C2×C4×D5—82no—
165C2×Dic10—21no—
166C2×Dic10—21no—
167C2×Dic10—21no—
168C2×Dic10C22×C412yes—
169C2×Dic10C2×D412yes—
170C2×Dic10C2×Q812yes—
171C5×M16—44no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8—21no—
175C2×C5×C8—21no—
176C2×C5×C8—22no—
177C2×C5×C8—22no—
178C2×C5×C8—21no—
179C2×C5×C8—21no—
180C2×C5×C8—22no—
181C2×C5×C8—44no—
182C5×C4⋊C4C22⋊C414yes—
183C5×C4⋊C4C22⋊C414yes—
184C5×C22⋊C4C22⋊C418yes—
185C42×C5C22×C414yes—
186C42×C5C2×D414yes—
187C22⋊Dic5—42no—
188#685—28no—
189C4⋊Dic5—24no—
190C4⋊Dic5—22no—
191C4⋊Dic5—22no—
192C4⋊Dic5—21no—
193C4⋊Dic5—21no—
194C4⋊Dic5—22no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—42no—
198C4⋊Dic5—82no—
199C4⋊Dic5—82no—
200C4⋊Dic5C22×C414yes—
201#683—28no—
202C4×Dic5—28no—
203C4×Dic5—42no—
204C22×#41—51no—
205C22×M16C2×D511yescharacteristic
206C23×C8C2×D511yescharacteristic
207#520—58no—
208#520—104no—
209C2×C4×Q8—52no—
210C2×C4×D4—52no—
211C2×C4×D4—102no—
212C2×#117—51no—
213C2×#117—101no—
214C2×#117C2×D511yescharacteristic
215C22×C4⋊C4—51no—
216C22×C4⋊C4—51no—
217C22×C4⋊C4—101no—
218#424—24no—
219C2×C4⋊C8C2×D512yes—
220C2×C4⋊C8C2×D512yes—
221#417—104no—
222#416—104no—
223C2×#103—51no—
224C2×#103—52no—
225C2×#103—51no—
226C2×#102—51no—
227C2×#102—52no—
228C2×#102—51no—
229#389—101no—
230#389—101no—
231#385—102no—
232#384—52no—
233#382—52no—
234#340—52no—
235#340—52no—
236#340—52no—
237#340—52no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5C2×C22⋊C412yes—
243C23×C5#12112yes—
244C23×C5C22⋊Q812yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—22no—
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—41no—
250C22×D5—41no—
251C22×D5—41no—
252C22×D5—41no—
253C2×C4×C5—22no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5C2×C22⋊C414yes—
258C2×C4×C5C2×C22⋊C414yes—
259C2×C4×C5C2×C22⋊C412yes—
260C2×C4×C5C2×C4⋊C414yes—
261C2×C4×C5#11714yes—
262C2×C4×C5#12112yes—
263C2×C4×C5#12114yes—
264C2×C4×C5C22⋊Q812yes—
265C2×C4×C5C22⋊Q814yes—
266C2×C4×C5C23×C411yescharacteristic
267C2×C4×C5C22×D411yescharacteristic
268C2×C4×C5C22×D411yescharacteristic
269C2×C4×C5C22×Q811yescharacteristic
270C2×C4×C5C2×#4111yescharacteristic
271C2×C4×C5C2×#4111yescharacteristic
272C22⋊D5—216no—
273C2×Dic5—22no—
274C2×Dic5—22no—
275C2×Dic5—22no—
276C2×Dic5—24no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—41no—
280C2×Dic5—41no—
281C2×Dic5—42no—
282C2×Dic5—41no—
283C2×Dic5—42no—
284C2×Dic5—41no—
285C2×Dic5—42no—
286C2×Dic5—41no—
287C2×Dic5—41no—
288C2×Dic5—84no—
289C2×Dic5—84no—
290D20—22no—
291D20—22no—
292D20—22no—
293D20C2×C4⋊C414yes—
294C4×D5—216no—
295C4×D5—88no—
296Dic10—22no—
297Dic10—22no—
298Dic10—22no—
299Dic10C2×C4⋊C414yes—
300C5×C8—41no—
301C5×C8—41no—
302C5×C8—42no—
303C2×#41—54no—
304C2×#41—54no—
305C2×#41—54no—
306C2×#41—102no—
307C2×#41—102no—
308C2×#41—102no—
309C22×Q8—51no—
310C22×D4—51no—
311C22×D4—101no—
312C23×C4—101no—
313C23×C4—101no—
314C23×C4—102no—
315C23×C4—101no—
316C23×C4C22×D511yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—22no—
320C22×C8—22no—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8C22×D511yescharacteristic
324C22×C8C22×D511yescharacteristic
325C22×C8C22×D512yes—
326C4×Q8—58no—
327C4×Q8—104no—
328C4×D4—58no—
329C4×D4—1016no—
330C4×D4—104no—
331#117—108no—
332#117—102no—
333#117—102no—
334#117C4×D514yes—
335#117C22⋊D514yes—
336C2×C4⋊C4—54no—
337C2×C4⋊C4—52no—
338C2×C4⋊C4—52no—
339C2×C4⋊C4—51no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—101no—
342C2×C4⋊C4—102no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—108no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—102no—
347C2×C4⋊C4—101no—
348C2×C4⋊C4—102no—
349C2×C4⋊C4—102no—
350C2×C4⋊C4D2011yescharacteristic
351C2×C4⋊C4D2011yescharacteristic
352C2×C22⋊C4—102no—
353C2×C22⋊C4—101no—
354C2×C22⋊C4D2012yes—
355C2×C42—102no—
356C2×C42—101no—
357C2×C42C22×D512yes—
358C4⋊C8—24no—
359C4⋊C8—24no—
360#103—102no—
361#103—102no—
362#103—104no—
363#102—102no—
364#102—102no—
365#102—104no—
366#95—102no—
367#95—102no—
368C22×C5—22no—
369C22×C5—22no—
370C22×C5—22no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5#38014yes—
375C22×C5C22×C22⋊C411yescharacteristic
376C22×C5C2×#12111yescharacteristic
377C22×C5C2×C22⋊Q811yescharacteristic
378C22×C5#57611yescharacteristic
379C22×C5#57711yescharacteristic
380C22×C5#57711yescharacteristic
381C22×C5#57811yescharacteristic
382C2×D5—24no—
383C2×D5—42no—
384C2×D5—42no—
385C2×D5—42no—
386C2×D5—42no—
387C2×D5—42no—
388C2×D5—42no—
389C4×C5—24no—
390C4×C5—24no—
391C4×C5#38014yes—
392C4×C5C22×C4⋊C411yescharacteristic
393C4×C5C2×#11711yescharacteristic
394C4×C5C2×#12111yescharacteristic
395C4×C5C2×C22⋊Q811yescharacteristic
396Dic5—24no—
397Dic5—44no—
398Dic5—81no—
399Dic5—82no—
400Dic5—81no—
401C24—101no—
402C24C2×D2011yescharacteristic
403#41—516no—
404#41—108no—
405#41—108no—
406#41—108no—
407C2×Q8—52no—
408C2×Q8—52no—
409C2×Q8—52no—
410C2×Q8—101no—
411C2×Q8—101no—
412C2×Q8—101no—
413C2×D4—52no—
414C2×D4—52no—
415C2×D4—52no—
416C2×D4—104no—
417C2×D4—104no—
418C2×D4—104no—
419C2×D4—101no—
420C2×D4—101no—
421C2×D4—101no—
422C22×C4—101no—
423C22×C4—104no—
424C22×C4—104no—
425C22×C4—104no—
426C22×C4—102no—
427C22×C4—104no—
428C22×C4—108no—
429C22×C4—101no—
430C22×C4—102no—
431C22×C4—101no—
432C22×C4—202no—
433C22×C4—402no—
434C22×C4—402no—
435C22×C4—401no—
436C22×C4—401no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4C2×C4×D512yes—
440C22×C4C2×D2012yes—
441C22×C4C2×D2012yes—
442C22×C4C2×D2011yescharacteristic
443C22×C4D4×D512yes—
444C22×C4#71112yes—
445C22×C4Q8×D512yes—
446C22×C4#71312yes—
447C22×C4C2×C22⋊D512yes—
448C22×C4C23×D511yescharacteristic
449M16—44no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—44no—
460C4⋊C4—54no—
461C4⋊C4—102no—
462C4⋊C4—104no—
463C4⋊C4—108no—
464C4⋊C4—102no—
465C4⋊C4—101no—
466C4⋊C4—101no—
467C4⋊C4—102no—
468C4⋊C4—101no—
469C4⋊C4—101no—
470C4⋊C4—202no—
471C4⋊C4—402no—
472C4⋊C4—402no—
473C4⋊C4#68514yes—
474C4⋊C4#68514yes—
475C22⋊C4—108no—
476C22⋊C4—202no—
477C22⋊C4#68518yes—
478C42—108no—
479C42—202no—
480C42C2×C4×D514yes—
481C42C2×C22⋊D514yes—
482C2×C5—24no—
483C2×C5C2×#38011yescharacteristic
484C2×C5#289711yescharacteristic
485C2×C5#289811yescharacteristic
486C2×C5#304511yescharacteristic
487C2×C5#304611yescharacteristic
488C2×C5#308311yescharacteristic
489C2×C5#308411yescharacteristic
490D5—44no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—102no—
496C23—102no—
497C23—102no—
498C23—201no—
499C23—201no—
500C23—201no—
501C23—201no—
502C23#404112yes—
503C23#404312yes—
504C23C2×#68512yes—
505C23C22×D2011yescharacteristic
506Q8—54no—
507Q8—102no—
508Q8—102no—
509Q8—102no—
510D4—54no—
511D4—1016no—
512D4—102no—
513D4—102no—
514D4—102no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—1016no—
523C2×C4—104no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—201no—
527C2×C4—202no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—202no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—202no—
535C2×C4—408no—
536C2×C4—404no—
537C2×C4—404no—
538C2×C4D5×C4⋊C414yes—
539C2×C4#403814yes—
540C2×C4#404112yes—
541C2×C4#404312yes—
542C2×C4C2×#68514yes—
543C2×C4C2×#68514yes—
544C2×C4C2×#68512yes—
545C2×C4#408414yes—
546C2×C4#409214yes—
547C2×C4C22×C4×D511yescharacteristic
548C2×C4C2×D4×D511yescharacteristic
549C2×C4C2×#71111yescharacteristic
550C2×C4C2×Q8×D511yescharacteristic
551C2×C4C2×#71311yescharacteristic
552C2×C4C22×C22⋊D511yescharacteristic
553C8—41no—
554C8—41no—
555C8—42no—
556C5#2065211yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—104no—
564C22—202no—
565C22—202no—
566C22—202no—
567C22—202no—
568C22—202no—
569C22—202no—
570C22#6527614yes—
571C22C2×#404111yescharacteristic
572C22C2×#404311yescharacteristic
573C22#6608511yescharacteristic
574C22#6608511yescharacteristic
575C22#6608611yescharacteristic
576C22#6608711yescharacteristic
577C22C22×#68511yescharacteristic
578C4—24no—
579C4—24no—
580C4—104no—
581C4—204no—
582C4—402no—
583C4—401no—
584C4—401no—
585C4#6527614yes—
586C4C2×D5×C4⋊C411yescharacteristic
587C4C2×#403811yescharacteristic
588C4C2×#408411yescharacteristic
589C4C2×#409211yescharacteristic
590C2—24no—
591C2—204no—
592C2#1061081911yescharacteristic
593C2#1061082011yescharacteristic
594C2#1061082211yescharacteristic
595C2#1061082311yescharacteristic
596C2C2×#6527611yescharacteristic
597C2#1061552811yescharacteristic
598C2#1061553711yescharacteristic
subgroupG/H
centerC23C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#424C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#424D5
hypocenterC5#20652
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
2. C2×C5×#424C2semidirect
5. #10613964C2semidirect
6. #10602778C2semidirect
7. C2×#64771C2semidirect
8. #10601279C2semidirect
9. #10601278C2semidirect
10. #10601273C2semidirect
11. #10601272C2semidirect
20. #65857C4semidirect
99. C4×D20C2×C4semidirect
101. C4×Dic10C2×C4semidirect
113. C2×#424D5semidirect
152. #709C4⋊C4semidirect
556. C5#20652semidirect

Maximal subgroups

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

subgroupindex
C2×#658572
C2×C5×#4242
C2×#654222
C2×#654022
#106139642
#106027782
C2×#647712
#106012792
#106012782
#106012732
#106012722
#206525