Finite Group Databasebeta

← #12100300

Subgroups — #12100300

Order 1280.

1458 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×#65838C211yesmaximal, characteristic
2C2×C5×#428C211yesmaximal, characteristic, complemented
3C2×#65422C211yesmaximal, characteristic
4#10613963C211yesmaximal, characteristic, complemented
5C2×#65291C211yesmaximal, characteristic
6#10602773C211yesmaximal, characteristic, complemented
7C2×#64749C211yesmaximal, characteristic, complemented
8#10601435C212yesmaximal, complemented
9#10601431C212yesmaximal, complemented
10#10601381C212yesmaximal, complemented
11#10601377C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×C4×C22⋊D5—22no—
17C2×#4072C2211yescharacteristic
18C22×C4⋊Dic5C2211yescharacteristic
19C23×#168C2211yescharacteristic
20#65838—24no—
21#65838—24no—
22C2×#4039C2212yes—
23C2×#4038—21no—
24C2×D5×C4⋊C4—21no—
25C2×#4033C2212yes—
26C5×#428—24no—
27C2×C5×#107C2212yes—
28C2×C5×#106C2212yes—
29#65422C418yescomplemented
30C2×#3953C2212yes—
31C2×#3948C2212yes—
32#65291—24no—
33#65262C2212yes—
34#65262C2212yes—
35C2×#3942C2211yescharacteristic
36C2×#3942C2211yescharacteristic
37C2×#3941C2211yescharacteristic
38C2×#3941C2211yescharacteristic
39#65231—22no—
40#65217—21no—
41#65217—21no—
42#64770C2212yes—
43#64770C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#20878—51nomaximal, Sylow
47C23×C4×C5—21no—
48C23×C4×C5C2311yescharacteristic
49C22×C22⋊D5—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×M16C2×C414yescomplemented
64C22×C5×C8C2312yes—
65C5×#117—22no—
66C5×#117—22no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4D414yes—
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C22⋊C4—21no—
74C2×C42×C5—21no—
75C2×C22⋊Dic5D412yes—
76C4×C22⋊D5—216no—
77C2×#685—22no—
78#4072—22no—
79#4072—22no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5C2×C414yescomplemented
83C2×C4⋊Dic5D411yescharacteristic
84C2×C4⋊Dic5D411yescharacteristic
85C2×C4⋊Dic5C2312yes—
86C2×#683—22no—
87C2×#683—22no—
88C2×#683—22no—
89C2×C4×Dic5—22no—
90C2×C4×Dic5C2312yes—
91C22×#168—22no—
92C22×#168—22no—
93C22×#168—22no—
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96#4039—24no—
97#4039—24no—
98#4038—28no—
99D5×C4⋊C4—28no—
100#4033—24no—
101#4033—24no—
102C5×#107—24no—
103C5×#106—24no—
104#3963—22no—
105#3963—22no—
106#3953C2×C418yescomplemented
107#3948C2×C418yescomplemented
108#3942—22no—
109#3942—22no—
110#3941—22no—
111#3941—22no—
112C2×#520—51no—
113C2×#428D511yescharacteristic, complemented
114C2×#417—51no—
115C2×#416—51no—
116#2316—51no—
117#1865—52no—
118#1864—52no—
119#1863—52no—
120#1862—52no—
121C2×#340—51no—
122#1729—51no—
123C24×C5C2×D411yescharacteristic
124C23×D5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—24no—
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—81no—
130C22×C4×C5—81no—
131C22×C4×C5C22×C412yes—
132C22×C4×C5C2×D412yes—
133C22×C4×C5C2×D411yescharacteristic
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×Q812yes—
136C22×C4×C5#4112yes—
137C22×C4×C5#4112yes—
138C22×C4×C5C2411yescharacteristic
139C2×C22⋊D5—24no—
140C2×C22⋊D5—24no—
141C2×C22⋊D5—24no—
142C22×Dic5—21no—
143C22×Dic5—22no—
144C22×Dic5—24no—
145C22×Dic5—81no—
146C22×Dic5—81no—
147C22×Dic5C2×D412yes—
148C22×Dic5C2×D412yes—
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—28no—
172C2×C5×C8—22no—
173C2×C5×C8—22no—
174C2×C5×C8C22×C414yes—
175C5×C4⋊C4—24no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4C2×D414yes—
182C5×C22⋊C4—42no—
183C42×C5—42no—
184C22⋊Dic5—24no—
185#685—28no—
186C4⋊Dic5—24no—
187C4⋊Dic5—22no—
188C4⋊Dic5—22no—
189C4⋊Dic5—22no—
190C4⋊Dic5C22×C414yes—
191#683—28no—
192#683—82no—
193#683—82no—
194C4×Dic5—28no—
195C4×Dic5—22no—
196C4×Dic5—22no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—44no—
204C22×#41—51no—
205C22×M16C2×D511yescharacteristic
206C23×C8—51no—
207#520—104no—
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—101no—
217C22×C4⋊C4C2×D511yescharacteristic
218#428—24no—
219C2×#107C2×D512yes—
220C2×#106C2×D512yes—
221#417—58no—
222#416—104no—
223C2×#103—52no—
224C2×#103—51no—
225C2×#103—51no—
226C2×#102—52no—
227C2×#102—51no—
228C2×#102—51no—
229#389—52no—
230#389—52no—
231#389—101no—
232#389—101no—
233#380—102no—
234#340—52no—
235#340—52no—
236#336—52no—
237#336—52no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5C4×D412yes—
243C23×C5#12012yes—
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—24no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—41no—
257C2×C4×C5—41no—
258C2×C4×C5—41no—
259C2×C4×C5—41no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5—84no—
263C2×C4×C5C2×C4⋊C414yes—
264C2×C4×C5C4×D412yes—
265C2×C4×C5C4×D414yes—
266C2×C4×C5#12012yes—
267C2×C4×C5C22⋊Q812yes—
268C2×C4×C5C22⋊Q814yes—
269C2×C4×C5#12314yes—
270C2×C4×C5C23×C411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C22×Q811yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C22⋊D5—216no—
277C2×Dic5—22no—
278C2×Dic5—22no—
279C2×Dic5—22no—
280C2×Dic5—24no—
281C2×Dic5—22no—
282C2×Dic5—22no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—22no—
287C2×Dic5—42no—
288C2×Dic5—42no—
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—24no—
301#168—41no—
302#168—41no—
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—21no—
313C23×C4—102no—
314C23×C4—101no—
315C23×C4—101no—
316C23×C4C22×D511yescharacteristic
317C2×M16—22no—
318C2×M16—22no—
319C2×M16C4×D514yescomplemented
320C22×C8—51no—
321C22×C8—51no—
322C22×C8—102no—
323C22×C8—102no—
324C22×C8—102no—
325C22×C8C22×D512yes—
326C4×Q8—104no—
327C4×Q8—104no—
328C4×D4—104no—
329C4×D4—1016no—
330C4×D4—104no—
331#117—22no—
332#117—22no—
333#117—108no—
334#117—102no—
335#117—102no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—54no—
339C2×C4⋊C4—52no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—51no—
342C2×C4⋊C4—108no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—101no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—102no—
347C2×C4⋊C4—102no—
348C2×C4⋊C4C22⋊D514yes—
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D512yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—52no—
354C2×C22⋊C4—102no—
355C2×C42—21no—
356C2×C42—52no—
357C2×C42—102no—
358#107—24no—
359#106—24no—
360#103—58no—
361#103—102no—
362#103—102no—
363#102—58no—
364#102—102no—
365#102—102no—
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#38514yes—
375C22×C5C2×C4×D411yescharacteristic
376C22×C5C2×#12011yescharacteristic
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—44no—
390C4×C5—81no—
391C4×C5—81no—
392C4×C5#38514yes—
393C4×C5C22×C4⋊C411yescharacteristic
394C4×C5C2×C4×D411yescharacteristic
395C4×C5C2×C22⋊Q811yescharacteristic
396C4×C5C2×#12311yescharacteristic
397Dic5—24no—
398Dic5—44no—
399Dic5—42no—
400Dic5—42no—
401C24—101no—
402C24D4×D511yescharacteristic
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—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—52no—
427C22×C4—52no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—108no—
431C22×C4—101no—
432C22×C4—104no—
433C22×C4—104no—
434C22×C4—104no—
435C22×C4—102no—
436C22×C4—104no—
437C22×C4—402no—
438C22×C4—402no—
439C22×C4—401no—
440C22×C4—401no—
441C22×C4C2×C4×D512yes—
442C22×C4#70912yes—
443C22×C4D4×D512yes—
444C22×C4D4×D511yescharacteristic
445C22×C4Q8×D512yes—
446C22×C4#71312yes—
447C22×C4C2×C22⋊D512yes—
448C22×C4C23×D511yescharacteristic
449M16—28no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—101no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—204no—
459C2×C8C2×C4×D514yes—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—54no—
467C4⋊C4—108no—
468C4⋊C4—102no—
469C4⋊C4—104no—
470C4⋊C4—102no—
471C4⋊C4—102no—
472C4⋊C4—402no—
473C4⋊C4—402no—
474C4⋊C4C2×C22⋊D514yes—
475C22⋊C4—42no—
476C22⋊C4—108no—
477C22⋊C4—104no—
478C42—42no—
479C42—108no—
480C42—102no—
481C42—102no—
482C2×C5—24no—
483C2×C5C2×#38511yescharacteristic
484C2×C5#294611yescharacteristic
485C2×C5#294711yescharacteristic
486C2×C5#301211yescharacteristic
487C2×C5#301311yescharacteristic
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#403912yes—
503C23#404212yes—
504C23#408312yes—
505C23C2×D4×D511yescharacteristic
506Q8—54no—
507Q8—102no—
508Q8—102no—
509Q8—102no—
510D4—54no—
511D4—1016no—
512D4—102no—
513D4—102no—
514D4—102no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—84no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4—102no—
528C2×C4—1016no—
529C2×C4—104no—
530C2×C4—102no—
531C2×C4—102no—
532C2×C4—102no—
533C2×C4—102no—
534C2×C4—102no—
535C2×C4—102no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—408no—
539C2×C4—404no—
540C2×C4D5×C4⋊C414yes—
541C2×C4#403912yes—
542C2×C4#404014yes—
543C2×C4#404212yes—
544C2×C4C4×C22⋊D514yes—
545C2×C4#408312yes—
546C2×C4#409214yes—
547C2×C4C22×C4×D511yescharacteristic
548C2×C4C2×#70911yescharacteristic
549C2×C4C2×D4×D511yescharacteristic
550C2×C4C2×Q8×D511yescharacteristic
551C2×C4C2×#71311yescharacteristic
552C2×C4C22×C22⋊D511yescharacteristic
553C8—24no—
554C8—201no—
555C8—201no—
556C5#2087811yescharacteristic, 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#6527814yes—
571C22C2×#403911yescharacteristic
572C22C2×#404211yescharacteristic
573C22#6610811yescharacteristic
574C22#6610911yescharacteristic
575C22#6611211yescharacteristic
576C22#6611311yescharacteristic
577C22C2×#408311yescharacteristic
578C4—44no—
579C4—81no—
580C4—81no—
581C4—104no—
582C4—202no—
583C4—202no—
584C4—204no—
585C4#6527814yes—
586C4C2×D5×C4⋊C411yescharacteristic
587C4C2×#404011yescharacteristic
588C4C2×C4×C22⋊D511yescharacteristic
589C4C2×#409211yescharacteristic
590C2—24no—
591C2—204no—
592C2#1061093211yescharacteristic
593C2#1061093311yescharacteristic
594C2#1061097411yescharacteristic
595C2#1061097511yescharacteristic
596C2C2×#6527811yescharacteristic
597C2#1061568011yescharacteristic
598C2#1061571511yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#428D5
hypocenterC5#20878
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×C5×#428C2semidirect
4. #10613963C2semidirect
6. #10602773C2semidirect
7. C2×#64749C2semidirect
8. #10601435C2semidirect
9. #10601431C2semidirect
10. #10601381C2semidirect
11. #10601377C2semidirect
29. #65422C4semidirect
63. C2×C5×M16C2×C4semidirect
82. C2×C4⋊Dic5C2×C4semidirect
106. #3953C2×C4semidirect
107. #3948C2×C4semidirect
113. C2×#428D5semidirect
152. #709C4⋊C4semidirect
319. C2×M16C4×D5semidirect
556. C5#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#658382
C2×C5×#4282
C2×#654222
#106139632
C2×#652912
#106027732
C2×#647492
#106014352
#106014312
#106013812
#106013772
#208785