Finite Group Databasebeta

← #12100188

Subgroups — #12100188

Order 1280.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10617632C211yesmaximal, characteristic
2C5×#2317C211yesmaximal, characteristic, complemented
3C2×#65294C211yesmaximal, characteristic
4C2×#65255C211yesmaximal, characteristic
5C2×#65253C211yesmaximal, characteristic, complemented
6#10602380C211yesmaximal, characteristic, complemented
7#10602371C211yesmaximal, characteristic, complemented
8#10600385C214yesmaximal, complemented
9#10600355C214yesmaximal, complemented
10C22×#709C2211yescharacteristic
11C2×C5×#117C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C2×#4072C2211yescharacteristic
15C22×#681C2211yescharacteristic
16C22×#681C2211yescharacteristic
17#65857—21no—
18#65857—21no—
19#65856—22no—
20#65855—21no—
21D5×#117—22no—
22#65850—21no—
23#65838C2212yes—
24#65838C2212yes—
25#65820—22no—
26C4×#709—22no—
27C5×#389—22no—
28C5×#385—22no—
29C5×#384C2212yescomplemented
30C5×#382C2212yescomplemented
31#65294C418yescomplemented
32C2×#3942—21no—
33#65255—22no—
34#65255C2214yes—
35C2×#3941—21no—
36#65253—22no—
37#65253C2214yescomplemented
38C2×#3940—21no—
39C2×#3939—21no—
40C2×#3937C2214yes—
41#64750C2212yescomplemented
42#64750C2212yescomplemented
43#64743C2214yescomplemented
44#20865—51nomaximal, Sylow
45C23×C4×C5—21no—
46C23×C4×C5C2311yescharacteristic
47C22×C22⋊D5—21no—
48C2×#709—21no—
49C2×#709—21no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709C2×C414yes—
53C2×#709D412yes—
54C2×#709Q814yes—
55C2×#709C2312yes—
56C22×D20D411yescharacteristic
57C22×C4×D5—21no—
58C22×Dic10D411yescharacteristic
59C5×#117—21no—
60C5×#117—21no—
61C5×#117—21no—
62C5×#117—21no—
63C5×#117—22no—
64C5×#117C2×C414yescomplemented
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—24no—
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C22⋊C4—21no—
76C2×C5×C22⋊C4D412yes—
77C2×C42×C5D412yes—
78C2×C42×C5C2312yes—
79C2×C22⋊Dic5—21no—
80C4×C22⋊D5—24no—
81C4×C22⋊D5—24no—
82#4072—21no—
83#4072—21no—
84#4072D412yes—
85#4072C2312yes—
86C2×C4⋊Dic5D411yescharacteristic
87C2×C4⋊Dic5D411yescharacteristic
88C2×C4×Dic5—21no—
89C2×#681—21no—
90C2×#681—22no—
91C2×#681—21no—
92C2×#681—22no—
93C2×#681—22no—
94C2×#681C2×C414yescomplemented
95C2×#681C2312yes—
96C2×#681C2312yes—
97C22×#168—21no—
98C22×#168—21no—
99#4039—22no—
100#4039—22no—
101#4038—22no—
102#4038—22no—
103D5×C4⋊C4—22no—
104D5×C4⋊C4—22no—
105#4033—22no—
106#4033—22no—
107#4027—24no—
108#4027—24no—
109D5×C22⋊C4—24no—
110#4023—24no—
111C4×D20—22no—
112C4×D20—22no—
113#4018—24no—
114C42×D5—24no—
115C4×Dic10—22no—
116C4×Dic10—22no—
117C5×#95—24no—
118#3942—24no—
119#3941—24no—
120#3940—24no—
121#3939—24no—
122#3937C2×C4116yescomplemented
123#3428—51no—
124C2×#428—51no—
125C2×#421—51no—
126C2×#417—51no—
127#2317D511yescharacteristic, complemented
128#1860—54no—
129#1859—54no—
130#1742—51no—
131#1733—51no—
132C24×C5C2×D411yescharacteristic
133C23×D5—21no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—41no—
137C22×C4×C5—41no—
138C22×C4×C5—82no—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D412yes—
145C22×C4×C5C2×D412yes—
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×Q812yes—
148C22×C4×C5#4112yes—
149C22×C4×C5#4112yes—
150C22×C4×C5C2411yescharacteristic
151C2×C22⋊D5—24no—
152C2×C22⋊D5—22no—
153C2×C22⋊D5—24no—
154C2×C22⋊D5—22no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157C22×Dic5—21no—
158#709—24no—
159#709—24no—
160#709—24no—
161#709—24no—
162#709—24no—
163#709—22no—
164#709—22no—
165#709C4⋊C418yescomplemented
166#709C22×C414yes—
167#709C2×Q814yes—
168C2×D20—21no—
169C2×D20—22no—
170C2×D20C4⋊C414yescomplemented
171C2×D20C2×D412yes—
172C2×C4×D5—24no—
173C2×C4×D5—22no—
174C2×C4×D5—24no—
175C2×C4×D5—22no—
176C2×C4×D5—24no—
177C2×C4×D5—24no—
178C2×Dic10—21no—
179C2×Dic10—22no—
180C2×Dic10C4⋊C414yescomplemented
181C2×Dic10C2×D412yes—
182C5×C4⋊C4—24no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4—22no—
189C5×C4⋊C4—41no—
190C5×C4⋊C4—41no—
191C5×C4⋊C4—84no—
192C5×C22⋊C4—24no—
193C5×C22⋊C4—22no—
194C5×C22⋊C4—22no—
195C42×C5—22no—
196C42×C5—22no—
197C42×C5—22no—
198C42×C5C22×C414yes—
199C22⋊Dic5—22no—
200C22⋊Dic5—22no—
201#685—24no—
202#685—24no—
203C4⋊Dic5—21no—
204C4⋊Dic5—21no—
205C4⋊Dic5C2×D412yes—
206C4⋊Dic5C2×D412yes—
207#683—24no—
208#683—24no—
209C4×Dic5—24no—
210C4×Dic5—24no—
211C4×Dic5—22no—
212C4×Dic5—22no—
213#681—24no—
214#681—22no—
215#681—22no—
216#681—22no—
217#681—42no—
218#681C22×C414yes—
219C2×#168—24no—
220C2×#168—22no—
221C2×#168—22no—
222C2×#168—42no—
223C22×#41—51no—
224C22×M16—51no—
225C22×M16—51no—
226#520—52no—
227#520—52no—
228#520—101no—
229#520—101no—
230#519—101no—
231#518—101no—
232#518—102no—
233C4×#41—102no—
234C4×#41—102no—
235C2×#117—51no—
236C2×#117—102no—
237C2×#117C2×D511yescharacteristic
238C2×#117C2×D511yescharacteristic
239C22×C4⋊C4C2×D511yescharacteristic
240#428—54no—
241#428—102no—
242C2×#107—101no—
243C2×#106—101no—
244#421—58no—
245C2×#104—54no—
246#417—54no—
247#417—102no—
248C2×#103—101no—
249C2×#102—101no—
250#389—22no—
251#385—22no—
252#384C2×D512yescomplemented
253#382C2×D512yescomplemented
254#343—52no—
255#343—52no—
256#339—54no—
257C23×C5—41no—
258C23×C5—41no—
259C23×C5C4×D412yes—
260C23×C5#12011yescharacteristic
261C23×C5#12011yescharacteristic
262C23×C5C22⋊Q812yes—
263C23×C5C22×D411yescharacteristic
264C22×D5—22no—
265C22×D5—24no—
266C22×D5—81no—
267C22×D5—81no—
268C2×C4×C5—24no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—42no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—44no—
276C2×C4×C5—42no—
277C2×C4×C5—82no—
278C2×C4×C5—82no—
279C2×C4×C5C2×C4⋊C412yes—
280C2×C4×C5C2×C4⋊C412yes—
281C2×C4×C5C4×D412yes—
282C2×C4×C5C4×D412yes—
283C2×C4×C5C4×D412yes—
284C2×C4×C5#12011yescharacteristic
285C2×C4×C5#12011yescharacteristic
286C2×C4×C5C22⋊Q812yes—
287C2×C4×C5C22⋊Q812yes—
288C2×C4×C5C22⋊Q812yes—
289C2×C4×C5#12312yes—
290C2×C4×C5#12312yes—
291C2×C4×C5C23×C411yescharacteristic
292C2×C4×C5C22×D411yescharacteristic
293C2×C4×C5C22×D411yescharacteristic
294C2×C4×C5C22×Q811yescharacteristic
295C2×C4×C5C2×#4111yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C22⋊D5—24no—
298C22⋊D5—24no—
299C22⋊D5—28no—
300C2×Dic5—22no—
301C2×Dic5—24no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—81no—
305C2×Dic5—81no—
306D20—22no—
307D20—24no—
308D20C2×C4⋊C414yes—
309C4×D5—24no—
310C4×D5—24no—
311C4×D5—28no—
312C4×D5—84no—
313C4×D5—84no—
314Dic10—22no—
315Dic10—24no—
316Dic10C2×C4⋊C414yes—
317#168—24no—
318#168—42no—
319C2×#41—54no—
320C2×#41—52no—
321C2×#41—54no—
322C2×#41—52no—
323C2×#41—102no—
324C2×#41—101no—
325C2×#41—102no—
326C2×#41—101no—
327C22×Q8—51no—
328C22×D4—51no—
329C22×D4—101no—
330C23×C4—21no—
331C23×C4—101no—
332C23×C4C22×D511yescharacteristic
333C2×M16—54no—
334C2×M16—52no—
335C2×M16—52no—
336C2×M16—101no—
337C2×M16—102no—
338C2×M16—101no—
339C2×M16—102no—
340C2×M16—102no—
341C22×C8—101no—
342C22×C8—101no—
343C4×Q8—102no—
344C4×Q8—102no—
345C4×Q8—102no—
346C4×Q8—102no—
347C4×D4—102no—
348C4×D4—102no—
349C4×D4—104no—
350C4×D4—104no—
351C4×D4—102no—
352C4×D4—102no—
353C4×D4—104no—
354C4×D4—104no—
355#117—22no—
356#117—21no—
357#117—21no—
358#117—21no—
359#117—21no—
360#117—52no—
361#117—52no—
362#117—104no—
363#117—104no—
364#117—102no—
365#117—102no—
366#117—101no—
367#117—101no—
368#117C4×D514yescomplemented
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—24no—
376C2×C4⋊C4—51no—
377C2×C4⋊C4—51no—
378C2×C4⋊C4—102no—
379C2×C4⋊C4—102no—
380C2×C4⋊C4C22×D512yes—
381C2×C4⋊C4C22×D511yescharacteristic
382C2×C4⋊C4C22×D511yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—104no—
385C2×C22⋊C4—101no—
386C2×C22⋊C4C22⋊D512yes—
387C2×C42—104no—
388C2×C42—101no—
389C2×C42C22⋊D512yes—
390C2×C42C22×D512yes—
391#107—104no—
392#106—104no—
393#104—516no—
394#103—104no—
395#102—104no—
396#95—24no—
397C22×C5—21no—
398C22×C5—21no—
399C22×C5—21no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—21no—
403C22×C5—42no—
404C22×C5—42no—
405C22×C5#38512yes—
406C22×C5#38512yes—
407C22×C5C2×C4×D411yescharacteristic
408C22×C5C2×#12011yescharacteristic
409C22×C5C2×C22⋊Q811yescharacteristic
410C2×D5—24no—
411C2×D5—42no—
412C2×D5—42no—
413C2×D5—84no—
414C4×C5—44no—
415C4×C5—42no—
416C4×C5—42no—
417C4×C5—82no—
418C4×C5#38512yes—
419C4×C5#38512yes—
420C4×C5C22×C4⋊C411yescharacteristic
421C4×C5C2×C4×D411yescharacteristic
422C4×C5C2×C22⋊Q811yescharacteristic
423C4×C5C2×#12311yescharacteristic
424Dic5—24no—
425Dic5—42no—
426Dic5—42no—
427C24—101no—
428C24C2×C22⋊D511yescharacteristic
429#41—54no—
430#41—54no—
431#41—58no—
432#41—102no—
433#41—102no—
434#41—104no—
435#41—104no—
436#41—104no—
437#41—104no—
438#41—104no—
439C2×Q8—52no—
440C2×Q8—54no—
441C2×Q8—102no—
442C2×Q8—101no—
443C2×D4—52no—
444C2×D4—54no—
445C2×D4—102no—
446C2×D4—104no—
447C2×D4—102no—
448C2×D4—104no—
449C2×D4—102no—
450C2×D4—101no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—82no—
456C22×C4—104no—
457C22×C4—104no—
458C22×C4—101no—
459C22×C4—102no—
460C22×C4—104no—
461C22×C4—104no—
462C22×C4—102no—
463C22×C4—101no—
464C22×C4—101no—
465C22×C4C2×C4×D512yes—
466C22×C4#70912yes—
467C22×C4D4×D511yescharacteristic
468C22×C4D4×D511yescharacteristic
469C22×C4D4×D511yescharacteristic
470C22×C4D4×D511yescharacteristic
471C22×C4Q8×D512yes—
472C22×C4#71312yes—
473C22×C4C2×C22⋊D512yes—
474C22×C4C2×C22⋊D512yes—
475C22×C4C2×C22⋊D511yescharacteristic
476C22×C4C23×D511yescharacteristic
477M16—54no—
478M16—102no—
479M16—104no—
480M16—102no—
481M16—102no—
482M16—202no—
483C2×C8—102no—
484C2×C8—104no—
485C2×C8—102no—
486C2×C8—202no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—52no—
497C4⋊C4—52no—
498C4⋊C4—84no—
499C4⋊C4—104no—
500C4⋊C4—104no—
501C4⋊C4—101no—
502C4⋊C4—101no—
503C22⋊C4—24no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—104no—
507C22⋊C4—104no—
508C22⋊C4—102no—
509C22⋊C4—102no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—104no—
514C42—104no—
515C42—102no—
516C42—102no—
517C42C2×C4×D514yes—
518C2×C5—21no—
519C2×C5—22no—
520C2×C5—21no—
521C2×C5—22no—
522C2×C5C2×#38511yescharacteristic
523C2×C5#302111yescharacteristic
524C2×C5#302211yescharacteristic
525D5—44no—
526C23—41no—
527C23—41no—
528C23—102no—
529C23—104no—
530C23—401no—
531C23—401no—
532C23C4×C22⋊D512yes—
533C23#408311yescharacteristic
534C23#408311yescharacteristic
535C23#409212yes—
536C23C22×C22⋊D511yescharacteristic
537Q8—54no—
538Q8—102no—
539Q8—104no—
540D4—54no—
541D4—104no—
542D4—104no—
543D4—108no—
544D4—102no—
545D4—104no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—102no—
558C2×C4—104no—
559C2×C4—104no—
560C2×C4—104no—
561C2×C4—108no—
562C2×C4—102no—
563C2×C4—102no—
564C2×C4—404no—
565C2×C4—404no—
566C2×C4—401no—
567C2×C4—401no—
568C2×C4D5×C4⋊C412yes—
569C2×C4D5×C4⋊C412yes—
570C2×C4#403912yes—
571C2×C4#403912yes—
572C2×C4#404012yes—
573C2×C4#404012yes—
574C2×C4#404212yes—
575C2×C4#404212yes—
576C2×C4C4×C22⋊D512yes—
577C2×C4#408311yescharacteristic
578C2×C4#408311yescharacteristic
579C2×C4#409212yes—
580C2×C4C22×C4×D511yescharacteristic
581C2×C4C2×#70911yescharacteristic
582C2×C4C2×D4×D511yescharacteristic
583C2×C4C2×D4×D511yescharacteristic
584C2×C4C2×Q8×D511yescharacteristic
585C2×C4C2×#71311yescharacteristic
586C8—104no—
587C8—202no—
588C5#2086511yescharacteristic, complemented, Sylow
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—42no—
596C22—42no—
597C22—104no—
598C22—202no—
599C22—202no—
600C22—404no—
601C22#6527812yes—
602C22#6527812yes—
603C22C2×C4×C22⋊D511yescharacteristic
604C22C2×#408311yescharacteristic
605C22C2×#409211yescharacteristic
606C4—44no—
607C4—42no—
608C4—42no—
609C4—82no—
610C4—104no—
611C4—202no—
612C4—202no—
613C4#6527812yes—
614C4#6527812yes—
615C4C2×D5×C4⋊C411yescharacteristic
616C4C2×#403911yescharacteristic
617C4C2×#404011yescharacteristic
618C4C2×#404211yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—204no—
624C2C2×#6527811yescharacteristic
625C2#1061512611yescharacteristic
626C2#1061514711yescharacteristic
subgroupG/H
centerC22C2×#4083
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2317C2
socleC22×C5C2×#120
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#2317D5
hypocenterC5#20865
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C5×#2317C2semidirect
5. C2×#65253C2semidirect
6. #10602380C2semidirect
7. #10602371C2semidirect
8. #10600385C2semidirect
9. #10600355C2semidirect
29. C5×#384C22semidirect
30. C5×#382C22semidirect
31. #65294C4semidirect
37. #65253C22semidirect
41. #64750C22semidirect
42. #64750C22semidirect
43. #64743C22semidirect
64. C5×#117C2×C4semidirect
94. C2×#681C2×C4semidirect
122. #3937C2×C4semidirect
127. #2317D5semidirect
165. #709C4⋊C4semidirect
170. C2×D20C4⋊C4semidirect
180. C2×Dic10C4⋊C4semidirect
252. #384C2×D5semidirect
253. #382C2×D5semidirect
368. #117C4×D5semidirect
588. C5#20865semidirect

Maximal subgroups

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

subgroupindex
#106176322
C5×#23172
C2×#652942
C2×#652552
C2×#652532
#106023802
#106023712
#106003852
#106003552
#208655