Finite Group Databasebeta

← #12121373

Subgroups — #12121373

Order 1280.

708 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10615514C211yesmaximal, characteristic
2#10610809C211yesmaximal, characteristic, complemented
3#10610553C211yesmaximal, characteristic
4C5×#1866C211yesmaximal, characteristic
5#10602802C211yesmaximal, characteristic
6#10601498C211yesmaximal, characteristic, complemented
7#10601488C211yesmaximal, characteristic, complemented
8#10601331C211yesmaximal, characteristic, complemented
9#10601169C211yesmaximal, characteristic
10Dic5×#104C211yesmaximal, characteristic
11#10601083C211yesmaximal, characteristic
12#10599536C211yesmaximal, characteristic, complemented
13#10599524C211yesmaximal, characteristic, complemented
14#10599438C211yesmaximal, characteristic, complemented
15#10599396C211yesmaximal, characteristic, complemented
16C2×C5×#131C2211yescharacteristic
17#41×Dic5C2211yescharacteristic
18C2×#4094C2211yescharacteristic
19#65851C2211yescharacteristic
20C2×C5×#108C2211yescharacteristic
21C5×#424C2211yescharacteristic
22C2×C5×#104C2211yescharacteristic
23C5×#416C2211yescharacteristic
24C2×#3969C2211yescharacteristic
25#65522C2211yescharacteristic
26C2×#3954C2211yescharacteristic
27#65408—21no—
28#65408C2211yescharacteristic
29#65408C2211yescharacteristic
30#65407—21no—
31#65407C2211yescharacteristic
32M16×Dic5C2211yescharacteristic
33#65395C2211yescharacteristic
34#65391—21no—
35#65391C2211yescharacteristic
36#65390C2211yescharacteristic
37C2×#3947C2211yescharacteristic
38C2×#3947C2211yescharacteristic
39C2×#3946—21no—
40C2×#3946C2211yescharacteristic
41C2×#3946C2211yescharacteristic
42#65287C2211yescharacteristic
43#65286C2211yescharacteristic
44C42×Dic5C2211yescharacteristic
45#65120—21no—
46#65119—21no—
47#65117—21no—
48#65117—21no—
49#65088C411yescharacteristic, complemented
50#65085C411yescharacteristic, complemented
51#65083C411yescharacteristic, complemented
52#65080C411yescharacteristic, complemented
53#65055C411yescharacteristic, complemented
54#65052C411yescharacteristic, complemented
55#65050C411yescharacteristic, complemented
56#65048C411yescharacteristic, complemented
57C5×#339C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#64778C2211yescharacteristic
60#64775C2211yescharacteristic
61#64775C2211yescharacteristic
62#64772C2211yescharacteristic
63#64752C2211yescharacteristic
64#64744C2211yescharacteristic
65#64743C2211yescharacteristic
66#64743C2211yescharacteristic
67#22983—51nomaximal, Sylow
68C2×C5×#41C2311yescharacteristic
69C5×#131—22no—
70C5×#131—41no—
71C2×C5×M16—21no—
72C2×C5×M16C2311yescharacteristic
73C2×C5×M16C2311yescharacteristic
74C22×C5×C8—21no—
75C22×C5×C8C2311yescharacteristic
76C22×C5×C8C2311yescharacteristic
77C5×#117C2311yescharacteristic
78C2×C42×C5C2311yescharacteristic
79#4094—22no—
80#4094—41no—
81Q8×Dic5C2×C411yescharacteristic
82Q8×Dic5C2×C411yescharacteristic
83D4×Dic5—21no—
84D4×Dic5—21no—
85D4×Dic5C2×C411yescharacteristic
86D4×Dic5C2×C411yescharacteristic
87#4072—21no—
88#4072C2311yescharacteristic
89#4072C2311yescharacteristic
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5C2311yescharacteristic
94C2×C4×Dic5C2311yescharacteristic
95C2×#681—21no—
96C2×#681C2311yescharacteristic
97C2×#681C2311yescharacteristic
98C22×#168—21no—
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101#4036C2×C411yescharacteristic
102#4036C2×C411yescharacteristic
103#4034C2×C411yescharacteristic
104#4034C2×C411yescharacteristic
105#4024—21no—
106#4024—21no—
107C4×Dic10—21no—
108C4×Dic10—21no—
109C5×#108—41no—
110C5×C4⋊C8—21no—
111C5×C4⋊C8—21no—
112C5×#104—22no—
113C5×#103C2×C411yescharacteristic
114C5×#103C2×C411yescharacteristic
115C5×#102C2×C411yescharacteristic
116C5×#102C2×C411yescharacteristic
117#3969Q814yescomplemented
118#3967C2×C411yescharacteristic
119#3967C2×C411yescharacteristic
120#3964C2×C411yescharacteristic
121#3964C2×C411yescharacteristic
122#3954—41no—
123#3950C2×C411yescharacteristic
124#3950C2×C411yescharacteristic
125#3949C2×C411yescharacteristic
126#3949C2×C411yescharacteristic
127#3947—21no—
128#3947—21no—
129#3947—41no—
130#3947—41no—
131#3946—21no—
132#3946—21no—
133#3946—21no—
134#3946—21no—
135#3946—21no—
136#3946—21no—
137#3946—21no—
138#3946—21no—
139#3946C2×C411yescharacteristic
140#3946C2×C411yescharacteristic
141#3946C2×C411yescharacteristic
142#3946C2×C411yescharacteristic
143#3946D411yescharacteristic
144#3946D411yescharacteristic
145#3946D411yescharacteristic
146#3946D411yescharacteristic
147C8×Dic5—21no—
148C8×Dic5—21no—
149#3940C2×C411yescharacteristic
150#3940C2×C411yescharacteristic
151#3939C2×C411yescharacteristic
152#3939C2×C411yescharacteristic
153#3935—21no—
154C4×#168—21no—
155#3076—51no—
156#2962—51no—
157#2906—51no—
158#1994—51no—
159#1950—51no—
160#1946—51no—
161#1946—51no—
162#1922—51no—
163#1922—51no—
164#1866—51no—
165#1866D511yescharacteristic
166#1767—51no—
167#1765—51no—
168#1765—51no—
169C4×#104—51no—
170C5×#41—21no—
171C5×#41—21no—
172C5×#41C2×Q812yes—
173C5×#41#4112yes—
174C2×C5×Q8C22×C411yescharacteristic
175C2×C5×D4—21no—
176C2×C5×D4C22×C411yescharacteristic
177C22×C4×C5—21no—
178C22×C4×C5—21no—
179C22×C4×C5C2411yescharacteristic
180C22×Dic5—21no—
181C22×Dic5—21no—
182C22×Dic5—21no—
183C22×Dic5#4111yescharacteristic
184C22×Dic5#4111yescharacteristic
185C2×Dic10—21no—
186C2×Dic10—21no—
187C5×M16—21no—
188C5×M16—21no—
189C5×M16—21no—
190C5×M16—21no—
191C5×M16—42no—
192C2×C5×C8—21no—
193C2×C5×C8—21no—
194C2×C5×C8—21no—
195C2×C5×C8—21no—
196C2×C5×C8—21no—
197C2×C5×C8—21no—
198C2×C5×C8—42no—
199C2×C5×C8C22×C411yescharacteristic
200C2×C5×C8C22×C411yescharacteristic
201C2×C5×C8C2×D411yescharacteristic
202C2×C5×C8C2×D411yescharacteristic
203C5×C4⋊C4C22×C411yescharacteristic
204C5×C4⋊C4C22×C411yescharacteristic
205C5×C22⋊C4—21no—
206C42×C5—21no—
207C42×C5—21no—
208C42×C5—21no—
209C22⋊Dic5—21no—
210C22⋊Dic5—21no—
211C22⋊Dic5—21no—
212C22⋊Dic5—21no—
213C4⋊Dic5—21no—
214C4⋊Dic5—21no—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5C22×C411yescharacteristic
218C4⋊Dic5C22×C411yescharacteristic
219C4⋊Dic5C22×C411yescharacteristic
220C4⋊Dic5C22×C411yescharacteristic
221#683—21no—
222#683—21no—
223#683—21no—
224#683—21no—
225C4×Dic5—21no—
226C4×Dic5—21no—
227C4×Dic5—21no—
228C4×Dic5—21no—
229C4×Dic5—21no—
230C4×Dic5—42no—
231C4×Dic5—42no—
232C4×Dic5C22×C411yescharacteristic
233C4×Dic5C22×C411yescharacteristic
234C4×Dic5C2×D411yescharacteristic
235C4×Dic5C2×D411yescharacteristic
236C4×Dic5C2×Q812yes—
237#681—21no—
238#681—21no—
239#681—21no—
240#681—42no—
241#681C2×Q812yes—
242C2×#168—21no—
243C2×#168—21no—
244C2×#168—21no—
245C2×#168—21no—
246C2×#168—21no—
247C2×#168—21no—
248C2×#168—42no—
249C2×#168C22×C411yescharacteristic
250C2×#168C22×C411yescharacteristic
251C2×#168C2×D411yescharacteristic
252C2×#168C2×D411yescharacteristic
253C2×#131—51no—
254C2×#131C2×D511yescharacteristic
255#533—51no—
256C4×#41—51no—
257#479—51no—
258#479—51no—
259#478—51no—
260#478—51no—
261#477—51no—
262#476—51no—
263#475—51no—
264#474—51no—
265#446—101no—
266#446—101no—
267#446—101no—
268C8×Q8—101no—
269C2×#108—51no—
270C2×#108C2×D511yescharacteristic
271#427—51no—
272#427—51no—
273#424—51no—
274#424—51no—
275#424—51no—
276#424—101no—
277#424—101no—
278#424C2×D511yescharacteristic
279#423—51no—
280#423—101no—
281C2×C4⋊C8—51no—
282C2×C4⋊C8—51no—
283C2×C4⋊C8—101no—
284C2×#104—51no—
285C2×#104C2×D511yescharacteristic
286#416—51no—
287#416C2×D511yescharacteristic
288#405—51no—
289#405—51no—
290C4×M16—51no—
291C2×#97—51no—
292C2×#97—51no—
293C43—51no—
294#341—51no—
295#341—51no—
296#339—51no—
297#339—51no—
298#339—51no—
299#339—51no—
300#339—51no—
301#339C2×D511yescharacteristic
302#336—51no—
303#336C2×D511yescharacteristic
304C23×C5—21no—
305C23×C5C2×#4111yescharacteristic
306C5×Q8—21no—
307C5×Q8C4×Q812yes—
308C5×D4—22no—
309C5×D4—22no—
310C5×D4—21no—
311C5×D4C4×Q812yes—
312C2×C4×C5—21no—
313C2×C4×C5—22no—
314C2×C4×C5—22no—
315C2×C4×C5—22no—
316C2×C4×C5—21no—
317C2×C4×C5—21no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5C23×C411yescharacteristic
321C2×C4×C5C22×D411yescharacteristic
322C2×C4×C5C22×Q811yescharacteristic
323C2×C4×C5C2×#4111yescharacteristic
324C2×C4×C5H(5,𝔽2)11yescharacteristic
325C2×C4×C52−1+411yescharacteristic
326C2×Dic5—22no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—21no—
330C2×Dic5—21no—
331C2×Dic5—22no—
332C2×Dic5—41no—
333C2×Dic5—41no—
334C2×Dic5—41no—
335C2×Dic5—42no—
336C2×Dic5—81no—
337C2×Dic5C4×D411yescharacteristic
338C2×Dic5C4×D411yescharacteristic
339C2×Dic5C4×D411yescharacteristic
340C2×Dic5C4×D411yescharacteristic
341Dic10—42no—
342Dic10—81no—
343C5×C8—21no—
344C5×C8—21no—
345C5×C8—21no—
346C5×C8—21no—
347C5×C8—41no—
348C5×C8—41no—
349#168—21no—
350#168—21no—
351#168—21no—
352#168—21no—
353#168—22no—
354#168—41no—
355C2×#41C22×D511yescharacteristic
356#131—22no—
357#131—41no—
358#131—102no—
359#131—201no—
360C2×M16—21no—
361C2×M16—51no—
362C2×M16—51no—
363C2×M16—101no—
364C2×M16C22×D511yescharacteristic
365C2×M16C22×D511yescharacteristic
366C22×C8—21no—
367C22×C8—51no—
368C22×C8—51no—
369C22×C8—101no—
370C22×C8C22×D511yescharacteristic
371C22×C8C22×D511yescharacteristic
372C4⋊Q8—51no—
373C4⋊Q8—51no—
374#125—51no—
375#125—51no—
376C22⋊Q8—101no—
377C22⋊Q8—101no—
378C4×Q8—51no—
379C4×Q8—51no—
380C4×Q8—101no—
381C4×Q8—101no—
382C4×D4—51no—
383C4×D4—51no—
384C4×D4—101no—
385C4×D4—101no—
386#117—51no—
387#117—51no—
388#117—101no—
389#117C22×D511yescharacteristic
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—101no—
393C2×C42—101no—
394C2×C42—101no—
395C2×C42C22×D511yescharacteristic
396#108—41no—
397#108—201no—
398#107—51no—
399#107—51no—
400#107—51no—
401#107—51no—
402#106—51no—
403#106—51no—
404#106—51no—
405#106—51no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8—51no—
409C4⋊C8—51no—
410C4⋊C8—51no—
411C4⋊C8—51no—
412C4⋊C8—51no—
413C4⋊C8—51no—
414C4⋊C8—51no—
415C4⋊C8—51no—
416C4⋊C8—101no—
417C4⋊C8—101no—
418C4⋊C8—101no—
419C4⋊C8—101no—
420C4⋊C8—101no—
421C4⋊C8—101no—
422C4⋊C8—101no—
423C4⋊C8—101no—
424#104—22no—
425#104—54no—
426#103—51no—
427#103—51no—
428#103C4×D511yescharacteristic
429#103C4×D511yescharacteristic
430#102—51no—
431#102—51no—
432#102C4×D511yescharacteristic
433#102C4×D511yescharacteristic
434#97—101no—
435#97—101no—
436#97—101no—
437#97—201no—
438#97—201no—
439C4×C8—101no—
440C4×C8—101no—
441C4×C8—101no—
442C22×C5—21no—
443C22×C5—21no—
444C22×C5—22no—
445C22×C5—41no—
446C22×C5—41no—
447C22×C5C2×C4×D411yescharacteristic
448C22×C5#55211yescharacteristic
449C22×C5#55411yescharacteristic
450C4×C5—22no—
451C4×C5—42no—
452C4×C5—41no—
453C4×C5—41no—
454C4×C5C2×C4×Q811yescharacteristic
455C4×C5#52011yescharacteristic
456C4×C5#54611yescharacteristic
457C4×C5D4×Q811yescharacteristic
458Dic5—21no—
459Dic5—21no—
460Dic5—21no—
461Dic5—42no—
462Dic5—41no—
463Dic5—41no—
464Dic5—41no—
465Dic5—41no—
466Dic5#44412yes—
467#41—21no—
468#41—21no—
469#41#71112yes—
470#41Q8×D512yes—
471C2×Q8—101no—
472C2×Q8—101no—
473C2×Q8C2×C4×D511yescharacteristic
474C2×D4—21no—
475C2×D4C2×C4×D511yescharacteristic
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—51no—
479C22×C4—51no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4C23×D511yescharacteristic
484M16—21no—
485M16—21no—
486M16—21no—
487M16—21no—
488M16—42no—
489M16—52no—
490M16—101no—
491M16—101no—
492M16—101no—
493M16—202no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—42no—
501C2×C8—51no—
502C2×C8—51no—
503C2×C8—51no—
504C2×C8—51no—
505C2×C8—101no—
506C2×C8—101no—
507C2×C8—101no—
508C2×C8—101no—
509C2×C8—101no—
510C2×C8—101no—
511C2×C8—202no—
512C2×C8C2×C4×D511yescharacteristic
513C2×C8C2×C4×D511yescharacteristic
514C2×C8D4×D511yescharacteristic
515C2×C8D4×D511yescharacteristic
516C4⋊C4—51no—
517C4⋊C4—51no—
518C4⋊C4—51no—
519C4⋊C4—51no—
520C4⋊C4—101no—
521C4⋊C4—101no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—101no—
526C4⋊C4—101no—
527C4⋊C4—101no—
528C4⋊C4C2×C4×D511yescharacteristic
529C4⋊C4C2×C4×D511yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—101no—
532C22⋊C4—101no—
533C22⋊C4—101no—
534C22⋊C4—101no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—51no—
539C42—51no—
540C42—51no—
541C42—51no—
542C42—52no—
543C42—101no—
544C42—101no—
545C42—101no—
546C42—101no—
547C42—101no—
548C42—202no—
549C42—202no—
550C2×C5—21no—
551C2×C5—21no—
552C2×C5—42no—
553C2×C5#236011yescharacteristic
554C2×C5C2×#44411yescharacteristic
555C2×C5#295611yescharacteristic
556C23—21no—
557C23D5×#4111yescharacteristic
558Q8—21no—
559Q8—202no—
560Q8—401no—
561Q8#403312yes—
562D4—22no—
563D4—22no—
564D4—21no—
565D4#403312yes—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—51no—
575C2×C4—51no—
576C2×C4—51no—
577C2×C4—51no—
578C2×C4—102no—
579C2×C4—101no—
580C2×C4—101no—
581C2×C4—101no—
582C2×C4—101no—
583C2×C4—102no—
584C2×C4—202no—
585C2×C4—201no—
586C2×C4—201no—
587C2×C4—201no—
588C2×C4—401no—
589C2×C4C22×C4×D511yescharacteristic
590C2×C4C2×D4×D511yescharacteristic
591C2×C4C2×#71111yescharacteristic
592C2×C4#414411yescharacteristic
593C2×C4C2×Q8×D511yescharacteristic
594C2×C4#415011yescharacteristic
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—41no—
600C8—41no—
601C8—101no—
602C8—101no—
603C8—101no—
604C8—101no—
605C8—102no—
606C8—201no—
607C5#2298311yescharacteristic, complemented, Sylow
608C22—21no—
609C22—21no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22C4×D4×D511yescharacteristic
614C22#6597011yescharacteristic
615C22#6598411yescharacteristic
616C4—22no—
617C4—42no—
618C4—41no—
619C4—41no—
620C4—52no—
621C4—101no—
622C4—101no—
623C4—101no—
624C4—202no—
625C4—201no—
626C4—201no—
627C4—201no—
628C4—201no—
629C4C2×#403311yescharacteristic
630C4#6588011yescharacteristic
631C4#6592511yescharacteristic
632C4#6596611yescharacteristic
633C2—21no—
634C2—21no—
635C2—42no—
636C2#1060612711yescharacteristic
637C2#1061162011yescharacteristic
638C2D5×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1866C2
socleC22×C5C2×C4×D4
prime radicalC2×C5×#41C23
cosocleC22×C4×C5C24
hypercenter#1866D5
hypocenterC5#22983
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1866

Complements

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

NH ≅ G/Ntype
2. #10610809C2semidirect
6. #10601498C2semidirect
7. #10601488C2semidirect
8. #10601331C2semidirect
12. #10599536C2semidirect
13. #10599524C2semidirect
14. #10599438C2semidirect
15. #10599396C2semidirect
49. #65088C4semidirect
50. #65085C4semidirect
51. #65083C4semidirect
52. #65080C4semidirect
53. #65055C4semidirect
54. #65052C4semidirect
55. #65050C4semidirect
56. #65048C4semidirect
117. #3969Q8semidirect
607. C5#22983semidirect

Maximal subgroups

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

subgroupindex
#106155142
#106108092
#106105532
C5×#18662
#106028022
#106014982
#106014882
#106013312
#106011692
Dic5×#1042
#106010832
#105995362
#105995242
#105994382
#105993962
#229835