Finite Group Databasebeta

← #12107150

Subgroups — #12107150

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10610927C211yesmaximal, characteristic
2#10610281C211yesmaximal, characteristic
3#10610274C211yesmaximal, characteristic, complemented
4C5×#1773C211yesmaximal, characteristic, complemented
5#10601515C211yesmaximal, characteristic
6#10601488C211yesmaximal, characteristic, complemented
7#10601454C211yesmaximal, characteristic, complemented
8#10600682C211yesmaximal, characteristic, complemented
9#10600680C211yesmaximal, characteristic
10#10600632C211yesmaximal, characteristic
11#10600604C211yesmaximal, characteristic, complemented
12#10599567C211yesmaximal, characteristic
13#10599563C211yesmaximal, characteristic
14#10599536C211yesmaximal, characteristic, complemented
15#10599523C211yesmaximal, characteristic, complemented
16C2×#4053C2211yescharacteristic
17C2×#4047C2211yescharacteristic
18#65856C2211yescharacteristic
19#65851C2211yescharacteristic
20C2×C5×#108C2211yescharacteristic
21C5×#427C2211yescharacteristic
22C5×#405C2211yescharacteristic
23C5×#405C2211yescharacteristic
24C2×C5×#97C2211yescharacteristic
25C2×#3957C2211yescharacteristic
26C2×#3954C2211yescharacteristic
27#65408C2211yescharacteristic
28#65407C2211yescharacteristic
29M16×Dic5C2211yescharacteristic
30#65402C2211yescharacteristic
31#65390C2211yescharacteristic
32C2×#3947C2211yescharacteristic
33C2×#3946C2211yescharacteristic
34C2×C8×Dic5C2211yescharacteristic
35#65291C2211yescharacteristic
36#65288C2211yescharacteristic
37#65287C2211yescharacteristic
38#65286C2211yescharacteristic
39C4×#683C2211yescharacteristic
40C2×#3937C2211yescharacteristic
41#65180C411yescharacteristic, complemented
42#65179C411yescharacteristic, complemented
43#65166C411yescharacteristic, complemented
44#65164C411yescharacteristic, complemented
45#65160C411yescharacteristic, complemented
46#65159C411yescharacteristic, complemented
47#65146C411yescharacteristic, complemented
48#65143C411yescharacteristic, complemented
49#64997—21no—
50D5×#97—21no—
51#64990—21no—
52#64990—21no—
53#64978—21no—
54C4×#676—21no—
55#64968—21no—
56#64968—21no—
57C5×#339C2211yescharacteristic
58C5×#339C2211yescharacteristic
59#64778C2211yescharacteristic
60#64775C2211yescharacteristic
61#64772C2211yescharacteristic
62#64770C2211yescharacteristic
63#64752C2211yescharacteristic
64#64752C2211yescharacteristic
65#64744C2211yescharacteristic
66#64743C2211yescharacteristic
67#21629—51nomaximal, Sylow
68C2×#709C2311yescharacteristic
69C2×C5×M16C2311yescharacteristic
70C2×C5×M16C2311yescharacteristic
71C22×C5×C8C2311yescharacteristic
72C22×C5×C8C2311yescharacteristic
73C5×#117C2311yescharacteristic
74C5×#117C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76#4072C2311yescharacteristic
77C2×#683—21no—
78C2×C4×Dic5C2311yescharacteristic
79C2×C4×Dic5C2311yescharacteristic
80C2×#681C2311yescharacteristic
81C2×#681C2311yescharacteristic
82C22×#168C2311yescharacteristic
83C22×#168C2311yescharacteristic
84#4053—22no—
85#4053—22no—
86#4047—22no—
87#4047Q814yescomplemented
88C2×#676—21no—
89C2×#676—21no—
90C2×C8×D5—21no—
91C2×C8×D5—21no—
92#4039C2×C411yescharacteristic
93#4039C2×C411yescharacteristic
94#4036C2×C411yescharacteristic
95#4036C2×C411yescharacteristic
96#4034C2×C411yescharacteristic
97#4034C2×C411yescharacteristic
98#4033C2×C411yescharacteristic
99#4033C2×C411yescharacteristic
100#4027—21no—
101#4027—21no—
102#4024—21no—
103#4024—21no—
104#4018—21no—
105C42×D5—21no—
106C4×Dic10—21no—
107C4×Dic10—21no—
108C5×#108—22no—
109C5×#107C2×C411yescharacteristic
110C5×#107C2×C411yescharacteristic
111C5×#106C2×C411yescharacteristic
112C5×#106C2×C411yescharacteristic
113C5×#97—21no—
114C5×#97—21no—
115C5×#97—22no—
116C4×C5×C8—21no—
117C4×C5×C8—21no—
118#3963—21no—
119#3957—22no—
120#3954—22no—
121#3953C2×C411yescharacteristic
122#3953C2×C411yescharacteristic
123#3948C2×C411yescharacteristic
124#3948C2×C411yescharacteristic
125#3947—21no—
126#3947—21no—
127#3947D411yescharacteristic
128#3947D411yescharacteristic
129#3947D411yescharacteristic
130#3947D411yescharacteristic
131#3946—21no—
132#3946—21no—
133#3946—21no—
134#3946—21no—
135#3946C2×C411yescharacteristic
136#3946C2×C411yescharacteristic
137#3946C2×C411yescharacteristic
138#3946C2×C411yescharacteristic
139C8×Dic5—21no—
140C8×Dic5—22no—
141C8×Dic5—21no—
142#3942C2×C411yescharacteristic
143#3942C2×C411yescharacteristic
144#3941C2×C411yescharacteristic
145#3941C2×C411yescharacteristic
146#3940C2×C411yescharacteristic
147#3940C2×C411yescharacteristic
148#3939C2×C411yescharacteristic
149#3939C2×C411yescharacteristic
150#3937—22no—
151C4⋊#168—21no—
152C4⋊#168—21no—
153#3935—21no—
154C4×#168—21no—
155#3076—51no—
156#2961—51no—
157#2875—51no—
158#1986—51no—
159#1950—51no—
160#1922—51no—
161#1922—51no—
162#1866—51no—
163#1866—51no—
164#1806—51no—
165#1773D511yescharacteristic, complemented
166#1772—51no—
167#1771—51no—
168#1765—51no—
169#1764—51no—
170C22×C4×C5—21no—
171C22×C4×C5C2411yescharacteristic
172C2×C22⋊D5—21no—
173C22×Dic5—21no—
174C22×Dic5#4111yescharacteristic
175C22×Dic5#4111yescharacteristic
176#709—21no—
177#709—21no—
178#709C2×Q812yes—
179#709#4112yes—
180C2×D20C22×C411yescharacteristic
181C2×C4×D5—21no—
182C2×C4×D5—21no—
183C2×C4×D5—21no—
184C2×Dic10—21no—
185C2×Dic10—21no—
186C2×Dic10C22×C411yescharacteristic
187C5×M16—21no—
188C5×M16—21no—
189C5×M16—21no—
190C5×M16#4112yes—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8—21no—
194C2×C5×C8—21no—
195C2×C5×C8C22×C411yescharacteristic
196C2×C5×C8C22×C411yescharacteristic
197C2×C5×C8C2×D411yescharacteristic
198C2×C5×C8C2×D411yescharacteristic
199C2×C5×C8C2×Q812yes—
200C2×C5×C8#4112yes—
201C5×C4⋊C4C22×C411yescharacteristic
202C5×C4⋊C4C22×C411yescharacteristic
203C5×C4⋊C4C22×C411yescharacteristic
204C5×C4⋊C4C22×C411yescharacteristic
205C5×C22⋊C4—21no—
206C5×C22⋊C4—21no—
207C42×C5—21no—
208C42×C5—21no—
209C42×C5—21no—
210C42×C5—21no—
211C22⋊Dic5—21no—
212#685—21no—
213#685—21no—
214C4⋊Dic5—21no—
215C4⋊Dic5—21no—
216C4⋊Dic5C22×C411yescharacteristic
217C4⋊Dic5C22×C411yescharacteristic
218#683—21no—
219#683—21no—
220#683—21no—
221#683—21no—
222#683—21no—
223#683—21no—
224#683—42no—
225C4×Dic5—21no—
226C4×Dic5—21no—
227C4×Dic5—21no—
228C4×Dic5—21no—
229C4×Dic5—21no—
230C4×Dic5C22×C411yescharacteristic
231C4×Dic5C22×C411yescharacteristic
232C4×Dic5C2×D411yescharacteristic
233C4×Dic5C2×D411yescharacteristic
234#681—21no—
235#681—21no—
236#681—21no—
237#681C2×Q812yes—
238C2×#168—21no—
239C2×#168—21no—
240C2×#168—21no—
241C2×#168—21no—
242C2×#168—21no—
243C2×#168—21no—
244C2×#168C22×C411yescharacteristic
245C2×#168C22×C411yescharacteristic
246C2×#168C2×D411yescharacteristic
247C2×#168C2×D411yescharacteristic
248#676—24no—
249#676—24no—
250C8×D5—24no—
251C8×D5—24no—
252C2×#131—51no—
253C2×#131—51no—
254#533—51no—
255C4×#41—51no—
256#479—51no—
257#479—51no—
258#478—51no—
259#478—51no—
260#477—51no—
261#476—51no—
262#475—51no—
263#474—51no—
264#446—101no—
265#446—101no—
266#446—101no—
267#446—101no—
268C2×#108—51no—
269C2×#108C2×D511yescharacteristic
270#427—51no—
271#427C2×D511yescharacteristic
272#424—51no—
273#424—51no—
274#423—51no—
275C2×C4⋊C8—51no—
276C2×#104—51no—
277C2×#104—51no—
278#416—51no—
279#416—51no—
280#405—51no—
281#405—51no—
282#405—101no—
283#405—101no—
284#405C2×D511yescharacteristic
285#405C2×D511yescharacteristic
286C4×M16—51no—
287C4×M16—101no—
288C2×#97—51no—
289C2×#97—101no—
290C2×#97C2×D511yescharacteristic
291C2×C4×C8—51no—
292C4×C4⋊C4—51no—
293#341—51no—
294#341—51no—
295#339—51no—
296#339—51no—
297#339—51no—
298#339—51no—
299#339C2×D511yescharacteristic
300#339C2×D511yescharacteristic
301#336—51no—
302#336—51no—
303C23×C5C2×#4111yescharacteristic
304C22×D5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5C23×C411yescharacteristic
313C2×C4×C5C22×D411yescharacteristic
314C2×C4×C5C22×Q811yescharacteristic
315C2×C4×C5C2×#4111yescharacteristic
316C2×C4×C5C2×#4111yescharacteristic
317C2×C4×C5C2×#4111yescharacteristic
318C22⋊D5—22no—
319C22⋊D5—22no—
320C2×Dic5—21no—
321C2×Dic5—22no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—41no—
325C2×Dic5—41no—
326C2×Dic5—41no—
327C2×Dic5—41no—
328C2×Dic5C4×D411yescharacteristic
329C2×Dic5C4×D411yescharacteristic
330C2×Dic5C4×D411yescharacteristic
331C2×Dic5C4×D411yescharacteristic
332D20—21no—
333D20C4×Q812yes—
334C4×D5—22no—
335C4×D5—22no—
336C4×D5—81no—
337C4×D5—81no—
338Dic10—21no—
339Dic10—81no—
340Dic10—81no—
341Dic10C4×Q812yes—
342C5×C8—21no—
343C5×C8—21no—
344C5×C8—22no—
345C5×C8—41no—
346C5×C8C22⋊Q812yes—
347C5×C8C22⋊Q812yes—
348#168—21no—
349#168—21no—
350#168—22no—
351#168—21no—
352#168—21no—
353#168—41no—
354C2×#41—51no—
355#131—54no—
356#131—102no—
357#131—102no—
358#131—102no—
359C2×M16—51no—
360C2×M16—51no—
361C2×M16—101no—
362C2×M16—101no—
363C2×M16C22×D511yescharacteristic
364C2×M16C22×D511yescharacteristic
365C22×C8—51no—
366C22×C8—51no—
367C22×C8—101no—
368C22×C8—101no—
369C22×C8C22×D511yescharacteristic
370C22×C8C22×D511yescharacteristic
371C4⋊Q8—51no—
372C4⋊Q8—51no—
373#125—51no—
374#125—51no—
375C22⋊Q8—101no—
376C22⋊Q8—101no—
377C4×Q8—51no—
378C4×Q8—51no—
379C4×Q8—101no—
380C4×Q8—101no—
381C4×D4—51no—
382C4×D4—51no—
383C4×D4—101no—
384C4×D4—101no—
385#117—51no—
386#117—101no—
387#117C22×D511yescharacteristic
388#117C22×D511yescharacteristic
389C2×C4⋊C4—101no—
390C2×C42—51no—
391C2×C42—51no—
392C2×C42—101no—
393C2×C42C22×D511yescharacteristic
394#108—22no—
395#108—102no—
396#107—51no—
397#107—51no—
398#107C4×D511yescharacteristic
399#107C4×D511yescharacteristic
400#106—51no—
401#106—51no—
402#106C4×D511yescharacteristic
403#106C4×D511yescharacteristic
404C4⋊C8—51no—
405C4⋊C8—51no—
406C4⋊C8—51no—
407C4⋊C8—51no—
408C4⋊C8—101no—
409C4⋊C8—101no—
410C4⋊C8—101no—
411C4⋊C8—101no—
412C4⋊C8—101no—
413C4⋊C8—101no—
414#104—102no—
415#104—102no—
416#103—51no—
417#103—51no—
418#103—51no—
419#103—51no—
420#102—51no—
421#102—51no—
422#102—51no—
423#102—51no—
424#97—22no—
425#97—21no—
426#97—21no—
427#97—51no—
428#97—51no—
429#97—51no—
430#97—51no—
431#97—101no—
432#97—101no—
433#97—101no—
434C4×C8—21no—
435C4×C8—21no—
436C4×C8—101no—
437C4×C8—101no—
438C4×C8—102no—
439C4×C8—101no—
440#95—101no—
441C22×C5—21no—
442C22×C5—21no—
443C22×C5C2×C4×D411yescharacteristic
444C22×C5#52911yescharacteristic
445C22×C5#53311yescharacteristic
446C2×D5—22no—
447C2×D5—41no—
448C2×D5—41no—
449C4×C5—41no—
450C4×C5—41no—
451C4×C5—41no—
452C4×C5—41no—
453C4×C5—81no—
454C4×C5C2×C4×Q811yescharacteristic
455C4×C5C4×#4111yescharacteristic
456C4×C5C2×C22⋊Q811yescharacteristic
457C4×C5#52511yescharacteristic
458Dic5—21no—
459Dic5—21no—
460Dic5—21no—
461Dic5—21no—
462Dic5—22no—
463Dic5—42no—
464Dic5—41no—
465Dic5—41no—
466#41—52no—
467#41—52no—
468#41—101no—
469#41—101no—
470C2×Q8—51no—
471C2×Q8—101no—
472C2×Q8—101no—
473C2×D4—51no—
474C2×D4—101no—
475C22×C4—21no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—101no—
479C22×C4—101no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4C23×D511yescharacteristic
483M16—21no—
484M16—21no—
485M16—21no—
486M16—52no—
487M16—104no—
488M16—101no—
489M16—104no—
490M16—101no—
491M16—101no—
492M16#70912yes—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—51no—
498C2×C8—51no—
499C2×C8—51no—
500C2×C8—51no—
501C2×C8—104no—
502C2×C8—101no—
503C2×C8—101no—
504C2×C8—101no—
505C2×C8—101no—
506C2×C8—101no—
507C2×C8—104no—
508C2×C8—101no—
509C2×C8C2×C4×D511yescharacteristic
510C2×C8C2×C4×D511yescharacteristic
511C2×C8#70912yes—
512C2×C8D4×D511yescharacteristic
513C2×C8D4×D511yescharacteristic
514C2×C8Q8×D512yes—
515C4⋊C4—51no—
516C4⋊C4—51no—
517C4⋊C4—101no—
518C4⋊C4—101no—
519C4⋊C4—101no—
520C4⋊C4—101no—
521C4⋊C4—101no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—202no—
526C4⋊C4C2×C4×D511yescharacteristic
527C4⋊C4C2×C4×D511yescharacteristic
528C4⋊C4C2×C4×D511yescharacteristic
529C4⋊C4C2×C4×D511yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—101no—
533C22⋊C4—101no—
534C22⋊C4—101no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—51no—
540C42—51no—
541C42—51no—
542C42—51no—
543C42—101no—
544C42—101no—
545C42—101no—
546C42—101no—
547C42—101no—
548C2×C5—21no—
549C2×C5—21no—
550C2×C5C4×C22⋊Q811yescharacteristic
551C2×C5#295611yescharacteristic
552C2×C5#295711yescharacteristic
553D5—42no—
554C23—101no—
555C23D5×#4111yescharacteristic
556Q8—52no—
557Q8—101no—
558Q8—401no—
559Q8—401no—
560D4—52no—
561D4—102no—
562D4—102no—
563D4—101no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—51no—
572C2×C4—51no—
573C2×C4—51no—
574C2×C4—51no—
575C2×C4—101no—
576C2×C4—102no—
577C2×C4—102no—
578C2×C4—102no—
579C2×C4—101no—
580C2×C4—101no—
581C2×C4—201no—
582C2×C4—201no—
583C2×C4—201no—
584C2×C4—201no—
585C2×C4—401no—
586C2×C4—401no—
587C2×C4C22×C4×D511yescharacteristic
588C2×C4C2×#70911yescharacteristic
589C2×C4C2×#70911yescharacteristic
590C2×C4C2×D4×D511yescharacteristic
591C2×C4C2×Q8×D511yescharacteristic
592C2×C4D5×#4111yescharacteristic
593C8—21no—
594C8—21no—
595C8—22no—
596C8—41no—
597C8—101no—
598C8—101no—
599C8—102no—
600C8—101no—
601C8—101no—
602C8—201no—
603C8#404212yes—
604C8#404212yes—
605C5#2162911yescharacteristic, complemented, Sylow
606C22—21no—
607C22—21no—
608C22—102no—
609C22—201no—
610C22—201no—
611C22#6586211yescharacteristic
612C22C4×D4×D511yescharacteristic
613C22#6591211yescharacteristic
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—101no—
620C4—101no—
621C4—101no—
622C4—101no—
623C4—102no—
624C4—202no—
625C4—201no—
626C4—201no—
627C4C4×#70911yescharacteristic
628C4C2×#404211yescharacteristic
629C4#6588111yescharacteristic
630C4C4×Q8×D511yescharacteristic
631C2—21no—
632C2—21no—
633C2—202no—
634C2C4×#404211yescharacteristic
635C2#1061162911yescharacteristic
636C2#1061163011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1773C2
socleC22×C5C2×C4×D4
prime radicalC2×#709C23
cosocleC22×C4×C5C24
hypercenter#1773D5
hypocenterC5#21629
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
3. #10610274C2semidirect
4. C5×#1773C2semidirect
6. #10601488C2semidirect
7. #10601454C2semidirect
8. #10600682C2semidirect
11. #10600604C2semidirect
14. #10599536C2semidirect
15. #10599523C2semidirect
41. #65180C4semidirect
42. #65179C4semidirect
43. #65166C4semidirect
44. #65164C4semidirect
45. #65160C4semidirect
46. #65159C4semidirect
47. #65146C4semidirect
48. #65143C4semidirect
87. #4047Q8semidirect
165. #1773D5semidirect
605. C5#21629semidirect

Maximal subgroups

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

subgroupindex
#106109272
#106102812
#106102742
C5×#17732
#106015152
#106014882
#106014542
#106006822
#106006802
#106006322
#106006042
#105995672
#105995632
#105995362
#105995232
#216295