Finite Group Databasebeta

← #12121599

Subgroups — #12121599

Order 1280.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2351C211yesmaximal, characteristic, complemented
2C2×#65317C211yesmaximal, characteristic
3C2×C4×#689C211yesmaximal, characteristic, complemented
4C2×C4×#688C211yesmaximal, characteristic
5#10605386C211yesmaximal, characteristic
6#10602426C211yesmaximal, characteristic
7#10602425C211yesmaximal, characteristic, complemented
8C4×#3967C211yesmaximal, characteristic, complemented
9#10600425C211yesmaximal, characteristic, complemented
10#10600418C211yesmaximal, characteristic
11#10600408C211yesmaximal, characteristic
12#10600407C211yesmaximal, characteristic, complemented
13#10600344C211yesmaximal, characteristic
14#10600333C211yesmaximal, characteristic, complemented
15C4⋊C4×#168C211yesmaximal, characteristic, complemented
16C2×C4×C5×Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C22×#689C2211yescharacteristic
20C22×#688C2211yescharacteristic
21C2×#4021C2211yescharacteristic
22C2×C4×D20C2211yescharacteristic
23C2×C4×Dic10C2211yescharacteristic
24C5×#389—21no—
25C5×#389—21no—
26C5×#389C2211yescharacteristic
27C5×#389C2211yescharacteristic
28C5×#382—21no—
29C5×#382C2211yescharacteristic
30C4×C5×C4⋊C4—21no—
31C4×C5×C4⋊C4C2211yescharacteristic
32C2×#3967C2211yescharacteristic
33C2×#3967C2211yescharacteristic
34C4×#689—24no—
35#65317C418yescomplemented
36C4×#688—24no—
37#65279—21no—
38#65278—21no—
39C2×#3942C2211yescharacteristic
40C2×#3942C2211yescharacteristic
41C2×#3942C2211yescharacteristic
42C2×#3941C2211yescharacteristic
43C2×#3941C2211yescharacteristic
44C2×#3941C2211yescharacteristic
45C2×#3940C2211yescharacteristic
46C2×#3939C2211yescharacteristic
47#65228C2211yescharacteristic
48C4×C4⋊Dic5C2211yescharacteristic
49#65218C2211yescharacteristic
50C2×C4⋊#168C2211yescharacteristic
51C2×C4×#168C2211yescharacteristic
52C2×C4×#168C2211yescharacteristic
53C2×C4×#168C2211yescharacteristic
54#64958—21no—
55#64955—21no—
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#64744C2211yescharacteristic
61#64744C2211yescharacteristic
62#23000—51nomaximal, Sylow
63C22×C5×Q8C2311yescharacteristic
64C22×D20C2311yescharacteristic
65C22×C4×D5—21no—
66C22×Dic10C2311yescharacteristic
67C4×C5×Q8—22no—
68C4×C5×Q8—22no—
69C4×C5×Q8—22no—
70C4×C5×Q8—41no—
71C4×C5×Q8—41no—
72C4×C5×Q8C2×C414yescomplemented
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4D411yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C42×C5—21no—
84C2×C42×C5—21no—
85C2×C42×C5—21no—
86C2×C42×C5C2311yescharacteristic
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2311yescharacteristic
89C2×#689—22no—
90C2×#689—22no—
91C2×#689C2×C414yescomplemented
92C2×#688—22no—
93C2×#688—22no—
94C2×#688C2×C414yes—
95C2×#685—21no—
96C2×#685—21no—
97C2×#685—21no—
98C2×#685—21no—
99C2×#685—21no—
100C2×C4⋊Dic5C2311yescharacteristic
101C2×C4⋊Dic5C2311yescharacteristic
102C2×#683—21no—
103C2×C4×Dic5—21no—
104C2×C4×Dic5—21no—
105C22×#168C2311yescharacteristic
106C22×#168C2311yescharacteristic
107C22×#168C2311yescharacteristic
108C22×#168C2311yescharacteristic
109#4021—22no—
110#4021C2×C414yes—
111C4×D20—22no—
112C4×D20—41no—
113C4×Dic10—22no—
114C4×Dic10—41no—
115C5×#95—21no—
116C5×#95—21no—
117C5×#95—21no—
118C5×#95—21no—
119C5×#95D411yescharacteristic
120C5×#95D411yescharacteristic
121#3967—22no—
122#3967D414yes—
123#3963—21no—
124#3963—21no—
125#3963—21no—
126#3942—22no—
127#3942—41no—
128#3942C2×C414yescomplemented
129#3941—22no—
130#3941—41no—
131#3941C2×C414yes—
132#3940—22no—
133#3939—22no—
134C4⋊#168C2×C414yescomplemented
135C4×#168—22no—
136C4×#168—22no—
137C4×#168—22no—
138C2×#464—51no—
139C2×C4×Q16—51no—
140C2×C4×SD16—51no—
141#2367—51no—
142#2351D511yescharacteristic, complemented
143#1958—51no—
144#1949—51no—
145#1934—51no—
146#1933—51no—
147#1880—51no—
148#1878—51no—
149#1865—51no—
150#1862—51no—
151C8×C4⋊C4—51no—
152C4×#103—51no—
153C23×D5—21no—
154C2×C5×Q8—21no—
155C2×C5×Q8—21no—
156C2×C5×Q8—21no—
157C2×C5×Q8C22×C412yes—
158C2×C5×Q8C2×D412yes—
159C2×C5×Q8#4112yes—
160C22×C4×C5—21no—
161C22×C4×C5—21no—
162C22×C4×C5—21no—
163C22×C4×C5—21no—
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×D411yescharacteristic
169C22×C4×C5C2×D411yescharacteristic
170C22×C4×C5C2411yescharacteristic
171C22×Dic5—21no—
172C22×Dic5—21no—
173C22×Dic5—21no—
174C2×D20—21no—
175C2×D20—21no—
176C2×D20—21no—
177C2×D20—21no—
178C2×D20—21no—
179C2×D20C22×C412yes—
180C2×C4×D5—42no—
181C2×C4×D5—42no—
182C2×C4×D5—81no—
183C2×Dic10—21no—
184C2×Dic10—21no—
185C2×Dic10—21no—
186C2×Dic10—21no—
187C2×Dic10—21no—
188C2×Dic10C22×C412yes—
189C5×C4⋊C4—24no—
190C5×C4⋊C4—22no—
191C5×C4⋊C4—22no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—21no—
195C5×C4⋊C4—21no—
196C5×C4⋊C4—21no—
197C5×C4⋊C4—42no—
198C5×C4⋊C4—42no—
199C5×C4⋊C4—42no—
200C5×C4⋊C4—42no—
201C5×C4⋊C4C22×C412yes—
202C42×C5—24no—
203C42×C5—21no—
204C42×C5—21no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5—42no—
208C42×C5—42no—
209C42×C5C22×C412yes—
210C42×C5#4112yes—
211#689—28no—
212#688—28no—
213#685—24no—
214#685—24no—
215#685—42no—
216#685—42no—
217#685—81no—
218C4⋊Dic5—21no—
219C4⋊Dic5—21no—
220C4⋊Dic5—21no—
221C4⋊Dic5C2×D412yes—
222#683—42no—
223C4×Dic5—42no—
224C4×Dic5—81no—
225C2×#168—21no—
226C2×#168—21no—
227C2×#168—21no—
228C2×#168—21no—
229C2×#168—21no—
230C2×#168—21no—
231C2×#168—21no—
232C2×#168—21no—
233C2×#168—21no—
234C2×#168C22×C412yes—
235C2×#168C22×C412yes—
236C2×#168C2×D412yes—
237C22×Q16—51no—
238C22×SD16—51no—
239C2×#124—51no—
240C2×C4×Q8—51no—
241C2×C4×Q8C2×D511yescharacteristic
242C2×C4×Q8C2×D511yescharacteristic
243C2×C4×Q8C2×D511yescharacteristic
244C2×C4×D4—51no—
245#464—58no—
246C4×Q16—104no—
247C4×SD16—104no—
248C2×#107—51no—
249C2×#106—51no—
250C2×C4⋊C8—51no—
251C2×#103—51no—
252C2×#103—51no—
253C2×#103—51no—
254C2×#103—51no—
255C2×#103—51no—
256C2×#102—51no—
257C2×#102—51no—
258C2×#102—51no—
259C2×C4×C8—51no—
260C2×C4×C8—51no—
261C2×C4×C8—51no—
262#391—51no—
263#390—51no—
264#389—21no—
265#389—21no—
266#389C2×D511yescharacteristic
267#389C2×D511yescharacteristic
268#388—101no—
269#388—101no—
270#385—101no—
271#382—21no—
272#382C2×D511yescharacteristic
273C4×C4⋊C4—21no—
274C4×C4⋊C4—51no—
275C4×C4⋊C4C2×D511yescharacteristic
276C4×C22⋊C4—101no—
277#340—51no—
278#340—51no—
279#340—51no—
280#340—51no—
281#336—51no—
282#336—51no—
283C23×C5C22×D411yescharacteristic
284C22×D5—22no—
285C22×D5—41no—
286C22×D5—41no—
287C22×D5—41no—
288C22×D5—41no—
289C22×D5—41no—
290C22×D5—41no—
291C5×Q8—22no—
292C5×Q8—22no—
293C5×Q8—22no—
294C5×Q8C4×D414yes—
295C2×C4×C5—22no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—22no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5—21no—
311C2×C4×C5—21no—
312C2×C4×C5—21no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5—41no—
319C2×C4×C5—41no—
320C2×C4×C5—41no—
321C2×C4×C5C4×D412yes—
322C2×C4×C5C4×D412yes—
323C2×C4×C5#13512yes—
324C2×C4×C5#13512yes—
325C2×C4×C5C23×C411yescharacteristic
326C2×C4×C5C22×D411yescharacteristic
327C2×C4×C5C2×#4111yescharacteristic
328C2×C4×C5C2×#4111yescharacteristic
329C2×C4×C5H(5,𝔽2)11yescharacteristic
330C2×C4×C52−1+411yescharacteristic
331C2×Dic5—22no—
332C2×Dic5—22no—
333C2×Dic5—41no—
334C2×Dic5—41no—
335C2×Dic5—41no—
336C2×Dic5—41no—
337C2×Dic5—41no—
338C2×Dic5—41no—
339C2×Dic5—41no—
340D20—22no—
341D20—22no—
342D20—41no—
343D20—41no—
344C4×D5—84no—
345Dic10—22no—
346Dic10—22no—
347Dic10—41no—
348Dic10—41no—
349#168—22no—
350#168—22no—
351#168—22no—
352#168—41no—
353C22×Q8—51no—
354C22×Q8C22×D511yescharacteristic
355C22×D4—51no—
356C23×C4—101no—
357C2×Q16—54no—
358C2×Q16—102no—
359C2×Q16—102no—
360C2×SD16—54no—
361C2×SD16—102no—
362C2×SD16—102no—
363C22×C8—51no—
364C22×C8—51no—
365C22×C8—51no—
366C22×C8—51no—
367#124—54no—
368#124—102no—
369C4×Q8—22no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—41no—
373C4×Q8—41no—
374C4×Q8—102no—
375C4×Q8—201no—
376C4×Q8C4×D514yescomplemented
377C4×D4—102no—
378C4×D4—201no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—51no—
385C2×C4⋊C4—51no—
386C2×C4⋊C4—101no—
387C2×C4⋊C4C22⋊D511yescharacteristic
388C2×C4⋊C4C22⋊D511yescharacteristic
389C2×C4⋊C4C22×D511yescharacteristic
390C2×C4⋊C4C22×D511yescharacteristic
391C2×C4⋊C4C22×D511yescharacteristic
392C2×C22⋊C4—101no—
393C2×C22⋊C4—101no—
394C2×C22⋊C4—101no—
395C2×C22⋊C4—101no—
396C2×C22⋊C4—101no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—101no—
401C2×C42—101no—
402C2×C42C22×D511yescharacteristic
403C2×C42C22×D511yescharacteristic
404C2×C42C22×D511yescharacteristic
405#107—102no—
406#106—102no—
407C4⋊C8—54no—
408#103—54no—
409#103—54no—
410#103—102no—
411#103—102no—
412#103—201no—
413#102—54no—
414#102—102no—
415#102—201no—
416C4×C8—102no—
417C4×C8—102no—
418C4×C8—102no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—101no—
424#95—101no—
425#95—101no—
426#95C22⋊D511yescharacteristic
427#95C22⋊D511yescharacteristic
428C22×C5C2×C4×D411yescharacteristic
429C22×C5D4211yescharacteristic
430C22×C5#54811yescharacteristic
431C22×C5C2×#13511yescharacteristic
432C22×C5C2×#13511yescharacteristic
433C22×C5#57711yescharacteristic
434C22×C5#57811yescharacteristic
435C2×D5—42no—
436C2×D5—42no—
437C2×D5—42no—
438C2×D5—81no—
439C2×D5—81no—
440C2×D5—81no—
441C2×D5—81no—
442C4×C5—24no—
443C4×C5—22no—
444C4×C5—22no—
445C4×C5—22no—
446C4×C5—22no—
447C4×C5—42no—
448C4×C5—41no—
449C4×C5—41no—
450C4×C5—81no—
451C4×C5—81no—
452C4×C5C2×C4×D411yescharacteristic
453C4×C5#52011yescharacteristic
454C4×C5#54711yescharacteristic
455C4×C5#55311yescharacteristic
456Dic5—42no—
457Dic5—42no—
458Dic5—81no—
459C24—101no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—52no—
464C2×Q8—101no—
465C2×Q8—101no—
466C2×Q8—101no—
467C2×Q8—101no—
468C2×Q8—101no—
469C2×Q8C2×C4×D512yes—
470C2×Q8D4×D512yes—
471C2×Q8#71312yes—
472C2×D4—52no—
473C2×D4—101no—
474C2×D4—101no—
475C2×D4—101no—
476C2×D4—101no—
477C2×D4—101no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4—101no—
485C22×C4—202no—
486C22×C4—202no—
487C22×C4—401no—
488C22×C4C2×C22⋊D511yescharacteristic
489C22×C4C2×C22⋊D511yescharacteristic
490C22×C4C2×C22⋊D511yescharacteristic
491C22×C4C2×C22⋊D511yescharacteristic
492C22×C4C2×C22⋊D511yescharacteristic
493C22×C4C2×C22⋊D511yescharacteristic
494C22×C4C23×D511yescharacteristic
495Q16—108no—
496SD16—108no—
497C2×C8—52no—
498C2×C8—52no—
499C2×C8—52no—
500C2×C8—101no—
501C2×C8—101no—
502C2×C8—101no—
503C2×C8—101no—
504C2×C8—101no—
505C2×C8—101no—
506C2×C8—101no—
507C2×C8—101no—
508C2×C8—101no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—52no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—202no—
526C4⋊C4C2×C4×D512yes—
527C22⋊C4—104no—
528C22⋊C4—104no—
529C22⋊C4—202no—
530C22⋊C4—202no—
531C22⋊C4—401no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—24no—
537C42—42no—
538C42—42no—
539C42—202no—
540C42—401no—
541C42C2×C4×D512yes—
542C42#70912yes—
543C2×C5#236111yescharacteristic
544C2×C5C4×#13511yescharacteristic
545C2×C5#294711yescharacteristic
546C2×C5#328211yescharacteristic
547C2×C5#328511yescharacteristic
548C2×C5#336411yescharacteristic
549C2×C5#336511yescharacteristic
550D5—82no—
551C23—102no—
552C23—201no—
553C23—201no—
554C23—201no—
555C23—201no—
556C23—201no—
557C23—201no—
558C23C22×C22⋊D511yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—102no—
563Q8—102no—
564Q8—201no—
565Q8—201no—
566Q8#403914yes—
567D4—102no—
568D4—102no—
569D4—201no—
570D4—201no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—102no—
598C2×C4—102no—
599C2×C4—201no—
600C2×C4—201no—
601C2×C4—201no—
602C2×C4—201no—
603C2×C4—201no—
604C2×C4—201no—
605C2×C4—201no—
606C2×C4—404no—
607C2×C4C4×C22⋊D512yes—
608C2×C4C4×C22⋊D512yes—
609C2×C4#409612yes—
610C2×C4#409612yes—
611C2×C4C22×C4×D511yescharacteristic
612C2×C4C2×#70911yescharacteristic
613C2×C4C2×D4×D511yescharacteristic
614C2×C4#414411yescharacteristic
615C2×C4C2×#71311yescharacteristic
616C2×C4#414711yescharacteristic
617C8—102no—
618C8—102no—
619C8—102no—
620C8—201no—
621C5#2300011yescharacteristic, complemented, Sylow
622C22—202no—
623C22—202no—
624C22—202no—
625C22—401no—
626C22—401no—
627C22—401no—
628C22—401no—
629C22C2×C4×C22⋊D511yescharacteristic
630C22D4×C22⋊D511yescharacteristic
631C22#6615111yescharacteristic
632C22C2×#409611yescharacteristic
633C22C2×#409611yescharacteristic
634C22#6617111yescharacteristic
635C22#6617211yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—202no—
647C4—202no—
648C4—401no—
649C4C2×#403911yescharacteristic
650C4#6583811yescharacteristic
651C4#6584111yescharacteristic
652C4#6584811yescharacteristic
653C2—402no—
654C2#1061389411yescharacteristic
655C2C4×#409611yescharacteristic
656C2#1061427811yescharacteristic
657C2#1061527611yescharacteristic
658C2#1061528011yescharacteristic
659C2#1061542011yescharacteristic
660C2#1061542811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2351C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#2351D5
hypocenterC5#23000
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C5×#2351C2semidirect
3. C2×C4×#689C2semidirect
7. #10602425C2semidirect
8. C4×#3967C2semidirect
9. #10600425C2semidirect
12. #10600407C2semidirect
14. #10600333C2semidirect
15. C4⋊C4×#168C2semidirect
35. #65317C4semidirect
72. C4×C5×Q8C2×C4semidirect
91. C2×#689C2×C4semidirect
128. #3942C2×C4semidirect
134. C4⋊#168C2×C4semidirect
142. #2351D5semidirect
376. C4×Q8C4×D5semidirect
621. C5#23000semidirect

Maximal subgroups

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

subgroupindex
C5×#23512
C2×#653172
C2×C4×#6892
C2×C4×#6882
#106053862
#106024262
#106024252
C4×#39672
#106004252
#106004182
#106004082
#106004072
#106003442
#106003332
C4⋊C4×#1682
#230005